SOME DIFFERENCE AND DIFFERENTIAL EQUATIONS USED IN BIOLOGY AND
ESTIMATION OF THE PARAMETERS OF THE DIFFERENTIAL EQUATIONS
by
ROBERT D SMALL
Institute of Stati3tics
Mi.meograph Series No 972
December 1974 - Raleigh
iv
TABLE OF CONTENTS
Page
PART I
vii
1.
INl'RODUC'l'ION
1
2.
REVIEW OF LITERATURE
3
2.1
2.2
4
3.
3. 1
6
Definition of the Model
Elementary Properties of the Model
Discussion of Numerical Results
6
8
10
NON.. ELEMENTARY PROPERTIES OF THE MODEL
13
4.1 Properties of the Exact Model
13
29
4.2
5.
3
MODEL AND ELEMENTARY PROPERTIES
3.2
3.3
4.
Evidence that Self Regulation Occurs
Theoretical Models . . . .
•.•.
Properties of the Model After Linearization
GENERALIZATIONS
5.1
5.2
5.3
5.4
5.5
5.6
5.7
...............
.
Definition and Examples of Reproductive Functions
Definition of the Generalized Model
Properties of the Generalized Model
Linearization of the Reproductive Function •
Properties of Model After Linearization
General Properties •
Conclusion . . . . . . • . • . .
35
35
39
40
41
44
47
49
6.
SUMMARY AND OVERVIEW OF OPEN PROBLEMS
50
7.
REFERENCES
52
8.
APPENDIX
53
8.1
8.2
53
The Solution of Equation (8) in Chapter 4
Solution of the Difference Equation of Section 5.5
55
PART II
57
1.
58
INTRODUCTION
v
TABLE OF CONTENTS C::ont.i.rmed)
Page
20
R.EVIEW OF LITERAl'URE
62
2.1
2 2
62
0
2.3
204
3
0
69
Cubic Splines and Their Properties
Least Squares Splines
Orthogonality of the Data and the Endpoint Problem. • .
0
••
0
•
•
81
4.1
81
4.3
Definition of the Sobolev Norm •
Estimating Parameters in a Differential Equation by
Mini.m:i zing a Sobolev No:r.m.~Like Expression
Fitting the Spline and Estimating the Parameter
Separately . ••
.••...•
Fitting the Spline and Estimating the Parameter
Simultaneously .
0
4,.4
•
•
•
•
•
82
84
•
85
86
ESTIMATING a IN Tl = aTl(l - Tl)
5 2
.503
0
Estimation of a in Tl = aTl(l - Tl) and Fitting the Spline
Separately
Estimating
and a Simultaneously . • • . •
Error Structu.res and Estimation of Variances and Bias
r
0
504
•
Logit Analysis • • . • . • . .
Other Applications in Bioassay
Autocatalysis, Enzyme Reaction and Haernolysis
0
a IN
d(,tn ]) ;:
dx
t:l
•
86
90
94
95
96
97
98
(l ,_ 'n)
IJ .
100
'I
A Different Form of the Logistic Differential Equation
The Variance of .tn(l1) • • .
Fitting the Spline and Estimating a in
d(.tn ..11)
~~: -' = a(l- Tl) Separately • • • •
104
Fitting the Spline and Estimating a Simulr:aneously
10.5
0
•
0
604
•
Results of Honte Carlo Simulation of Data
ESTIMATING
6.1
6.2
6.3
0
0
5.301
5 3.2
5.303
60
69
71
77
THE SOBOLEV NOR.1\f AND SPLINES
402
50
63
65
67
LEAST SQUARES SPLINES .
:3 01
3.2
3.3
4.
Applications of the Logistic Equation
Estimation of the Parameters of the Logistic.
Error Structures.
Fitting Splines to Data
• • • • •
100
101
vi
I'ABLE OF C01\1I'ENI'S (.:ontinued)
Page
605
606
Er.:':'or Stx"ucture and Estimations of Bias and Vari.ance •
Results of Monte Carlo Simulation of Data
106
108
'7,
Sl;"IYftVfARY AND OVERVIEW OF OPEN PROBLEM.S •
110
8.
REFERENCES
112
9.
APPENDIX
115
9.1
9.2
9. 3
9.4
1\
The E:~:pe::red '1/ 0 1(te uf S
Apprl.'x:hnar:8 Expected Variance
The E:\i:peccation of
The \/arlance of
1\
S
e in
1\
the Lugaritblf1:1.C Models •
in the Logarithmic Models
116
124
127
129
1\
ThE: gxpected Value of Swith One Ter.m in the Binomial
App:roximation
•....
130
1.
INTRODUCTION
How could a species evolve the ability to regulate its own population
size at: a level 'below the ma}Qimum detennined by the environment?
Although
there are species that seem to show this property, it is difficult to see
how they could have evolved this ability in competition with other nonregulating species.
A group that was regulating its own population would
have to have a reproductive rate that decreases as population size
increases, while a non-regulating group would have a reproductive rate
that was always greater than
one~
As the population increased the self-
regulating type's rate would have to fall to or below one before
environmental deterioration occurs, while the rate of the nOD-regulating
type would be greater than one at all times.
BY' definition this is a
selective advantage for non-regulating types, unless density independent
factors keep the population at low enough levels
&0
that the self-
regulators can be competitive.
A model will be presented in which there are two types, regulators
and non-regulators.
1~e
only difference between the two will be their
reproductive rate as a function of population size.
The population
will be kept from unlimited increase by crashes caused by the population
size exceeding an environmental maximum (carrying capacity).
Because
the population size will be returned to low levels after a crash, a
type with a high reproductive rate at low population sizes will be
able to compete with and even overcome a type with a constant reproductive
rate.
The most important thing about the model is that it shows how a
species can evolve the ability to regulate its own population size in a
manner completely consistent with the assumptions of Darwinian evolution
theory.
2
The model given is for haploid individuals only.
However, computer
simulations indicate that a similar model for diploid individuals gives
similar results.
Also the summary discusses a method of extending the
results for the haploid model to populations made up of diploid individuals.
Chapter 2 contains a review of the biological literature which
indicates that species do control their own population sizes.
It also
The mudel is defined in Chapter .3 and some of H:s elementary
properties are derived.
Section .3.3 gives a discussion of some of the
numerical results gained by varying the parameters of the modeL
A detai led discussion of the model is given in Chapter 4.
results can be divided into two kinds.
These
In the first section, the
results follow from an approximation to the model while the second
section gives exact results.
The generalization in Chapter 5 shows that the model is quite
robust, with regard to the form of the reproductive function of the
regulators.
The results are about the same whenever this function is
a decreasing function of population size.
Chapter 6 summarizes the results and suggests possible applications
of the methods used.
It also outlines a method of extending the results
to diploid popul.ations.
3
2.
2.1
RE.VIEW OF LITERATURE
Evidence That Self Regulation Occurs
There is ample evidence that many mammals regulate their own
populati.on sizes.
Christi an and Davis (1964) cite several examples,
in mannnals, of endocrine factors regulating the growth of population.
Watson and Moss (1970) and Wynne-Edwards (1962) come to similar
conclusions.
Rodentia.
Most: of these examples involve species of the order
1 t is quite prevalent in this order though.
Snyder (1961)
cites various population regulating mechanisms in Norway rats, voles,
mice, hares, and rabbits"
Snyder also reports on an extensive study
of woodchucks in which the rodents maintai.ned their populati.on at a
constant level despite strong efforts by the researchers to change
the population size.
In addition to evidence that some mechanism is controlling the
population size there is some indication as to what the mechanism is.
Snyder (1961) cites inverse relationship between reproductive function
and densi ty and a direct relationship between adrenocortical activity
and population density.
Christian (1961) gives evid8nce that population
growth and decline is "regulated by a series of feedback mechanisms,
particularly involvi.ng the pitiutary-adrenocortical and pitiutarygonadal systems .. ,"
In a later paper Christian et aL find
"Endocrine responses serving to decrease natality have
been enumerated and have been shown to affect all stages
in the reproductive process; so that the net effect of
increased density is to inhibit recruitment into the
population. Inhibition of. reproduction involves a
number of mechanisms which result in inhibition of
maturation, diminishe((-fertility, increased intrauterine
4
mortality and developmen tal abnormali ties, and inadequate
I aetation.
We have also seen that species di fferences may
be rather marked."
Andrews (1962) found evidence for similar mechanisms in Alaskan
lemmings.
2.2
Theoretical Models
I t has appeared to many researchers that the evolution of a selfregulating mechanism violates the ideas of Dar·winian evolution.
Various
efforts have been made to explain the evolution of such a mechanism.
Wynne-Edwards (1962) proposes a change in evolutionary theory.
He
suggesrs that a non-Darwinian group selection can overcome Darwinian
selection on individuals.
In addition to being quite vague this idea
suffers from the lack of any experimental backing.
Williams (1966) argues that all of the apparent examples of
self.,regulation are really examples of the population being limited
by something as yet undiscovered such as a trace element or unknown
dise,a:::H.>.
This may be a factor.
However, in light of Snyder's work
(1961) wi th woodchucks, Andrews work with lemmings (1968) and numerous
others it does not seem to be a major factor.
Chitty (1967) suggests that there is a polymorphism in the population with one morph selected for at low and the uther morph selected for
at high densities"
These opposing factors woald srabilize the population.
This is a satisfactory explanation of the experimental resulcs, the
only criticism being that a delicate balance must be maintained over a
long period and in apparently many different species.
5
Wright (1969) gives a model of the Volterra-Latka type to deal with
the problem.
the model requires that the self-regulating type enjoy a
very high mutati0n rate or that the non-regulating type is becoming
extinct in order for the mutants to win out.
TheBe are possible but
certainly fortuitous.
Williams (1975) proposed the basic model to be considered here and
in €extensive COlllp1..1ter simulations showed the possibility of such a
process
oc~urring.
She also made some of the suggestions for further
study mentioned in the summary.
OnE of the most: important things considered here is the intricacies
of difference equation models.
These often give quite different results
from the corresponding differential equation model.
In a paper on model
building, van der Vaart (1973) discusses some of these differences and
gives several examples of models in which the two types were confused
with some serious consequences.
6
3.
MODEL AND ELEMENTARY PROPERTIES
<3, 1
Definition of the Model
There will be two
~ypes
in the model.
One, called mutants, will
regulate its own population by reproducing in a logistic manner.
The
other, called normals, will reproduce at a constant rate thus having
exponential growth,
Reproduction is by discrete generations.
When
the pupulation size (normals plus mutants) exceeds the environmental
maximu;n, a crash occurs and the population is reduced to a constant
minimum size before the next reproductive period.
The post crash
populat::Lon is mad8 up of normals and mutants in the same proportion
as the pre-crash population.
Finally, it is assumed that sampling
error during the crashes will eliminate a type at sufficiently low
percentage.
The model will be defined in terms of the following quantities.
th
number of mutants in the g- generatiun
y(g)
x(g)
-
n(g)
th
number of normals in the g-- generation
y(g)
+ x(g) = total population size in the
g!.b.
generation
M
maximum population size that the environment can support
m
post crash population size
C -
constant repruductive rate of the normals
A,B
pusitive constants that define the reproductive function
of the mutants"
The model can be written
[A'~Bn(g-I)Jy(g-l)
x (g) -
r::x (g - 1)
neg)
x(g)
+
y(g)
if
neg - 1)
~M.
7
"
1) In (g
y(g " 1,)
[y (g -
"
[x(g-l)/n(g-l)]m
,
. ~.
x{g-l)
-1)]
m
if
y{g)
1\
LA
- Bm ] y(g
- 1)
x(g)
C x(g - 1)
neg - 1) > M
1\
To ensure that the model is realistic and interesting, certain
r€strict:ions must be imposed on the constants.
a)
C:<: 1
If not the normals would become extinct.
b)
A ,- Bm > C
If not the normals would always have a selective
advantage.
c}
A/B:2 M
If not: [A-Bn(g)], the reproductive rate of the
mutants, could become negative.
d)
A2 /4B s: M
This ensures that the mutants are self-regulating.
e)
A
< 4
If A > 4 then the population size of the mutants
can become negative.
The above equations together with the restrictions a through d will
hereafter be called
O.
Restrictions a and b are evident upon inspection of the model.
R8stxiction
<::
is necessary since if the population size, n(g),
were a110\",ed to exceed AlB then A - B n(g), the reproductive rate of the
mutants, wor,ld become negative.
Reatl' fetion d is the key to the model.
If the mutants are to be
self··T.egul ating then a population of mutants only can never exceed H.
A populatiun of all mutants would l'eprod1..lce by
y (g + 1) = [A - B Y (g)] Y (g) .
The right hand side is a function of y(g) and takLng the derh;ative
and
SE
tUng itt-) zeru gives A/2B as the point: that will give the
maximum 'value ':Jf y(g+ 1).
Inserting A/2B intu the expression on the
8
right hand side gives
Thus A2 14B is the maximum population size of the mutants.
Restriction e is necessary to prevent unstable oscillations even
if all other Landi tions are meet.
If the mutants in a mutant only
population did at tain the maximum population size of A2 /4B and A > 4
then the population size in the next generation is
The, last expression is negative if A > 4.
3.2
Elementary Properties of the Model
Some elementary properties of the model will now be derived.
Property
Proo£'
1.
A-Bn(g):2 C
~
neg)
~
(A-C)/B .
A~Bn(g):2 C ~ A-C:2 Bn(g) ~ (A-C)/B:2 neg)
0
Thus, in any generation in 'which the population is smaller than
(A - C) /B, the mutants have the selective advantage; and they lose it
as soon as the population exceeds this point.
Property
Proof.
4C > O.
1.
Under 0, A2 /4B > (A - C) /B.
Since B > 0, then A2 /4B > (A·· C)!B is equivalent to A2
On cumpleting the square, this becomes
A2
-
4A
+ 4 > 4-
4C
or
(A - 2/'
> 4(1- C) .
Ci,)
-
4A+
9
SincE: C > 1 the RHS is negative whi Ie the LHS is a square and therefore
always positive.
Thus ("'') is always true and it is equivalent to the
indqualitj in the conclusion.
Property
ProoL
1..
Vndt':r 0 i f neg)
SuppDse that neg)
0
< (A ~ C) /E then neg + 1) < M.
< (A-C)/B, and recall that
n(g+l) = [A-Bn(g)Jy(g) + Cx(g).
Nuw neg) < (A- C)!B ~ [A- Bn(g)]
n(g+l)
> C; thus
< [A-Bn(g)]y(g) + [A-Bn(g)Jx(g)
[A~Bn(g)][y(g)
+
x(g)] = [A-Bn(g)]n(g).
This last. expression is a quadratic in neg) with a max of A2 /4B so
n (g
+ 1) <
A2 /4B .
By n, A2 /4B < M and thus neg + 1) < M. 0
Pro.E.§,r.,Si ,~"
Proof.
Under 0, m < (A- C)/S.
0 requires that A - Bm > C which is equivalent to
m
<
(A~C)/B.
LJ
Properr:ies 2, 3, and 4 restrict the ways in which the population
can grow from m to M.
By 0 and Property 4 there must be at least one
generation in whi<.:h the mutants haVE the sE::lective advantage.
Property
2 and :5 say that· there must be at: least one generation in which the
n(J:r.~I!dls
ha'v'€: the st::lectiv8 advantage.
Sin,~e
each type has a generation
in which it has an adv am: age , it may not be possible to elimi.nate one
of the t.,yp8S.
In fact, it is possible to give a 10"rer bound on the
numl:'E:::r: of normals.
'~his
will be done in ChapLer 4.
10
3.3
Discussion of Nmoerical Results
In order t.o get some idea of the behavior of the systems various
values of the parameters were chosen and run on the computer.
The
behavior is qualitatively the same through the complete range of O.
If the equations are initialized with a small number of mutants
and x(O)
+
yeO) =
Tn
the population grows until it exceeds N, a crash
occurs, the population is reset to m and the numbers of mutants and
nonnals is recalculated.
The population then begins to grow again
until another crash occurs; these growth crash cycles occur
repeatedly.
If the initial number of mutants is small enough the
number of mutants irmnediately after the first crash is greater than
x(O).
As the growth crash cycles continue, the percentage of
mutants at the end of each cycle continues to increase until a
maximum value is reached.
Once this happens, the percentage of
mutants oscillate about this maximum point in an irregul ar manner.
The oscillation is not periodic, and the number ·of crashes on either
side of the maximum point 'Jarys.
The only consistency is that the
numbe'c of generations in a cycle in which the per(>entage of lllutants
decreases is always great.er than in a cycle. i.n which the percent.age
of mutants in.creases.
Figure A is a plot of percentage of mutants against generations.
At g = 0 the mutants are introduced at a low
h~v,~L
The cusps
correspond to ..:rash generations (and alsu in U:ial gimerations).
'The
first generation after the maximum curresponds to t:he first generation
such that n (g) > (A'~ C) lB.
11
The fact that there is always one such generation imples that
there is always one generation between the maximum and the cusp.
One disturbing feature of the mL)del is the possibility of
crcssing of solutiun paths.
Thus, if y(g) > y' (g) i t is possible
that y(g+y) < y'(g+y) even though neither population has crashed.
12
Percentage
of
Mutants
Number of
Generations
Figure A
13
4.
NON-ELEMENTARY PROPERTIES OF 'TIlE MODEL
In this chap ter some non-elementary properties of the model are
developed.
In particular bounds that are functions of the parameters
of the model are derived.
These give so;:ne idea as to when the mutants
would be able to overcome the normals.
Also the idea of crossing is
introduced and some rules are given to determine its incidence.
Crossing occurs when the solution of a difference equation wi.th a smaller
initial ·value becomes larger than a solution wi.th a larger initial value.
This occurs more frequently in difference equati.ons than in differential
equations.
Finally, in the second section, a linearization of the model
is carried out and it is shown that crossing isn I
t
a problem in most
practical cases.
4.1
Properties of the Exact Model
The first theorem gives a lower bound on the number of normals.
Theorem L~. 1.
------
Under 0, if K /t~B < M(l-~)
111'
K > 0 then ,,rhen neg) > M,
neg) > KM./m.
Proof.
Since neg) > M then M < x(g)
written as x(g) 2: M"· y (g).
+
y(g)
=
neg).
This can be re-
Using the formula for y(g) in the definition
of 0 this is
x(g) > !vI •• y(g)
= M - y(g - 1)(A- Rn(g - 1))
M - y(g - 1) (A- By(g - 1) - Ex(g - 1))
~ M -
y{g .. l)(A .. By(g - 1),
v1here the fact that neg - 1) ::;: x(g - 1)
x(g)
:?:
+
yeg .. 1) has bee.n used.
M .. y(g .. l)[A- By(g - l)J
So
14
and the second term on the RHS is a quadratic in y(g) with a maximum
of A2 /4B.
Subt.racting the maximum will give a minimum so
x(g) ~ M - A2 /4B.
But
A2 /4B
<
M(l - Kim)
so
x(g)
~
K
M - M(l- -) = MK/m.
m
[J
Since the number of normals at an initial generation
(i..~.,
neg)
is given by
x(g)
x(g)
x(g)
+
y(g)
'm
the theorem gives a lower bound on the number of normals surviving a
crash.
The percentage of normals after a crash can be rewritten as
x(g)
1
x(g) + y(g)
1 +
YiB.2.
x(g)
where the Theorem 4.1 was used to get the bound.
Thus,
MK
x(g)
m
x(g) + y(g) ~ MK
- + y(g)
m
and recalling that y (g)
<
i~ 14B under (2
MK
x(g)
x(g)
+
i'1nlt:iplying the LHS by
In
y(g) ~ MK
In
A2
+4B
"ill
gives the number of normals surviving
the crash, and thus m times the RHS is a lower bound for these.
= m)
15
Thus,
MK
2
MK
A
+4B
m
is a lower bound for the initial number of normals at each cycle.
It would be nice if the number of mutants at any number of generations
g after an initial generation were a monotone function of the number at the
initial generation.
Thus, i f n(y)
=
n' (y)
=
m and y(y)
> y' (y) a useful
property would be y(y+g) >y'(y+g)Vg (or at least Vg ~ n(y + g)
n'(y+g) <
M).
,
Unfortunately, this is not the case as the following
example shows.
Let
A = 3.8
B
.1
C
1. 001
ill
=12.
Consider the two cycles given by the following tables.
Cycle I
x(g) + y(g)
g
y(g)
x(g)
o
9,999
2.001
12
1
25.9974
2.003001
28.000401
25.9963575
2.00501301
27.9992782
Cycle II
+
g
y(g)
x(g)
x(g)
o
9.998
2.002
12
1
25,9945
2.004002
27.998802
2
25.99791477
2.006006002
28.0021563
y(g)
16
Using a subscript to denote the cycle,
yr(O)
= 9.999 > 9.998
and
y r (2)
= 25.9963575 < 25.99791477 = Y11(2).
This crossing of paths by the solutions of the difference equations
causes quite a bit of trouble.
Bounds are harder to derive, it is more
difficult to determine the number of generations in a cycle but worst
of all it becomes possible to eliminate the mutants by increasing the
ini tial number.
Fortunately these bad cases seldom arise.
For example,
in the example given C is barely above one, and A is close to four at
which point an instability is introduced.
In the rest of this section some criteria for non-crossing will
be given and properties of non-crossing models will be derived.
of the results are exact,
i.~.,
no approximations are made.
All
In the
next section approximations are made and further results obtained.
Definition 4.2.
Under 0 a cycle with fixed parameters A, B, C, m and
M is the set of initial values x(g), y(g), neg) and the values taken
on by n(g+y), y(g+y), x(g+y) Vy:;, n(g+y - 1) < M.
Notation.
To identify the cycle write C(A,B,C,x(g) ,y(g) ,m,M) or C when
it is clear that all parameters are fixed.
When comparing two different cycles of O(A,B,C,m,M) with g
initial generation of C
and g
the initial generation of C
1 2 2
x(g ) by x (0) and x(g + 1) by x (1), etc.
1
'
1
1
1
1
the
denote
17
Defjnition~.3.
Under O(A,B,C,m,H) two cycles C
v (0) and n (0)
. 2
and C
with y (0) >
121.
1
n (0)
= M,
(g) >
Y2
2
Y1
are non-'crossing if and only if
(g) Vg 3 n (g - 1), n (g - 1) < M.
It shuuld be noted that any
1
c~ossing
2
that occurs not only occurs
to the mutant: population size but i.s due to the mutants.
were no mutants t:here would be no crossing.
Thus, if there
This is easily seen.
all normal population would reproduce exponentially.
An
Thus i f x (0) >
1.
x (0)
2
x 1 (g).
x (g)
2
for any g.
This property is true when there is a large percentage of normals.
This is true since if there are a large number of normals in two cycles
C
1
and C
2
the difference between them will grow quickly (in fact,
exponentially).
The mutants on the other hand will grow slowly and so
will the difference.
change sign.
Sin(;e difference grows slowly it is not able to
The following theorem makes this expli cit and tells just
how large a number of normals there must be.
large at~ all.
In fact it is not very
First, though, a lemma is necessary.
A/2B is an important
In the discussion of restriction Cd) it was seen
quanti.ty in the lemma.
that if a populat:ion consisting of only mutants were to attain their
size in generatiun g
.~
I then the population size at generation g would
be the mai{imum.
Lemma 4.4.
I.e t C
be a cycle under O(A,B ,C ,m,M) and suppose that
2
x (0) > 2(M - A/2B)/CY
2
where y is such that
18
(A - Bm) Y-l ill < A/2.B
(A - Bm) Ym
~
then for any g
y(g)
~
(A - Bx(g» /2B.
PrLlof"
First consider the case that M > A/2B.
[A~' Bx(g) J/2B"
Suppose that y (g) >
Then 3:g such that
Y(g) > A/2B -
'21
x(g)
Add x(g) to both sides of this to get
neg)
=
x(g)
+
y(g) > A/2B
+ '21
(1)
x(g)
Using the lower bound on x (0) given in the hypothesis
2
Inserting this into the extremes of (1) gives
neg)
> A/2B + Cg(M - A/2B)
cY
Now
Y is
the number of generations it would take the population size to
reach A/2B if every i.ndividual reproduced at (A
~
Bm) for every generation.
Under (2 (A - Bm) is the largest possible reproductive rate so y is the
smallest number of generations possible to reach A/2B.
So since neg) >
A/2B than g > y and
g
n(g)
> A/2B + C {M .. A/2B > 2.B
A +'1
A
M
1'- 2B- , .
But this is a contradi.ction since Vg in any c:ycle, neg) < M.
19
Foe the ease A/2B
neg)
>
M note that (l) becomes
> A/2B + ~ x(g) > ~ x(g)
and again this is a contradiction. 0
Theorem 4.5.
C and C be cycles under 0 with y (0) > Y (0).
Let
1
2
1
2
Let
x (0) >2(M-A/2B)/CY
2
where Y is such that
(A - Bm) Y-l m
< A/2B
~
(A - Bm) Ym
then the cycles are non-crossing.
~roof.
It must be shown that under these conditions for every integer
g ~ 0, .y 1 (g) > .y 2 (g).
The proof will be by mathematical induction on g .
The first inducti.on step is trivial, since for g
assumption.
y
• 2
= 0,
y (0) > Y (0) by
1
2
So it must be shown that y (g) > Y (g) implies y (g + 1) >
1
2
1
(g + 1) .
By definition
y (g + 1)
[A - Bn (g) Jy (g)
2
2
=
[A - Bx (g)
2
By
2
2
since n (g
+ 1)
= x (g + 1)
+
Y (g
2 2 2
+ 1).
(g)Jy (g),
2
Regrouping the last expression
gives
y (g + 1)
2
[(A-Bx (g»
2
By (g)Jy (g).
2
(2)
2
The right hand side of this expression is a quadratic function in y (g)
2
and i.s monotone increasing on the interval [0, (A- Bx (g»/2B]'
Lemma
2
4.4 showed that y (g) is in this i.nterva1.
2
The as sump ticn is that
20
y (g) > Y (g) so if y (g) is replaced by y (g) in the monotone increasing
1
a
a
1
function on t:he right hand side of (2) it will become larger.
y (g+ 1)
a
Since Y (0) > Y (0),
1
a
xl
< [(A- Bx (g» - By (g)Jy (g)
a
(3)
a l l
(0)
x (g)
Thus,
<
x
a
(0) and
cgx (0) > Cgx (0)
2
1
x (g) .
1
Thus replacing x (g) by x (g) in (3) gives
a
1
y (g + 1) < [(A - Bx (g»
- By (g) Jy (g)
a l l
1
and the right hand side of this inequality is y (g + 1).
1
Thus the
induction step has been completed. 0
If Y (0)
1
> y (0) are the initial values of mutants for
a
two cycles, then
y (g + 1) > y (g + 1) 1[g :7 n (g) < A/2B .
<
Proof.
a
1
a
In the proof of Theorem 4.5 it was shown that there is no
crossing as long as
n
a
(g) < A/2B +
x
a
(g)/2
This is satisfied if
n (g)
a
Cord'! arv 4.7.
Proof.
If M
< A/2B . 0
< A/2B all pairs of cycles are
non·~crossing.
By definition of a cycle neg) < M for all except the last
generation in the cycle.
Thus for all g except the last
21
neg) < M
< A/2B
and by Corollary 4.6 there is no crossing. 0
Remark.
A:3 /48
M < A/2B if and only if A < 2.
< M so
it follows that A2 /4B
This is true since under 0
< A/2B for A < 2.
By Corollary 4.7 and the remark following it, crossing is impossible
as long as A < 2.
The linearization in the next section will allow us to
make a simi.lar comment for A < 3.
The next theorem gives a criterion
for non-crossing when A > 2.
Theorem 4.8.
Proof.
Let
If M
C
< A(4 - A) /4B all cycles are non-crossing.
C
and
be two cycles under the same
:3
1
O.
Let y_ ('1)
>
.l
y ('1) be the mutants at corresponding generation and suppose that the
:3
first: cross occurs at generation '1 + 1.
I f n ('1) > n ('1) the proof is trivial for then
2
1
Y ('1+ 1) - Y ('1)(A- Bn ('1»
1
> y ('1) (A- Bn ('1»
1 1 1
>
y ('1)(A - Bn ('1»
:3
2
2
y ('1 + 1)
:3
and no crossing has occurred.
So assume that
Then, since a cross occurs,
Neglecting they' s straightforward manipulati.on gives
y ('1)(A-Bn ('1»
< y ('1)(A-Bn ('1»
1 1 2
A(y
- y ) - By n
1:3
1 1
< - By
:3
:3
n
2
22
A(y
2
A (y
"1
x (g)
= y + So
and
n
1
=x +y
1
Replacing y
1
By n
1
<
1
n
(4)
1
cgx (0) > Cgx (0)
> x (0) and x (g)
1
2
=x +y
n
l'
by y
;3
<
2
Yl
;3
1
2
;3
Since x (0)
2
By n
- Y )
1
B
Let y
+
- Y )
1
2
;3
then
n
;3
1
2
< n
2
+
1
S •
- e: in (4) gives
or
since n < n
1
:3
+ e:.
Taking the extremes of this
-
1 [AB
y
since v
. 1
-
- n
[ 1:B
n
J" <
2-
1
1
>a
l<y
2-
1
or
A
-<y+n.
B
Under 0, n
2
<
1
2
M since y is not the last generation and y.
< A2 /4B
~
so
A
-B
2
+
< .v1 + n 2 s: ~
4B
N.
(5)
23
The extremes of this give
A(4 - A)
4B
<
M.
rhus if
M
s:
A(4 - A)
4B
(6)
a cross cannot occur. 0
Remark.
As A gets close to 4 the RHS of (6) goe,s to zero so that the
inequali ty gets harder to satisfy.
satisfied for A
~
This inequality can only be
2 for
A(4 -.& ~ M .... A'2 =>
4B
.::: 4B
2 ~ A.
Theorems 4.2 and 4.8 show that there are many cases where there
In fact, Theorem 4.2 says that unless xeD)
are non-crossing cycles.
is quite small there is no crossing.
Theorem 4.9.
Let P (g) and P (g)
1.
represent the percentage of mutants
'2
in a pair of non"crossing cycles.
y (g)
~
Then
y (g)
1
~
P (g)
~
1
'2
P (g).
'2
Proof
_
-- .
.
First define
Pi (g)
x.. (g)
l··P. (g)
1
1.
is a (l - 1) increasing function of p. (g) so that
1
P (g)
1
~
P (g)
2
~
Q (g)
1
~
Q (g).
2
24
Sin~e
the cycles are
oon·~crossing
y (g) > Y (g)
1
a
~
x (g) < x (g)
1
:3
thus
y (g)
Q] (g)
= _1
X
1
Rema:t~.
y (g)
>
:3
(g)
X
1
>
(g)
y (g)
_2_ _
= Q (g).
0
2
(g)
X
:3
The number of generations between crashes varies.
The next two lemmas and theorems show that there is a strong
relation between the number of generations and whether the percentage
of mutants increases or not.
Lemma 4.10.
Let
(A - BM) M
then i f 3:y
in a cycle
~
P (y)
>
A .. C
-B- ,
> P (y + 1) then
Vg > y, P (g + 1) < P (g) .
Proof.
First note that if
M
>
n (y)
A-C
> -B- , P (y) > P (y + 1) •
Recall that Q is a (l - 1) function of P (y) and
Q(y
+
= y(y)(A-Bn(y»
1)
x(y)C
I'here fore,
Q(y
+ 1) < Q(y)
~
An Bn(y) < C
~
A-C
ney) > -B-
25
So let n(y) > (A - C) /B and thus P (y) > P (y + 1).
A-I
neg) > (A·· C)/B Vg > y.
If n(y) :s: -B-
n(y
If n(y) >
,(A~ 1)
+ 1) > n(y) >
It must be shown that
then
A-C
-B-
then
n(y+ 1)- (A-Bn(y)y(y)+Cx(y)
> [A-Bn(y)]y(y) + [A-Bn(y)]x(y)
since n(y) > (A-B)/C.
This is
n(y+ 1) > [A-Bn(y)][y(y) + x(y)] = [A-Bn(y)]n(y).
Two cases must be considered:
A < 2 and A> 2.
Let A> 2.
Then
n(y) > (A- 1)/B > A/2B.
On (A/2B,oo) the quadratic (A-Bn(y»n(y) is decreasing.
Thus,
n(y+ 1) > (A- Bn(y»n(y) > (A- BM)M
since M > n(y).
By hypothesis
(A - BM)M > (A - C) /B
and thus
n(y+ 1) > (A- C)/B.
Let A < 2.
n(y) > A/2B.
In this case (A - 1) /B < A/2B so n(y) < A/2B or
If n(y)
n(y+1)
< A/2B, then
> (A-Bn(y»)n(y) > (A_S(A; 1) ) A; 1
A- 1
=-B
26
since the quadratic in n(y) is increasing on
[~
B
Since C
>
A-I
A-C
B> --B-
1,
~J
'2B
and thus
A-C
n(y+ 1) >
-B-
If
A
n(y) > 2B '
then on
(2~
,
M)
the quadratic is decreasing and as in the case A
n(y+ 1) > (A- EM) M >
Lemma 4.11.
A-C
0
-B-
Let n (g ) and n (g ) be the initial population for two
1
non-crossing cycles.
a
1
a
If
+y ) >
n (g
111
M
and
+ Y ) s:
P (g
P (g )
111
then
+y ) s: (A- C)/B.
n (g
2
Proof
0
> 2
Suppose not,
2
.b.~.,
1
n (g
a
a
+ y ) < (A - C) /B
1
y
n (g
a
2
+ Y.. ) >
elm
.!.
since
Yfy3 g
s:y S:y,
1
+y) < (A-C)/B,
n (g
a
2
:3
0
Then
27
and thus
A - Bn (g
2
2
+ y) >
C
and the population is thus reproducing at a rate higher than C.
p (g )
~
p (g
Since
+Y ) ,
111
n (g
+Y ) <
111
y
C
1
m
since the mutants have averaged less than C per individual per generation.
Thus
n (g + Y ) > Cy 1 m > n (g + Y ) > M > A - C
221
B
111
This is a contradiction. 0
The cycle change in percentage for cycle C , denoted
Definition 4012.
1
by C P is defined by
1
C P
=
P (g
+ y) -
P (g)
1 1 1
where g + y is the last generation in C .
1
Remark.
The following theorem shows that if two cycles have percentages
that cross and the one with the initially larger percentage has a
negative C P then the crossing was caused by an additional generation
:1
between crashes.
Nu assumption is made about whether C P was positive
2
or negative.
The next theorem adds the hypothesis that C P is positive.
:3
Theorem 4.13.
Let (A-BM) M> (A-C)/B and y (g ) > y (g ) represent
1
1
initi.al value of mutants for two non-crossing cycles.
M. and n (g
+y ) >
22:3
2:3
Let n (g +y ) >
111
M be the population values at the next respective
28
crashes and P (g
+Y )
and
P (g
11121
If C P < 0 and P (g
the crashes.
Proof.
+Y )
122
Suppose not.
be the percentage of mutants at
1
+Y ) >
P (g
2
+Y )
111
y
then
1
> Y
a
.
By the assumption of non-crossing
y (g
+y ) > y
+y )
(g
11122
1
and thus by Theorem 4.11,
P (g
+Y ) >
P (g
11122
n (g +y )
By lemma 4.11 since
1
1
~
+ Y ).
1
> M then n (g +y ) > (A-C)/B.
221
The proof of limma 4.10 showed that i f (A - BM) M > A; _<2
total population size was non-decreasing.
that (A - BM) M
n (g
2
>
2
A; C
+ y) >
then the
Since the assumption here is
then
(A - C) /B ify,
Y
1
:s;
Y :s; Y .
2
Thus,
(A - Bn (g
2
2
+ y)
such y and, therefore, P (g
2
P (g
+Y ) >
P (g
11122
+ y)) <
C if
form a decreasing sequence for
2
+Y ) >
1
P (g
+Y +
221
1)
This contradicts the hypothesis that
P (g
+Y ) > P
(g
22211
+ Y ).
1
[J
> ... >
P (g
+ Y ).
222
29
Theorem 4.14.
Let y (g ) > y (g ) be initial values of mutants in two
1
1
cycles and suppose that y
:3:3
1
and yare the first generations such that
:3
n (g + Y ) > M and n (g + Y ) > M.
111
:3
2
:3
If
c1 P < a
and
ca P > a
then
Y1
Proof.
:?:
Y2 •
Since P (g + Y ) > P (g ) the mutant and therefore population
222
reproduction rate> C per individual per generation and
sin~e
P (g + Y ) < P (g ) the mutant and therefore population reproduction
111
rate < C per individual per generation.
Thus y
1
must be at least as
big as y . 0
2
4.2
Properties of the Model After Linearization
In this section a linearization of the model will be carried out.
The previous section indicated that if crossing were to occur then it
could not occur until a few generations had elapsed.
This section will
show that (under the restrictions imposed by the linearization) if
crossing occurs it must occur a few generations before the crash.
Thus
the two sections together will show that crossing can only occur during
the middle generation.
30
Let C
and C
1
:3
the cycles.
be cycles with subscripts on the variables denoting
Instead of using y (g) etc. make a transformation to new
1
variables.
By (g) - (A- 1)
x
(g)
2
x
Bx (g)
1
1
N (g)
1
By (g) - (A - 1)
Y (g)
1
2
Bx (g)
a
(g)
2
Bn (g) - (A - 1)
N (g) = Bn (g) -
1
2
(A - 1) .
2
The transformed variables satisfy similar relations to the old ones.
Y (g) - Y (g) = By (g) - (A - 1) - By (g) + (A - 1)
1 2 1
2
= B(y (g) - y (g»
1
2
so that
Y (g) > Y (g)
~
Y (g) > y (g).
1 2 1
2
Also,
X (g)+Y (g) =Bx (g)+By (g) - (A-I) =B(x (g)+y (g»
1
1
1
1
1
1
- (A-I)
=Bn (g) - (A- 1) =N (g).
1
1
The idea is to determine when
implies that
(y (g + 1) - y (g
1
2
+ 1»
> O.
Now consider the difference equation for the difference between
the y. r s.
1
31
(y (g + 1) - Y (g + 1»
1
-
2
(y (g) - y (g»
1
2
= [A-1-Bn (g)Jy (g) - [A-l-Bn (g)Jy (g).
1
1
2
~,
Using the difference operator
2
this is
My (g)-y (g»=[A-l-Bn (g)Jy (g)-[A-l-Bn (g)Jy (g).
1
2
1
1
2
2
Introducing the transformed values,
~(y
1
(g) -y (g»
(Nl(g)
2
= [ A-l-B
B
~A -
-
+
(A-
1»1\ (g) + (A- l)l
B
1 - B (N (g)
y (g)
= -N (g) [ 1
+
+
B.'
I
(A - 1»
(A - 1)J
+N
B
1
Y (g)
+
(A - 1)
B
2
[Y (g)
(g) .....2 '"--
+
J
(A - l)J
_
B
2
Multiplying by B gives
My (g) 1
Y2 (g»
= - N (g)
1
[Y1 (g) +
(A- l)J+N (g)[y (g) + (A- l)J
2
2
.
Since
N (g)
=
X (g) + Y (g)
222
this is
My
1
(g) - Y (g»
2
= - (A - l)(Y (g) - Y (g»
1
2
+ (A -
l)(X (g) - X (g»
2
1
+ [y2 (g) _ y 2 (g) + Y (g)X (g) 2
1
a
Y (g)X (g) J
a l l
Now it will be shown that the equation can be linearized under
certain conditions.
To do this it is necessary to ignore the squared
(7)
32
and cross product terms in the last expression in brackets.
Under what
conditions are these second order?
Consi.der the term in brackets; for convenience the g will be
ignored for the present.
[y2 _ y2
2
1
= y 2 _ y2
+Y X - Y X ]
2 2
1 1
2
+ Y (N - Y ) _ Y (N _ Y )
1
222
111
=NY -NY.
2 2
Now it is always true that Y. < N..
~
(8 )
1 1
~
So in order that the products
in (8) be second order, it must be that
i = 1, 2.
-1<Y.<N.<1
~
~
N. < 1 implies that
~
Bn. - (A - 1) < 1
~
or
(9 )
But under
(2,
M <
A
B
so (9) is always true.
In order that -1 < Y. it must be that
~
-1
< By. - (A - 1).
~
This gives
A-2
B
< y.
~
i = 1, 2.
Thus as long as
A-2
Yi > -B-
i = 1, 2
the linearization of the difference equation can be carried out.
33
It is interesting to note that the limits of
A~2/B
to AlB can be
written as
A --1+ 1
B
- B
Thus, as long as the n's and y's are in an interval of liB about the
stable point at (A - 1) IB the linearization is a reasonable approximation.
Thus under the linearizati.on, the difference equation becomes
t:.(Y (g)-Y (g))=-(A-1)(Y (g)-y (g))+(A-1)(X (g)-y (g)).
1
2
1
2
2
(10)
1
The appendix gives the solution as
It is also shown there that this solution remains positive as long
as A < 3 and thus there is no crossing.
This can be summarized in the
following theorem.
Theorem 4.15.
If Y (g)
1
>
Y (g) and y.(g) and n.(g), i = 1,2 are in a
2
1
1
neighborhood of radius liB about the stable point (A - 1) IB then
difference equation (7) can be approximately linearized.
The solution
of the linearized equation is positive for A < 3 and thus there is no
crossing under these conditions.
When combining the results in this section with those in the
previous section, the following inequalities should be kept in mi.nd.
For A> 2,
For A
< 2,
34
For A :::; 2,
A
A-1
2B
B
Corollary 4.6 in section 4.1 showed that no corssing is possible
< A/2B.
if n
The linearization shows that no crossing is possible if
2
Y
l'
Y
<2
and therefore nand n
1:3
A> 2, A/2B > (A- 2)/B.
are greater than (A - 2) /B.
But for
The cycles are non-crossing in their earlier
stages by the results of Section 4.1 and non-crossing in their later
stages (as long as A < 3) by Section 4.2.
If A :s;: 2, the process is
always in the band that allows linearization.
Thus it is not
surprising that the example of crossing given in Section 4.1 occurred
with an A = 3.8.
35
5.
5. 1
GENERALIZATIONS
Definition and Examples of Reproductive _Functions
As with mosi of the models in biology, the one under discussion
here is not an absolute law but an approximation to the real world.
Thus it is necessary to investigate the performance of the model if
some of its assumptions fail to hold.
In Chapter 4 it was shown that
slight variations in the parameters of the model resulted in slight
variation in the output of the model.
This chapter will investigate
changes in form of the model.
In the model 0,
the mutants reproduce according to the formula
y(g + 1) = y(g) (A - Bn(g»
y(g)(A- B(x(g)
+
y(g») .
This has the general form
y(g+ 1) = f(n(g))y(g) .
Since the mutants are to be self-regulating, f must be monotone
decreasing.
This is not, however, a sufficient condition for self-
regulation.
There are decreasing functions, f(y), such that yf(y)
is unbounded.
regulation.
Obviously such functions will not produce selfSo f must be such that yf(y) is bounded.
Now, since f is monotonically decreasing it can only be di scontinuous at a countable number of points, and all of the discontinuities are simple
at these jumps.
(.!.~.,
jumps).
Also, f is differentiable except
Thus, if f is' taken to be differentiable on (O,M)
only rather pathological functions are excluded.
This seems to be
an example of a very smooth functi.on being more realistic biologically
than a non-continuous one.
36
Defini tion 5. 1 describes a class of funct1.()oS that: can be used in
t:he model.
O1h:Icusly fey) must always be positive and to be self-
regulating yf(y) must be bcunded.
Another requiremenc is that the
mutant.s rf:producing with a function like f should be able to compete
wi th an expunent Lally reproducing population.
f (y) should cause increases
when the population is small and stability or decreases when the population
1 s large.
This -:au be accomplished by requiring that: yf(x + y) (x is the
number of normals) be i.ncreasing when x + y is small and decreasing when
x+ Y
~1
And x + y must be used because an overriding assumption is
s large.
chat the only difference between normals and mutants is the way that
they reproduce.
The mutants "know" that the total population is getting
too large so they stop reproducing.
The only point: left is the position of the division between yf(x+y)
as a function of y being monotone increasing and monotone decreasing.
Since f is to have a derivative,
the transition point will be the point
at which
d(yf(x+
dy
The point depends on x and is
~(x)
y»
=
o.
in the definition.
Definition :5.1
Let fey)
Le t
Slip
0Q,~(x)
yf (v)
~
0 be monotone decreasing and differentiable on (m,M).
s < '.'10
Let f be such that yf(x+ y) be mono-increasing on
and mono-decreasing on
(~(x),M)
d(yf(~+y)
dy
where
o .
~(K)
is the solution of
(1)
I'hen f is called a selfuregulating reproductive function on (O,M) and
37
If f(m) > C > 1 thf:m f is compenUve with Co
Of
LU,~rSE:-~,
Write
the reproduccive function A-By used in (2 is in g(H),
If,:1der (2 uf Chap ter .'3,
fev)
A"' By
dnd
A
cp(x)
2B
Nute thaI: the quantity cp(x) pla~Y's an important: l'ule in the crossing
theore:n uf Ch ap ter 4"
As lung as
y (g)
s: cp(x (g»)
2
2
there is no crossing.
Let
f(y)
First
th~
c;.f{~.;)
=
A exp(-by)
(2)
0
is bounded and tty) is U1onotonE:c dec.reasing.
Solving
Equation (1) gives
or
1 - ABy -= 0
~ .. ;.: cp(x) .. ,~
\'
'l.S a.
n,)Y:Jidl.s,
CG:·~stdflt
"
AB
.
w"TTt to :x.o
Thus iq a c'J)[lpeti cion w:i th
lJIutanc:,s repCOd'.ll'ing with a Eunr.'tiun likf.', (2) '"r,)uld be using
38
Independf:ci: of the
nU{lnto1"
of nonuals in the populati(ln.
For mutants
using thLs fUflccion, che turning point comes when
A exp ( ~·Bn)
C
c
-~
exp (-Bn) = A
~
C
£TJ A
-Bn
1
C
B £0 Ii:
£0, -A-- £n- C
B
This is sLni.lar to the curning point A "h
1. ~~
~~
.
H:@o,mU{TI
()
f
f")
:,/1.:;.
"
'113
,.
atT~al,ne
for the original model O.
B
dat
' y = 1 an d '1S A/'S"
' C,cor
'
f' to
,C.
self-regulating
AI (Be) <
The p"do!:
dt:
which fCy)
r,'
','~
,c•. .
1'' , (,'y' ',J
= A
M.
1 is
y
(.en A) /B.
Tu find the turning point, put
V '" (
8"'",,-.-D,'
",,,2), •
fey)
:=
C.
A exp ( _D~/3)
",'"
cIs
"
~.
)
find the poin':: at wh:i en. the. !I!.uLant:s 'wuuJd start tel de.:rease, put:
y
l.n A
39
To get the i;aIue at which yf(y) attains its sup, its derivative is set
to zero.
Thus
or
1 - 2By
. z = 0,
~
y = J}zB .
Thus,
sup yf(y) = Aj}zB exp(-}z)
'i'his must be less than M.
To find cp(x) , the equation
d
dy Ay exp(-B(x+ y)z)
WJst bt solved.
=
0
This is
A[exp(-B(x+y)Z) - 2B(x+y)y exp(-B(x+y)z)]
=
0
or
1 .. 2B (x + y) Y = 0
yZ
+ xy
- ~B = O.
This is a quadratic in y with solution
-x
y
+
-
J
xZ
2
+ -B
2
The positive root is cp(x).
5.2
Definition of the Generalized Model
The m.odel 0 can be generalized now to a model 0' with a general
reproductive function.
0' is defined by the equations:
40
x(g + 1)
Cx(g)
y(g + 1)
f(x(g)
n (g + 1)
x(g
+
y(g»y(g)
= x (g + 1) +
c(?S.W.)
neg)
+ 1)
<
M
i f neg)
~
M
Y (g + 1)
m
= f(m)(~~:D
y(g + 1)
i f neg)
m
and f(m) > C where m is some number less than M.
This model will be
called 0'.
5.3
Theorem .5 • .s.
Properties of the Generalized Model
If f(m) > C and yf(y) < M for all y ~ 0, and
K
yf(y} s: M(l --)
sup
m
ye:(m,M)
then
neg) > M ~ x(g)
Proof.
neg) > M
= x(g)
~ ! M.
m
~
M - y(g)
or
x(g)
~
M-y(g-l)f(x(g-l)
+ y(g-l».
*
Since f € g ,
x(g)
~
M-y(g-l)f(y(g-l».
By hypothesis the last term on the RHS is less than M( 1 -;) so
41
In this proof only monotonicity of f was used and not the stronger
'k
requirements for f to be in g .
5.4
Linearization of the Reproductive Function
If it is assumed that f has a Taylor Series, then a linearization
can be carried out as was done in Section 4.
seen that all f
fCy ) = 1.
o
E:
In this way it will be
g have an equilibrium position,
i.~.,
3: Yo €(O,M) :7
As certain parameters vary, the type of equilibrium will
vary from stable to unstable and the cycles lose the property of
non-crossing as the parameters approach the value that gives an unstable
equiLibrium.
Let
f(y)
a
=
1 and
fey)
=
f(y) + f/(y )(y-y) +R (y'Yo)
o
0
0
<3
be the Taylor Series about Yo.
R<3 (y,yo) is the remainder term.
since f is such that f I (y ) s; 0, we have
o'
f (y) = 1 -
For y
<
Yo' fCy)
> 1
I f' (Yo) I (y -
Yo) + R (y Y ).
<3 1 0
and thus
y(g+ 1) = y(g)f(y(g»
>
y(g).
To investigate the behavior as y(g) approaches Yo' let
Y(g)
Then using the first two terms in the Taylor Series for f
Y(g
+ 1)
= y(g + 1) _ 1
Yo
y(g)(l-I9(y(g»
~ Yo)
~
1
Also,
42
where
e=
This is
I£'(y
)/.
. 0
Y(g+ 1) =
.Y.W.
Yo
(1- e(y(g) - y » - 1
0
(y(g) + 1) (1_.l.- (
Yo
=
(y(g) + 1)(1
.YiBl ..
1»
- 1
Yo
_.l.- Y(g»
- 1
Yo
= Y(g) +
1 -
.l.-
.l.- Y(g) - 1
y2 (g) -
Yo
If y(g) > 0 then Y(g)
Yo
= Yi&l .. 1 > - 1 and if y(g) < Y then Y(g) > O.
Yo
0
Also if y(g) < 2y , then y(g) < 1.
o
Discarding the second order terms
Y(g + 1) = Y(g)(l
_.l.- )
Yo
Thus if,
1.
y(g)
oscillates sign but grows
II.
y(g)
oscillates sign but damps
III.
Y(g)
goes to zero with no sign change
IV.
Y(g)
grows.
43
These can be expressed as
2 <-..t
Yo
I.
< -..t < 2
I
II.
Yo
0 <-..t < I
III .
Yo
IV.
.i... <
0
Yo
Case IV can be excluded since 6 and y
o
> O.
In Case I after the
oscillations have grown, then IY(g)1 is no longer less than one and
the linearization fails.
I.
II.
III.
2y
<
Y
If' (y 0 ) I
<
o
This can be put in terms of f as
If
I
(y )
growing oscillations
I<
damped oscillations
2y
0 0 0
o<
If
I
(y )
o
I<
y
damped non-oscillating.
0
Case I can be further split.
f(y)
~
If
0 'Iy
then
yf(y)
~
O.
Here the oscillations might be large but yf(y) never becomes negative.
A
However, if f(y)
A
=
0 for some y > Y , it is possible that there is a
o
y such that
A
Y > Y > Yo
and
yf(y) < O.
44
In fact, i f fey)
A - By, Y
A<3/4B and A ~ 4, then
A<3 /4B(A- B(A<3 /4B»
5.5
Properties of Model After Linearization
First the conditions for linearization will be investigated.
The
equations of the model are
+
y(g) = y(g - l)f(y(g - 1»
neg)
x(g - 1)
+ y(g) .
x(g)
A theorem similar to Theorem 4.15 will be given now.
This shows
that t.he linearization carried out in Chapter 4 can be carried out for
the more general model 0'.
Theorem 5.6.
If Y (g) > Y (g) and y. (g) and n. (g), i
1
<3
1
1
= 1,2
are in a
neighborhood of radius 1/[3 about the stable point n , then the
o
difference equation which gives (y (g + 1) - y (g
1
approximately linearized.
positive for [3n
o
+ 1»
can be
:2
The solution of the linearized equation is
< 1 and thus there is no crossing under these conditions.
The Taylor series for f about
where fen )
o
1,
is
fen)
=
1-1£'(n )I(n-n)
o
0
+
R (n,n).
2
0
The model equation can be approximated by
y(g) == y(g-l)(l-If'(n )\(n(g-l) ~n)
o
0
45
The quantity of interest is the difference between two solution
curves.
This can be approximated by
Y (g) - Y (g)
1
2
=Y
1
(g - 1)[1- S(n (g - 1) - n )
1
0
J - y 2 (g -
1)[1- S(n (g - 1) - n )
2
0
J
where
S=
If' (n o ) I .
The var iables can be transformed as follows:
Yi(g)
Y. (g) = - - - 1
~
n
0
n. (g)
~
N. (g) = - 1.
n
-1
,
0
x. (g)
1
X. (g)
n
1
0
Then the equation for the difference becomes
[Y (g) - Y (g) In
1
2
0
=
[n Y (g - 1)
01
- [n Y (g - 1)
o
2
Cancelling the common factor n
[Y (g) - Y (g)
1
2
J = [Y1 (g
o
+ n
+
0
J[1 - Sn 0N1 (g
n J[1- Sn N (g - 1) ] .
0
0
2
and expanding, this is
- 1) - Sn Y (g - l)N (g - 1)
01
1
- [Y (g - 1) - Sn Y (g - 1)N (g - 1)
2
- 1) J
02
2
+
1-
+1-
en
N (g - 1)
01
J
en N (g - 1) J
02
Grouping terms and recalling that
=
N. (g - 1)
X. (g - 1
+
Y. (g - 1),
1 1 1
[Y (g) - Y (g) ]
1:3
= (1-
en )(Y (g - 1) - Y (g - 1»
0
1
., en (X (g - 1) - X (g - 1»
2
0
1
2
- Sn (Y (g - 1) N (g - 1) - Y (g - 1) N (g - 1».
°
1
1
2
2
(i()
46
As long as 0 < n.(g) < 2n
1.
then
0
n. (g)
1.
=-=-n
N.(g)
1.
is between -1 and 1.
- 1
o
Since
o<
y.(g) < n.(g),
1.
1.
Y. (g) :is aIsc' between -1 and 1.
So if
1.
neg - 1) < 2n
o
then
-1 < Y. (g - 1) < N. (g - 1) < 1.
1.
1.
Thus, the last term on the RHS of (*) is second order and a further
approximation can be made.
[y (g) - Y (g)] = (1 - Sn )[Y (g - 1) - Y (g - 1)]n
1
2
0
g
+ ~n c - l
o
1
2
(X (0) - X (0».
2
1
This is a linear, non-homogeneous equation in
Y (g) - Y (g) = l:I(g) ,
1
2
where
6(0)
=Y
(0) - Y (0)
1
2
=X
2
(0) - X (0)
1
The solution is
and i f Sn
o
< 1 then l:I(g) > O.
(See Appendix for proof.)
0
47
5.6
General Properties
In Chapter 4 it was shown that in the model
[2
if the numbers of
mutants at an initial generation was not too large then no crossing
would occur during the cycle.
[2
I
A similar theorem can be proved for
though at first the hypothesis looks a little different.
The
inequality 7 defines y the smallest number of generations that it
could take a population of all mutants to reach the point at which
yf(y) would become decreasing.
The conclusion of the theorem is that
if the number of normals is large enough that in the same number of
generations they could become larger than M- cp(M),
i.~.,
the maximum
population size minus the turning point in a population of M normals,
then there will be no crossing.
Theorem 5.7.
Let
"/e
e g (m,M,C)
f
and let
y (0)
1
> Y (0)
2
represent the initial values of mutants for two cycles.
Let
x (0) > (M-cp(M»/CY-Z
2
where
Y is
3
(7)
Then the cycles are non-crossing.
Proof.
The proof is by induction.
Suppose y (g) > y (g) and n (g),
1 2 1
n (g) < M.
By definition
2
y (g + 1) = f(x (g)
2
2
+ Y (g»y (g).
2
2
48
Since f
g~'(m,M,C), y f(x +y ) is increasing on [O,cp(x )J.
€
222
Assume
2
then that
o < Y2 (g) <
cp(x (g».
2
Then
y (g + 1)
2
=
f(x (g) - y (g»y (g) < f(x (g) + Y (g»y (g)
2
2
since y (g) > Y (g).
1
2
2
1
1
Now f is decreasing and x (g) > x (g), since
2
2
1
x (0) > x (0), su
2
1
Y (g+l)
2
< f(x (g)-y (g»y (g) < f(x (g)+y (g»y (g) = y (g+l).
211
111
1
Thus, by induction, for any g such that
° < Y (g) < cp(x (g»,
2
2
the cycles are non-crossing.
Assume then that
y (g) > cp(x (g».
2
2
Then
y (g) > cp(x (g»
2
=n
2
+ cp(x (g».
(g) > x (g)
2 2 2
Y is the number of generations needed for the population to grow
from m to cp(O) if all individuals grew at f(m) > c per generation.
Since the population does not grow that fast, then g
~
by the hypothesis of the theorem, x (0) > (M-cp(M»/CY,
2
x (0) >!1-W(M).
2
Cg
y.
Thus, since
49
(If not, then n = n + y > M
cp is a decreasing function and M > x (g).
2
and the cycle is completed.)
2
2
2
So
cp(M) < cp(x (g))
2
and
(0) >
X
2
x (g)
M - cp(x 2 (g))
M - cp(M)
> ----:~-g
g
C
C
cgx (0) > M - cp(x (g))
2
2
2
Thus
n
(g) >x (g)+cp(x (g)) >M-cp(x (g)) +cp(x (g))
M.
2 2 2 2 2
So i f
y (g) > cp(x (g))
2
2
then the cycle is already complete.
5.7
0
Conclusion
The linearization and theorems in this Chapter give strong support
to the idea that the
behavio~
of the model 0 defined in Chapter 3 would
not be changed if the fecundity function of the mutants, A - Bn, were
replaced by almost any monotone decreasing function.
Thus, if the
mutants were to decrease their reproductive rate according to any of
a large class of functions, there exists parameters for which the mutants
could win out.
50
6.
SUMMARY AND OVERVIEW OF OPEN PROBLEMS
The most important result given in the first part of this work is
the development of a model which demonstrates a method by which a
population could evolve a self-regulating mechanism.
It has been
shown that the model is not overly sensitive to changes in the
parameters of the model.
changes in form.
Chapter 5 showed that the model even allowed
Thus for a wide range of parameters and even a wide
class of reproductive functions it is possible for a population to
evolve a self-regulating mechanism.
Another point that was made is that the success (or failure) of
the mutants in the evolutionary struggle did not depend on unrealistic
behavior of the solutions to the equation of the model,
i.~.,
crossing
did not play an important role in the model, the model was well behaved
around its stable point and round off error caused no qualitative
differences.
All of these results are consistent with and support the
extensive computer work of Williams (1975) on this and similar problems.
The model is so well behaved that certain generalizations suggests
themselves for future study.
The individuals could be made diploid and
the two types interbreeding.
This suggests the use of two decreasing
reproductive functions and one constant reproductive function.
The
results of Chapter 4 seem to depend on the differences between the
reproductive functions so that similar results might be possible for
any two straight line functions.
Sometimes the crash is a too severe reduction in the population
size.
A slower reduction may occur over two or three generations.
51
In such a case the first generation of reduction is the same as a crash
generation in the model
("2.
Thus one need add to the present results any
further consequences of yet another reduction.
An extension of the idea of gradual crashes is the idea of
oscillations.
When there are oscillations in the population, sometimes
the mutants have the advantage then later they would not.
An
investigation of the possibility of evolving self-regulating mechanisms
under oscillations could have great value, for oscillations in the
population size are common.
If it were true that a self-regulating
mechanism could be evolved in the presence of relatively mild
oscillations then processes similar to the one discussed here would be
quite common.
It may be possible to extend these results to diploid populations.
If the numbers of the alleles were considered instead of the numbers of
the individuals then only two quantities need be considered since there
are only two alleles.
The resulting difference equations may not be
very different from those considered here.
52
7.
REFERENCES
Andrews, R.
1968. Daily and Seasonal Variations in Lemming Adrenal
Metabolism. Physio. Zoo1. 41(8):89-94.
Chitty, D. 1967. The Natural Selection of Self-Regulatory Behavior
in Animal Populations. Proc. Ecol. Soc. of Australia. 2: 51-78.
Christian, J. J.
1961. Phenomena Associated with Population Density.
Proe. N. A. S. 47:428-448.
Christian, J. J., and D. E. Davis. 1964. Endocrines, Behavior, and
Populations. Science 146:1550-1560.
Christian, J. J., J. A. Lloyd, and D. E. Davis.
1965. The Role of
Endocrines in the Self-Regulation of Mammalian Populations.
Recent Progr. Hormone Res. 21:501-571.
Miller, K. S. 1968.
Inc., New York.
Linear Difference Equations.
W. A. Benj amin,
Snyder, R. L. 1961. Evolution and Integration of Mechanisms that
Regulate Population Growth.
van der Vaart, H. R. 1973. A Comparative Investigation of Certain
Difference Equations and Related Differential Equations:
Implication for Model-Building. Bull. Math. BioI.. 35: 195-211.
Watson, A. and, R. Moss.
1970. Dominance, Spacing Behavior and
Aggression in Relation to Population Lincitation in Vertebrates.
p. 167-181. In Watson (ed.) Animal Population in. Relation to
Their Food Resources. Blackwell Scientific Publi.cations,
Oxford.
Williams, G. C. 1966. Adoptation and Natural Selection.
University Press, Princeton, New Jersey.
Williams, M. B.
Mechanism.
Wright, S. 1969.
pp. 160-162.
Princeton
1974. A Model for the Evolution of a Self-Regulatory
In Press.
Evolution on the Genetics of PopUlation, Vol. 2,
University of Chicago Press, Chi.cago, Illinois.
Wynne"Edwards, V. C. 1962. Animal Dispersion in Relation to Social
Behavior. Oliver and Boyd, London, England.
53
8.
8.1
APPENDIX
The Solution of Equation (8) in Chapter 4
To prove Theorem 4.15, we need to show that the solution of
(y (g
+ 1) - Y (g + 1)
- (Y (g) - Y (g»
1
:3
1:3
is positive for A < 3.
Let 2(g)
= - A(Y
(g) - Y (g»
1:3
=Y
Y (g).
(g)
+ (A - l)Cg~(O)
Then the equation
2
1
is
Z (g
+
- AZ (g) + (A - l)CgZ (0)
1) - 2 (g)
or
2(g+ 1) + (A- 2)Z(g) = (A- l)CgZ(O).
This is a non-homogeneous first order difference equation.
Any
solution is the sum of a general solution to the homogeneous equation
and a particular solution to the full equation.
To get the particular
solution, let
2 (g)
P
where K is a constant.
Dividing B/C
g
=
KC
g
If this is a solution
this is
KC
+ (A - 2)K
(A-l)Z(O).
Solving for K
K
=
A- 1
]
Z (0) [ A+ C _ 2 _ .
The auxiliary equation for the homogeneous part is
III
which gives m
- (A - 2)
2(g)
= (2
+
- A)
(A - 2)
=:
0
Thus any solution is of the form
A-I
g
A+ C _ 2 Z(O)C +
K:3
(2 - A)g.
54
We would like
Z(O) = Y (0) - Y (0),
1
i.~.,
2
the function should take the value Z(O) at g
O.
So
A- 1
Z(O) = A+C _ 2 Z(O) + K
2
Thus,
Z(g)
=
A:~ ~ 2
Z(O)C
g
+
Z(O)
[A~~ ~ 2 J(2 -
A)g
is the solution.
Now, 1 < C < A so A + C - 2 > O.
be negative.
positive.
If 2
~
Thus if A < 2, Z (g) can never
A < 3 and g were even, then all terms are
So let 2 S A < 3 and g be odd.
If the solution Z(g) is
to be less than 0:
or
(A - 1)C
g
+
(C - 1)(2 ~ A)g
A -- 1
CuI
< 0
(2 ~, A)g
cg
-- <---Since A < 3, A - 2 < 1, we have
A-I
C- 1 < -
(2 - A)g
Cg
(A--2)g
1
<--g<1o
g
C
C
Since C > l, both sides can be multiplied by C - 1
55
A-l<C-l
A<C
which is a contradiction.
8.2
Thus, i f A < 3, Mg) > O.
Solution of the Difference Equation of Section 5.5
The difference equation is
Mg) =
0 - Sno )Mg -
1)
+ Sn 0 MO)C g "
1.
As in Section 8.1, the solution is the Sum of a particular solution and
a general soluti.on to the corresponding homogeneous equation.
The
particular solution i.s
6. (g)
P
=
g
K C .
1
Substituting this into the difference equation gives
g
Dividing through by C - 1 gives
K C
1
= 0-
Sn )K + Sn Mo).
0
1
0
Solving for K gives
1
Sn
0
\ = [C - 0_
Sn ) ] 6.(0)
o
Thus the particular solution is
The general solution of the corresponding homogeneous equation is
derived from the auxi.llary equation.
Thi.s is gotten by replacing Mg)
56
with m in the difference equation.
So
m - (l - ~n)
0
o
gives
m
=
1-
~n
o
.
The solution is
Mg)
Thus the most general solution is
The right hand side must be 6(0) when g
~n
L:>( 0)
=
0 so
o
~n
K
2
6(0) [
l-C_(l_o~n)J
o
Thus
Note that i f
~
=
.
B and no
A-I
= -B-
the difference equation considered in
this section is the same as that in Section 8.1.
58
1.
INTRODUCTION
The logistic equation is defined most generally by the differential
equation
(1.1)
y(x)
where
0',
biology.
~,
= ~y (x)
y(cd~) =
(1 - y (x) /k),
k are positive constants.
.5
This equation has many uses in
It was first proposed by Verhulst (1845) as a model for
population growth, in which the argument of y is time.
~
In this case
is the intrinsic reproduction rate of the individuals in the population,
k is the so-called carrying capacity, and
0'
is a constant of integration
that determines the inflection point.
With k = 1 the equation has been used by Berkson (1944) and others
(1972) in the bioassy of quantile responses.
In this case the form
commonly used is the integrated form
1
y (x) = 1 + exp (- (0'+ ~x)
,
and x represents the dose or strength of a stimulus or drug such as a
poison or X-rays and y(x) is the percentage of subjects receiving dose
x and having a response.
The response is of the all or nothing type
such as death or survival and
O'/~
is the LD
50
or dose at which exactly
50% of the subjects show the response.
The equation can also be used in work involving autocatalysis,
enzyme reactions, and hemolysis (Burn et a1. (1950), Makler and Corder
(1966».
In this work several methods to estimate
~
are developed.
The
significant idea in the methods is that the differential equation (1.1)
59
does nct have to be integrated.
In the past one method of estimating
the paramE.ters of the equation has used the integrated equation and
sume variant.
~Jf
non··linear least squares (Oliver (1964)).
method assumes that the errors on the observations Y(x )
Another
i = 1, ... ,
i
£1
are independent and normal and then solves the resulting non-linear
maxiu,u:.n lLkeli.hood equations by numerical methods (Oliver).
Lastly
in an interesting method due to Berkson (1953) t.he maximum X2 estimates
are obtained by numerically solving a non-linear set of equations.
All
of these methods use the integrated form of the equation.
If the differential equation is written
y(x) = Sy (x)
- yy2 (x)
it can be seen that though it is a non-linear differential equation, it
is linear in the parameter S and y.
at points x ,
1
points.
... ,
x
£1
Usually values uf y are observed
.
but i t is not possible to observe y at these same
The key step to all of the methods of estimating the parameters
of the logistic: in this work is the use of the derivative of a cubic
sp line in order to approximate y.
'.i:he observations are used as est:imates of "(x.)
and the derivatives
J
~.
of a cul)ic spUn;;; are estimates of }r(x.).
~
Thus i.t is desirable to
minimize the distance between the true function, yex), and its estimate
as well as the distance between the true derivative, y(x), and its
estimate.
I~he
Sobolev norm measures distance between functions in terms
of the difference in the functions as well as the difference in the
derivatives of the functions.
Of course, it is usually not possible to
l)t,serve the func,tion but only its valu.es at discrete points.
Thus a
60
d iscrc:r:e version of the Sobolev norm is minimized to obtain the estimate
uf~.
This quantity is
E (y (x.)
1
i
l,sing this instead of the usual least squares emphasi.zes the fit of the
approximated derivative values as well as the fit of the Y(Xi) values.
In Chapter 2 a review of SWle of the known methods of estimating
parameters of the logistic is given.
Also there are examples of some
of the many uses of the logistic.
Chapters 3 and 4 give a discussion of elementary cubic spline
theory, the Sobolev norm and their connection.
Also a least squares
cubic spline is derived.
In Chapter 5, the parameter
~
in the equation
y = ~y(l - y)
(1. 2)
is estbnated by fitting a spline to the observations y(x.), . " , y(x )
1
n
then differentiating it and using the result as an estimate of y(x.).
1
'Lewo prucedures are discussed.
used.
In the first a two step method is
A cubic spline is fitted to the data by least squares methods.
This spline is then differentiated to give
derivative at the x, I s.
~-(x.), an estimate of the
1
These estimates are inserted i.nto equation
1
(1.2) and l:hen the discrete version of the Sobolev norm is minimized
1:0 fi nd ~ only.
If y.
=
yex.) and
~.
1 1 1
the Teast squares estimator of
~
y.
l
=
~
1
in the model
~Y. (l - y.)
1
~(x.)
1
+ O.1
this gives essentially
61
w\-.,,~·re
',~n"
o.'I.
t.he
"'<-~(',md
a:ct:~
Ve"cedure m:Lnimi,zes 6 simultaneously picking the spline
....,
('-'81t1.::1<':'1 [3
and
FxLen,51;;c:
The
aC'Jiat:l:::J£ls due to random error in thE: observations.
1\
S.
~'1u;).!.:e
las~~ 8t~'..:tion
Carlo studies vf chesa methods were carried out.
of Chapter 5 rep::>rts on the results
0
r,:hapter 6 uses a di.fferent fOem of the h)gis 1:ic. equation.
If
[3y (1 - y)
y
then
1.
(1.3)
y i o.
S(l-y),
y
d(.e~ _,
dt
(.en
y)
S:) Eqr,aU,J£l (1.3) can be wri tten as
.e
S(l·-y).
":he tiN,) pC(K:edures used in Chapter 5 are repeated hut now the spline is
fit:t~~d
t,J .eLl (-".) instead of v.
~~he
, 1:
0
- 1
last section repGrts on Monte Carlo studies of these methods.
c:hap ::81' 7 gives a smmnary of the work and suggests other applications
ok these methcds
i~
the field of biology.
62
2.
ES'~duse
~Hl
~L6
REVlf,w OF LITERATURE
0
of Lht:: many uses of the logistic equation, the literature
applicatiuns a.nd on estimation of its parameters is quite
Unfortunately thc:::re seems tu be no comprehensive review of
di'"erse,
tither the applicaU.ons or of the estimation problem.
so,nE::
id""a of the many us,'s of the logisric was given.
In Section 1,
Section 2
rev iews some of the Ii tf:rature on the estimation problem.
No doubt
due to the difficulty of the problem the literature here is quite
large,
A goud bibliography of bioa3sy app lications appears in Ashton
(1912),
1.'he third section deals with the problem of the error
s Lructure
La
h' used,
There seems to be l i ttle work in this area.
The last section gives a brief survey of the literature on fi.tting
splin~s
to data.
2. 1
The
Applications of the Logistic
logLst~c
.~.9.uation
equation has a long histot'y of applications in
hiclogv, economics, and demography
is that of Verh:,dsr (184'»
~
One of the earliest known uses
who dpparendy named the equation.
Pearl
and R"t-,d (lY2U) used it to describe the population growth of the
Lni ted Scales"
In addition
Lu
populatl.Gn g:cowth, the equation has been
used to describe thE growth of bacterial colonies, Gf an individual
[at, and sunflower's
earl:,
·U&8.S
0
Lotka (1956) gives a good discussion of the
of the eq'latioD
Co
describe growth,
i,'ht-: t::quaU.on is aJso ',lsed as a model fur various pb.enomena in
The percentage of blood cells l:'i'sed by di fferent:
l..u:1<.entt at iuns ,-,f a hemulysin lui
!UvIS
a 'logistic law (Ashton, 1972),
63
c"w r'f;,:d.2·"·s;:n- i ids:3elhach equation which descrj.bes the ti.tration of a
weak
(Jt'
a~id ~ith
a strong base gives a logistic curve when concentration
hase is plc'l:ted against pH (l1akler and Corder 1966).
The process of
aul:ocal:.alysi.., is described by the logistic curve (Ashton 1972).
Cc;'...;'l:?
rea •.,tions buch as the hydrolysis of sucrose into glucose give a
!)g is L:~,~ c'Jl'";e when concentration is plotted agains t time.
11;,,,',;
Certain
CC:LI1
S:'.\i1
Some of these
a tL 0 o:cetical hasiswhile othern ha-Je only empirical
Never t:hele.3s in all of these cases experimental data usually
i> a:king.
fit the curve with very little error.
()c'e of the most important uses of the logistic equation is logit
al'a~y:::; i;:;
il' the bioassay of quantile respoD.se.
ldf~a
BccksOll (19".'+) this
1964).
':0
First proposed by
originally caused some controversy (Finney
Host recent opinion seems to he that there is little reason
prefer the rno'.:"e classical probit analysis (Finney 1971).
'.L'he ea.cU.eoSt method of estimating the parameters seems to be the
T!l(~dlOd
due 1:0 Pearl (1950).
n
{ y.e
-b x.
l
,,7),C:"c~
(:';.,
J.
'~'.)
'J.
('
OL
O _ (6b)x.)+
(y. _l)e
~
a}
2
~
are ohservations oa
[(x)
ann. b
This is an iterative procedure that
1
-;-+e-:(a+b~) ,
ie, an initial gutossed 'faIlle of b.
After a and 61> are estimated
64
t C'l-'il'
(::'
0
n,
a
(lc'W
b is calculated b:y b a
puh'~E:d
S:h,lr.z (19.30)
+ Ll.b
and the process is repeated.
out that Pearl's method did not give true least
'':;.1'' ::it':''') t's riu1ates and prl)posed minimizing
r:[Vo
, 1
wh·.::re a
l.'
dod b
0
f(x ) - f'Ca )Ll.a- E'(b )Ll.bF
0
o·
0
dte initial values and Ll.a and Ll.b are corrections.
This
'ned,od uften tdkes many iCErations to conVE',rge.
DavIs (1941) sL:ggested writing the differential equation as
lEY
Y dt
l~ht:n t:
S(l-y)
st l:l1dtl ng the derivative by Ll.y / Ll.x giving
lJur.
y Ll.x
then using the usual least squares approach.
Oli~cr
(1964) has criticized all of these methods and urges the
as,,=, (if the !lH,t-.hud of ful1. least squares because of its equivalence to
Iraxl'Fll'al Li'kc:lihood in the case of independence and homoscedastic
nurmd,Iit:~,'o
Unfut'tL.nately there is no reason to make such an assumption
and b.iog: L analysis just the opposite muse be assumed (Finney 1964).
H,:t:ksCi(j whv fil'st suggested that the logist:ic be used Ln bioassay
suggi-;SLS using :njniJm.nn chi",square estimat:c.rs (1953)
This results in
0
(l-l.!.) is the
L
65
probability of response where Pi is the sample estimate of Po
Thi.s method
though giving good results is only useful in the analysis of quantile
responses where the dependent. variable is the parameter of a binomial
distribution.
2.3
Error Structures
In almost all uses of the logistic equation observations are made on
the dependent variable y in
y =
~y(l-
y).
Thus it is important to know the error structure of these variables.
Unfortunately there has been little work on this subject.
Indeed
apparently not much thought has been given to the subjecL
The bioassay of quantile responses is the one area that has a welldeveloped theory for the error structure.
In this appli.cation of the
logistic y is the percentage of individuals receiving a given level of
stimulus and showing a response"
Because of the assumption that
associated with each dose is a constant probability of response in each
individual, y is a binomial random variable
y gives the variance of Yo
Knowing the distribution of
Finney discusses the theory in both of his
books covering the subject (1964, 1971).
detailed
0
Ashton (1972) gives a less
di~~ussion.
[n other appl ie.ations ease of computation or expediency seem to be
the deciding factors as to the choice of error structure.
For example,
Oliver (l961.) argUES that the least squares estimators of the parameters
are. che best because of their equivalence to maximum likelihood estimators.
66
Rut least squares estimators are equivalent t.o MLE only under the assump-
thln of hOflloscedasticity,
Finney (196/+) makes some suggestions for bioassay uses.
He suggests
duing a simple analysis of variance to determine whether there is
tvidence against pooling.
If there is, then a transformation can be
used and he: recommends the arc sin.
",,"ith
d
c, f)Sr~dClt\/ariance if
y(1<>y).
This transformation will give data
the original variance \>;dS prop0rtional to
This, of course, is exactly the form of the variance in
quantile response bioassays.
Since the logistic is often used as a model for p0pulation growth
one might think that the stochastic versions of the theories would give
some information on the error structure.
Feller (1939) in a c1 assic
paper was the first to give a stochastic version of the logistic.
More
recent literature seems to support the idea that: Feller's failure to
give an expression for the population variance as not a minor oversight.
There' seems to be no expreSSion for the variance in the
literat:ure..
Attempts at deriving t.he variance fur logistic-like
populations (Bartlett: et al. 1960) all make simp] i f::ling assumptions
such as the population is close to the upper asymptote and a first
order approx imation wU 1 do.
0 thers such as Tsukos (1973) assume that
the limit on the populations is not the
so~(:alh'd
the logisclL: but.: a predetor or competitor.
carrying capacity of
When Lhe other species is
removed, the formula for the variance beCl:jmE's usel t'''s.
AnothEx example of neglecting the error SLrucune is a method by
Rhodes (1939),
Essentially Rhodes modifies the integrated form of the
logistic to obLdin
67
1
yt
Y + Y (-)
o
where y , Y ,
1
0,.
1
are population values at equally spaced time intervals.
Z
He then uses the least squares estimators of
Y1
and y .
a
Unless one
assumes that the error in l/y HI is independent of the error in l/y t'
these estimators are biased.
problem,
Neither Leslie nor Rhodes mention thi.s
Of course, in some bioassay problems this independence could
be assumed.
Again there is little reason to assume that the errors in
the l/y's are homoscedasti.c:.
Thus, Finney's suggestion of a variance that has the form
er2 (y)
0:
Y (1 - y)
or a homoscedastic variance seem to be the only choices considered in
the literature.
204
Fitting Splines to Data
The idea of a spline function is relatively new having been
discovered by Schoenberg (1946) in 1946.
DespH:e their nice properties
they did not immediately catch on so that mos t of t.he literature is
quite recent.
Papers by Poirier (1973) and by Wold (1974) give a good
review of [he uses of splines.
Most of the applications of splines
are in the area of numerical analysis with not much acr:encion paid to
statisci~al
aspects.
In this work, rhe derivative of the spline will be used to
8stima.te the d"rJvative of a function.
Ht-:t'shej and Zakin (1968)
investigated several methods of smoothing and differenr:iaLing data
68
and
fo~nd
splines to be superior to all other methods tested and to give
good results with errors in the data of up to ten percent.
Dunfield and
Read (1972) applied spline functions to data and then differentiated them
to obtain reaction rates in certain chemical processes.
good results.
They got: quite
Bell (1973) used spline-like functions and other methods
to estimate derivatives and parameters in di.fferential equations.
He
found that his spline-like function did as well as the usual methods.
The estimation of derivatives with splines is complicated by two
sources of error.
The derivatives of the spline converge to the
derivative of the function as the number of knots go to infinity.
However, in any application there will be a finite number of knots and,
in general, some deviation from the true values
bounded (see Ahlberg et al. (1967)).
0
This error can be
In fact, Secrest (1965) gives a
bound on the deterministic error that is a function of the position
of the knots only.
of the error in
S is
This bound could be used to determine how much
attributable to the deterministic error.
However, in the application to be considered here, the observation
contains random error.
There seems to be no work on the exact nature
of the error in the estimate of a derivative under the assumption of
random error in the observations.
69
:J
0
LEAST SQUARES SPLINES
In this chapter the cubic spline function is defined and some of its
properties are given.
Also the idea of a least squares spline is
developed and some problems arising in the choice of: the conditions are
discussed.
3.1
CGbic Splines and Their Properties
The discussion here follows that of Ahlberg et al. (1967) and
Greville (1969).
De fini tion:
A cubic spline function on [a,b] with knots gl' ... , gk is a
function Sex) such that on any segment [~., ~.+_] Sex) is a cubic
J
J 1
(or less) degree polynomial and at ~j' the functiuns S, S', and
SII
are defined and continuous for all j = 1, ... , k.
There are a few things to note about this definition.
Any po1y-
nomia1 of degree three or less is a cubic spline; in this case the
spline function can be considered as having no knots or an arbitrary
number with a:britrary locations.
In general, the third derivative of
a cubic spline is a step function with jumps at the knots.
In fact, a
spline function can be defined as the three time integral of a step
Eunctiuno
If there are m knots, then there are k
~
1 intervals.
Since the
spLine is a cubic on each interval, this gives 4(k - 1) parameters to
specify.
However, the cubic on the intErval [xc '.1' x.] and the cubic
]",
J
on the in~:erval [x ' x + ] must: have the samE:~ value and first and
j
j l
second derivati.ve at x..
J
Thus, there are three less degrees of
70
Thus, the 4(k- 1) degrees of freedom
freedom for each interior knot.
are reduced by three (continuity conditions) times (k - 2) interior
knots lea-dng k+ 2 degrees of freedom.
If the value of the spline is
now specified at each of the k knots, only two degrees of freedom
are left.
These two usually appear as boundary conditions, one at
each end of the interval [a, b J.
In fact, the following theorem can
be proved (Ahlberg et al. (1967)
< S
Suppose a = S
1
or (Grevil1e (1969».
< Sk = band f(Si) = Vi and suppose that
2
one of the following holds
f
I
(a)
f
I
(b)
f"
(a)
=
(i)
or
(ii)
or
2f
I
(a)
+ IJ. f' (S )
o
2
f (S ) - f (t; )
= K
1
2
s
2
-S 1
1
(iii)
Then there is a unique cubic spline that satisfies these requirements.
This means that the values at the knots plus the specification
i, ii or iii at the endpoints completely determines the cubic spline.
The choice of end point conditions will be discussed at greater length
later in this chapter in connection wi th least squares splines.
71
Splines have very nice convergence properties.
Indeed splines are
being used here to fit data because of the strong convergence properties
of the spline as well as the derivatives of the spline.
In general, as
the nurribe.r of knots increases, the sp line converges to the function it
i.s inrerpolating.
In addition, the derivative of the spUne and second
derivative of the spI ine converge to the first and second derivative of
the function.
This convergence is quite strong.
(1967) that i f tl
k
It is shown in Alberg et al.
is a mesh of the interval [a, b ] with k knots, and
Stl (x) is the spline approximation to f(x) e C2 [a,b], then
k
f (x) - Stl (x)
k
= 0 ( Iltl k 11 2 )
and
f' (x) - S~ (x)
k
uniformly as the mesh norm, IItl
k
ll,
= O(lltlkll)
goes to zero if the spline has either
(a) or (b) above for end conditions.
If the assumption that f(x) e
the function and its derivative respectively.
The remarkable thing
in this ease, (f e C3 [a,bJ) is that the second derivative converges as
O( Iltlkll) and the third derivative of the spline converges to (" (x) as
0(1) despite the fact that the third derivative of the spline is
discontinuous.
3.2
Least Squares Splines
In the discussic.n in Section 3.1, it was shown that for a spline
with k knots
then~
are k+ 2 free parameters.
These consists of the
72
values at each of the k knots plus a constant for each of the two
boundary condi.tions.
The value of the spline at any point can be
written as a linear combination of these k+ 2 parameters.
For data
that '_:ontain random error it is desirable to use the k + 2 parameters
that give the best fit in some sense.
The following definition gives
a spline that is best in the usual least squares deviation manner.
Definition
Let y (x. ) , i
i
knots at
= 1,
" ' , n be observations and assume these are
Sl' "', Sk' k < n -·2.
Then the least squares spline is the
spline, S (x), for which the values at each of the k knots and the two
boundary condition constants are chosen so as to minimize
n
(y (x .) ~ S (x . ) ) 2
E
1=1
~
•
~
The concep t i s slightly different than that used by Poirier (1973)
and by Wold (l914) who do not: estimate the boundary conditions from the
data.
The derivation that follows is similar to that of Poirier's with
modifications being made so that the boundary condi.tions can be
estimated.
Also the first: rather than second derivatives are used
here.
Let x , ... , x
1
observed
n
ordinat~s.
be the abscissas and y , ... , y
1
Let S_' ... , Sk' k
l.
conditions
2'J
~ 'J
r = [y l ' 111' ... ,11 k ,
be estimated.
(1967) "
The general end
.
1, . " , k are the values of the first: df>xivati.ves at the
1
Let
< n he knots.
the ... orresponding
102
where Vo, i
knots.
+
n
YkJ
T
Then cubic spline theory
be the vector of parameters to
givE:~s
that: (see Alberg et
1!l.
73
2
Ie
J1. o
2
I
0
Ie
a
a
............
a
J1. 1
0
................
a
2
fL
:3
0
~
0
"
., . .. ..
.
..........
0
0
-JA
__-.1.
h
1
0
2
Ak .. 2
....
....
.,
(Ieh
J1. k - 2
A_
k 1
0
"
1
a
....
:3
,,<,>"" ......
1
It
\I
2.
\I,
K
........................................................ 0
3J1.
h
3J1.)
., h
~
~~
I
o
a
2
:3
C
-3A _
k 2
h _
k 2
Ak - 2 _ 3J1.k"2)
h _
h _
k 2
k 2
3f1 _
k 2
h _
k 1
and A 1 are constants, h.
kJ
h.
1
h. 1
.J -
+
and
J1. J'+1 = 1 - AJ'+ l
This matrix equatioQwill be written
/1.\1 =
®I
h.
J
11 I
11 k
1
0
where fL
0
YI
Yk
I
..~
74
It can bE: snO\,Jn that 11. Is non-singular as long as A.. 1 and fl.
f.(-
.
0
< 4 (see
Ahlberg et .aI. (1967).
Thus,
under these cCJf!dit:ions.
Now define two matrices P, Q with Pen X k) and
Q(n X (k+ 2)).
For S. 1
1."
~
x.
~
g.,
J1
(g.-x.)2(x·-g· )
1.
J
J
1.- 1
:i .. 1
h~1.- 1
-(x·-g· )2(g.-x.)
J
1.- 1
1.
J
i
h2
i.-1
o
ow
(g. - X.)2. (2(g. - x.)
1.
J
1.
J
+ k.
1-
1
h~1.- 1
(x .., g. 1)2 (2(g. - x.)
J
1.1.
J
+ h·1- 1
o
L+ 1
h~1.- 1
o
ow
From cut-Ie spline theory the value of the spline at any puiut x. can
J
bewriti:.:en in vector notation. as
.§. (~)
p'J -t. Q[
75
but
so
(3.1)
or
Thus, this equati.,m gives the values of the spline at any point x.
J
as a linear function of the values at the knots
11
,
••• ,
1
two constants
11k
and the
y l ' Yk which represent the end conditions.
The usual least squares estimator of
I
is
where y is the vector of observations at the x's.
The LS estimator of the value of the spline then is
T-_ - . T
= WIII = W(W-W)
1W y.
II
~(x)
(3.2)
1'0 get the deri.vative of (3.1) define two matrices P I and Q I .
S1.'-l
~ x.J ~
S.L
(g. - x.) (2g. 1+ g. - 2x.)
1
J
11
J
£ = i - 1
h~_l
(g. 1- x .)(2g.+g .. 1- 3x .)
I
P'i,
J
1-
J
1
1-
J
£ = i
h~1- 1
o
ow
For
76
-6(lc-g.)(S.-x·
J
1
1- 1
J
)
i
h~.1- 1
6 (x. (
1
I
g.) (g. - x.
~j,e
J
J
1-
1)
i
h~_l
o
+1
ow
p' is n X k and Q' is n X (k + 2) .
Now
Th'lS,
the derivative of the least squares spline is
or defining
W'
=P
I
!I. -1
e+ Q
I
and the estimated values of the spline and the derivatives of the spline
are 1 iXlear Eunet ions of the observations y.
The expression for the derivative given here is not the only one
possible.
For example, Ahlberg et. al.
(1967) suggest: fitting a spline
to data then differentiating it and using the values of the derivative
at the knots as dat:a points for a new spline.
as thE derivative estimate.
spI Lne fuc the derivative.
more straightforward.
This new spline is used
This has the advantage of giving a cubic
The method used here seems simpler and
Also it gives the deri.vative as well as the
spline itself as a linear function of the vector X.'
n
~'he
8plines derived in this manner give t.he usual lease squares fit
of the data points (x., y.).
.1
1.
derived has no such property.
However, the derivative of the spline so
"
This is, as it has been derived, S/(X) is
not guarant:eed a best fit eo Lhe derivative of t.he underlying funccion
being estimaced by che spline.
In Chapter 4 a method that attempts to
deal with this problem will be introduced.
It should 'be observed that more can be said in certain Cdses.
=
the limiting case of no error in the data if y.
1.
f (x.) for some f e: C I
1.
then the spline and the derivative of the spline converge to f and
respectively.
f'
Secondly if there is random error in the daca but every
x. is equal to a
1
estimator of
In
I
~.
J
(i.e., every observation is at a knot) then the LS
--
leaves y
and y k free and sets T). equal to the mean of
J
1.
the observations at
S..
Unless the data are very poorly behaved these
J
means will converge to the true value of the dependent variable.
Lastly
if there are observations which are not at knots then the estimates of
the T). are correlated.
However, Powell (1969) bas shown that i f the
J
knots are equally spaced then this correlation is exponentially damped
as the distance from the knot increases.
'r
covariance matrix (W-W)
T
rows of (W w)
.. "
T
1W
In this case, the variance-
_
1
can be checked for hi.gh correlations and the
can be che eked to see if a large weight is bei.ng
given to a point far fruTH a knot.
:3.3
Orthogonali ty of the Data and the Endpoint
Pr()blL~Ill
Con.sider the ease in which every x. is a knot aod there is at:
J
Least one ubservation at each knot.
In this case the ','1atr:ix: P = 0
and ther.efore W :::- Q, which has the form
(w+
2 columns)
where n. is the number of reps at x.
J
J
S..
J
ro~
Then,
0
n
0
0
1
0
'T'
W-W
n
19-
0
0
0
Thus the normal equations are
WTW~
or
-
k
WTy
79
0
Ov 1
n
n
1
Tl"
1 1
E Y
j=l d
n
nk~k
=
k
E Y .
k
j=l
J
Ov k
0
. .t: h e J
. th
.
were
y .. 1.8
- 0 b servat10n
at t h
e '1th
- k not.
h
1J
The solution leaves
Y1 and Y arbitrary while
k
T"l.1
y.,
1.
which is intuitively pleasing except for the fact that Y1 and Yk are
undetermined.
When this situation arises, there is no choice but to bring other
knowledge into play.
In most of the applications to be discussed here,
the function is the logistic with values between 0 and 1 where
are also asymptotic values as x --. -
<Xl
or
+
<Xl
respecti.vely.
a
and 1
In addition
the differential equation is
y = ay (1 - y)
which gives
upon differentiating.
All ,)f this is preamble to the following suggestions on how to
pick Y
and Yk"
If the endpoints are far out from the inflection
1
point so that Yl. (orYk.) is approaching zero (or unity) then the
80
derivatl\7E::S will be approaching zero.
Thus, it would seem best to pick
the end conditi.on that implies zero second derivative unless
suspected of being large.
~
is
That condition is
-11)
3('11
+ \)
2\)
1
1
=
2
h
:3
1
Thus, y= 3 (11
1
- 11 ) Ih.
The same argument at the other end gives
121
In case the data points are clustered about the center and it is
not felt that the choice of a zero second derivative at the endpoints
is sufficient an estimate can be gained from the formula
y
=
~y
(1 - y) (l - 2y) .
The data are usually clustered only when there is some preliminary
estimate of the parameter.
This can be used to approximate
~lll
~
y (1 -
P
1
Y1 ) (1 -
2y )
1
and the end condition
+ \)
2\)
1
3 (11
1
2
- 11 )
2
h
h y//
1
1
2
1
can be used.
J t: is seen that unless y// is quite large, the choice
1
yH
" 1
0 will not be very different.
81
4.
4.1
THE SOBOLEV NORM AND SPLINES
Definition of the Soholev Norm
A Sobolev space of order m on [a,b] consists of the set of functions
on [a,b] with absolutely continuous
~
derivative and the norm defined
by
(4.1)
This is not the most general definition of the Sobolev norm possible
but it will suffice for the purposes of this thesis.
In fact, a
simpler version of (4.1) will commonly be used here.
I t was pointed out earlier that as the number of knots increases
not only does an interpolating cubic spline converge to the function
being interpolated but the derivatives of the spline up to the third
order converge to those of the function.
Thus, from the definition
of Sobolev space and norm a spline will converge to any memeber of the
space that it is interpolating.
In the rest of this chapter a method of estimating parameters in
differential equations by minimizing a discrete version of the Sobolev
norm will he described.
Since the equations of interest here involve
only the first derivative only a norm of order one is considered.
This
discrete version can be written
ilfl!2 _.
n
n
] + E
E [f (x.)
~
i=l
i=l
[f '
(x.)
~
J .
-2
Note that a Soholev norm of order zero is just the L
norm and the
<3
discrete version is just the sum of squares at the points x ,
1
.. .
)
x .
n
82
4.2
Estimating Parameters in a Differential
Equ.ation By Minimizing a Sobolev Norm-Like Expression
Suppose that y , ... , Yn are observations on a variable that satisfies
1
a differential equation
y(x) == g (y(x); (3)
Let x , ... , x
where 13 is a parameter that appears linearly in g.
the points suel] that y. == v(x.)
L
J
1
1
+ e.1 where e.1 is random error.
n
be
Let the
knots be at Sl' ... , Sk' k + 2 < n.
Let W, Wi be the matrices and
3 such that Wr
and
w'r
r
be the vector defined in Chapter
gives the values of t.he cubic spline at x 1 ,
••• ,
xn
gives the values of the derivative at x , ... , x .
n
1
Then the problem can be written in the following form:
(4.2)
where
A
is a vector of quantities to correct for the fact that there
is noise in the observations and for lack of fit.
y"
L
are avai lable but there are none of the y..
Observations of the
If observations of the
1
Yi were available then (4.2) would be a linear model and the usual least
squares estimators of
I
could be used.
Since
r
specifies a spline this
would give estimate of a spline from functional values and derivative
values.
However, since y(x) = g(y(x); (3)
by g(y(x); (3)
(4.3)
then Y(x ) can be replaced
i
so (4.2) can be written
l~(:;sJ
1--- w
=
1
lW/J
r
+ l~_
83
where E.(I;I3) represenr:s a vector whose i
Now 13 an d I
found.
ro
th
element is given by g(y.;I3).
l.
·
'
' ..
are t h e unknowns
and
estl.mators
th
at ml.nl.mJ.ze
ATA
..!:!
..!:!
can b e
this end write
[
. T
T
~
T
[y Y + g g] - [ylW + g Wi] r
+ [ gg
T + g Tw/ -r _ rTw 'g + -rTw/Tw/r]
[y = WI]T[y -
WI] + [g - W/I]T[g - W/IJ
This is just the discrete version of the Sobolev norm of order one for
the difference of a function y(x) and a cubic spline
W(x)I.
The first
and last members of the above can be written
(4.4)
This E::xpression can now be minimized with respect to 13 and
estimators.
r
to obtain
The restriction that: g(y; (3) be linear in 13 allows for
easy analytic handling but it is not necessary if une is willing to
use non,··linear methods.
Equatbn (4.3) has the usual form of a linear model except that
one of the parameters, is on thE: left-hand side.
13,
This d,)es not keep us
from solving for 13 but the normal equations are not: now useful written
in the usual matrix form such as
84
(::'05)
In the problem at hand
£,
in (4.5) is replaced by
in (4.'»
I
and y is replaced by
contains the unknown
easier tLl attack the expression (4.4) directl.y.
S.
It will prove
Two ways to do this will
be outlined in the next two sections.
:1.0 3-li~
th~~.Line
and Estimating the Parameters Separately
Let Z(x) be a vector of observations at the points x , ... , x
1
n
and
suppose that the least squares cubic spline is derived by one of the
methods described in Sections 2.3 and 3.2.
The estimated value of the
dependent variables and of the derivatives can then be expressed as a
By equations (3.2) and (3.3),
linear function of the observations.
"
T
/I
-,
T
Y = WI = W(W-W) '"Wy
and
wit
These estimates can be substituted into (404) and then the value of ~
.
..,
that
ITll.n:unlzes
ATA
.l:}. ~
d
.
d
can b eerl.ve.
Since
S appears only in the second
term of the right hand side of (4.4), the desired
S
is the one that
minimizes
(4.6)
[ g (z; S) -VI I I
'1'
l-[ g (y; S)
- WI I]
-T-
[g (y ; S) - DyJ [g (y; S)
where
Since [he original differential equati.on was
~ Dol J
85
y
and
£
=
g (y; S)
is a vector uf observations on y, then the expression to be
minimized is just the expression minimized in the linear model
Dy = g(y;S) +
~
Here the usual least squares estimate of
s;l'J:
~
~
-. 'h'IS
WhlC.
.
S is the one that: minimizes
'I
8xacty
t'he quantIty (/6)
~.
.
0
4.4
Fitting the Spline and
,Estimating the Parameter Simultaneously
The approach that was introduced in Section 4.2 gives expression
(4.4) as the quantity to be minimized to obtain an estimator of the
parameter of the linear model (4.3) with the minimization being taken
jointly over
S and I.
(4.7)
To do this the derivatives
and
ha','e to be set equal to zero and sol'Jed for the estimators
As 10::-1g as g (y;
S
"
and
I.
"
S) is linear in S the estimator.s are the solutions of
a linear system.
This method also cvercomes a problem mentioned in Chapter 3.
In
the case of the logistic, when the data are orthogonal, then the endpoints
cannot be estimated by the least squares spline technique.
this Lase the solutions of (4.7) will always exist.
However, in
86
11
ESTEvJAl'ING ~ IN
5.
= ~11 (l -
11)
In this chapter the two methods outlined at the end of Chapter 4
11 = 13110-11).
are used to estimate ~ in the logistic equation
5.1 develops che prL)cedure fO'r estimating
Se...:t:ion
I
separately from
5.2 estimates the two simultaneously.
Section
~
Section
and
5.3 discusses
the possible error structures and their connection with estimating
the error variance and the bias that arises in
~.
The last section
gives several numerical examples.
5.1
e in
Estimation of
] = 6] 0
- 'D)
and Fitting
the Spline Separately
Yn]T be observations on
Le., y.
-
-
1
=
y(x.); i
1
= 1, ... ,
necessarily all distinct.
n.
x.
1
[a,b].
E:
11
at the values xi'
The x. are not
1
Let Si' " . , Sk' k < n determine a
Some or all of the Si may be equal to a x '
j
partition of [a,b].
I}
Let
11.1
x..
.1
be the derivative of the least squares cubic spline for
11
at
This estimate can be substituted into the di.fferential equati.on
to get
Ii
11 1.
(5" 1)
I'::j
Q",.
I-'J 1
(1 -
v.
)
. 1
i
1, ... , n.
The problem is to estimate ~ in (5.1).
As was mentioned in
section 4.2, this can be accomplished by treating (5.1) as a linear
'Ilhjdel and using the usual least squares estimator of S.
g,)8.1 in nd.nd, let: Y =:
i
l1 i +
si where
l1 i
is the true value and si is
a x'and"m error assoc::iated with the observation
y
c
i '(1"'
· .'{ i').
Then.
With that
y ••
., 1.
Define z
i
87
'rj. (l - 'rj. )
+
€. (l - 2'rj. - €.)
11111
c. + e.
1.
C.I
where
= 'rj. (l .- 'rj.)
1
1
and e.
I
€. (l - 2'rj. - €.).
1.
1
1.
With these definitions
1.
(5.1) can be written
I}
scc.
'rj. ~
(5.2)
I
I
+ e.).
1
~
Each 'rj is a linear combination of the Y so (5.2) can be written in
i
matrix form as
s .£. +
Dy ~
S~ .
This equation has the funn of a linear model except that the usual
error term is now
S~.
.
1
....
Invo yes mInImIzIng
Nevertheless, minimizing the error term still
T
~ ~.
Thus fur ubservational data y , ... , y
1
squares (or min Soholev nann) estimator of
(5.3)
"
S=
n
and z ,
1.
the least
S is
T"'1 T
(~~)
~ Dy
'I'
~-Dy
-T
(~ ~)
Unfortunately since z is correlated wit:h y,
S.
S is a biased estimator of
Equation (5.:;;) (see Appendix) can be written
where
88
The expectEd value of
S
cannot be calculated directly because of
the presEnce of quotients of random variables in both the second and
third terms
0
Using a binomal expansion of
can give an approximate value (see Appendix I).
Using 'Jne term of the expansion given in the Appendix, we have
A n n
e(s) = S, - S ( E (1- 2n.)cr~ + E cr~)
o
0 i=l
1
1
i=l 1
n
+ E
i=l
(1- 211.) D.. cr~
1
11 1
where cr~ is the vari anee of y at x ..
1
1
For two terms the expression is in Section 8.1.
The bias in the estimator of
S
is a function of the variance so
that it would be nice if small variances produced small bias.
In fact,
the following theorem shows that in a certain case the variance and
therefore the expected value of the bias go to zero as the sample size
increases.
Theorem 5.1.
Tf
g ,
1
.,,,,,
, Sk determine a mesh of 'L-a,b] with ':>j+1
I='
h V.J and ·1
v , •. 0, ". are observations on
.1 nk
x , ....
1
"
'1"1
'I
at x 1 ,
o.
_I='
':>j
=
0, xn'k with
, xn
and ILk = N (io~., all observations are at equidistant: 'knots and there
are an !;.'qual
Tlurf11H<:'
of observations at each knot and the chosen end
89
cuodi t iuns can be expressed as a linear combi nation of the y.' s then
1
"
13
U)l1verges i
P rOuf
~l
probabi 1 Lt.y
=
T
(~z)
-1
T
~ DY
tL)
Since all the observations are at knots then
0
where Y-. " is the mean of the observations at the i
th
1
knot and y. and
1
Yk
are Ii near combinations of they.' s.
1
Now the estimatur of the derivative is
,"
Dy =W r
in
this
~ase
(see equation (1.1».
Then
pIirn
rr-=
r
T
plim ,(Z T2) '"1 zDIl
0-000
+
(z Tz) ~'l z TDe
l
The term in brackets is a continuous function in z and
theut'ur· (,0eW;Jl<s, p
1 irrr: t.
i .. ~.,1.
and~!.
•
€ and by Slutsky's
12)2: and ~ can be replaced by Lheir p:robabiJity
90
"
plim 13
Care has
[:0
be used in interpreting this theorem"
in pr.::bab:ility to ~
of bias.
of the
a
.
The estimator
However, there are really two types
One can be called stochastic bias and this is the concern
tht;;On:llL
The other type results from r.he fact: that the spline
and the derivatives
L)[
the spline converge to the real values only as
the number of mesh points becomes large,
absence of error, the estimator of
~
~
o
of~.
Hence, even in the
would not be exact.
More is
said about this problem in Section 5.4.
To ge l the val' i ance of the estimator of
expansion as is used to get:
"
e(~)
~
the same kind of
can be applied, however, only the
first: term in the binomial series is used in this work.
The calculations
are in the AppEmdix.
S.2.
Estimating
L
and
~
Simultaneously
As has heen mentioned before, the derivative of the least squares
spline derived in Chapter 3 has no proven best fic pl'operlies of its
own.
Since what is really wanted is the best approximation to both the
fune:t:ioD and t:he derivative, an es timator that would minimize an expression
Like the disl-rete version of the Sobolev norm discussed in Chapter 4
suggests itself.
The fact that the min Soholev norm estimator is
derivable fri)H! a least squares type estimator gives added support to
thi s
j
d ea.
An'Jt:bel: reason fur seeking a new estimator is the fact that the
mEthud discllssed in Section 5.1 does nut give an estimator for the
91
paca..nete.rs in the boundary conditions when the data are orthogonal.
This
prolllc,I) dueS nut exist with the min Sobolev norm estimator in this case.
Let 1 bEj a veeeor of observations on
11
i
at xi'
.<~.,
Y(xi)
= Yi
and let
11 satisfy the differential equation
11 =
1+:::
I
be a
\7<:'([0('
~)f
\{ a 1.unLt:ion of t1-le
W [hus making
w'r
~11 0-
11)
parameters such that:
Xl
sand S's only.
wr
And let W' be the derivative of
the derivative of WI.
I, Wand WI could be just the
quanti t:ies defined for splines in Chapter 3.
(,thee quant,ilies
w'r
t00.
ies derivative.
is a smc'othing functi.on with
However, they could be
For example, WI could represent a polynomial and
II
Then
W'r
is an estimate of
11.
So this can be
expressed as
=
If
I
g(z:S) in the derivation in Section 4.3 is replaced by Sz, then the
fullmvi.ngresults
[]
wheC8 ~ i ~ ,,'
J:l'(1
a linea:'." m,,del in
[J
I+~
v 'I and ~ represents a vector of de'vi at::ions.
.,1i"
Sz
and
I
and the estimat:ors
to be
rd nhrize the sum of squares of the deviations or
weire
T
This is
used are those that
~~..
Tu this end,
92
T
YvJ
Wi
T
I
r
J
- W -I
I
y
I
S?ij
T
W
r
I
i
r
l'
I
_Wi _
1
W I
,I
l'
y
Wi
S~ i
- j
+ l'
l'
VI
T
W -j
[
WI]
W'
'I'
b(~y
'"
T
_~'WI
T T
T T
r w Y + I ·w wI}
rh is las t member is the discrete version of the Sobo1t·v norm mentioned
earlier,
1\
'I\J fi.nd the estimat.ors Sand
dedvatl"\'€S
1\
r
that minimize thi.s, the
dL::.1'MoS and oL::.TL::./or have to be set equal to zero,
o
and
rr
2SW/_~
+
ilt
2Iii
/J
'WT '" 00
93
,.r
-w
(k X k)
IT
---I
I
I
I
Z
(k X 1)
T
r
-w 1:
\
I
-=
,I
-z
T
z z
T .. I
w
IL.
S
(l X 1)
inv~:cse
The.
{w·~w
+
1
W / 'W· / )c· l
~
1) X (k
+
!
J
i
••.
Ck +
I)
.4
1)
of this partit.ioned matrix exi.sts whenever
1
e:x:ists
If
o
1
M
a
(k X k)
(k xl)
lV[
M
M
3
4
(1 X k)
(l X 1)
is the i fl\'f,:<,'se pan:itiooE:d l.n the same manner as the original, then
r-,
[
,
i
-l
M
]v[
J
2
I
l
~I
I
I!
I!
wl~n
I
0
M
'1
I
I
I
J
4
l'
M W.1] •
3
r--·-
94
T ".:.ur, bE' shown (see kao) that
-3TW I (WTw
+ W'TW') ~l
---------
~I'Cw' CwTW + W,TW')W')W/ T ~ I)"~
iI
S-
l.'t)
isis of thE. SclH'" general form as the estimator of
S
given in Section
where here
A.€,ai (\ ttL! s est:Lnate is biased and contains the quotient of two random
,Jar! aIles, thus making calculations of the expected value difficult.
;'OE
wet:hod as was used in Section 5.1 can be used to get an
Sa'l'F
d:pprvxlrnat:e va] lie for the bias and the variance of
5.3
iI
S
(see Appendix),
E:rJ':or SLt'"ctUrf'::s and Estimati.ons of Variances and Bias
II
f:xpet:ted value elf
j""",
S
and the variance of these estimaturs all
dr'" fun,':l,DS (if the variao<.'es of the Lndividual obser;atilJDS,
1'\
w.dE.C [\. caL:,!lace t:he bias in the estl.mates ur che vaci ance of the
esdi'dt. r,
'.'
Thus,
SU.\·
a'
1
dl,'LuSStd
i.e is ne,essary to have infurmati..:;.n on the variances of the
1.5,
lit the fvJl(Jwing, several uses "f i:he IngisUl: are
cwd, in edl.'h ,asl2"
e:cror struct:ures are suggt,:sLed"
95
In t::hi s app I.icat:lon different levels of stimuli. which are represented
by tbt~ independent" variable x are given to groups of subjects.
The
response tJ this stimulus in any single individual is of an all or
n(;thi.ng nat1Jre and T]. is the fraction of individuals receiving stimulus
.1
x,
1
t"hat show the characteristic response.
a:l
Some examples follow.
I'he s t:imulus consists of different levels of an insecticide.
The response of the insect is death and T]. i.s the fraction of
1
insects receiving dose x. that die.
1.
b)
The stimulus is radiation.
The response of the fruit fly
eggs is non-hatching (or non-maturation of larvae).
T]. is
1.
the fraction Df eggs that do not hatch after receiving a
lE::vel of radiation equal to x .•
1.
c)
An interesting example arises when the stimulus is time and
the response is the occurrence of some event such as death
or
~nenarche
(Milicer, 1968).
Tn examples of this type it is assumed that there is a probability
'!l(x. ) that an individual that receives a dose Xi wi.ll show the response.
1
Th:'iS ~
the pl'otabiHty that r. out of T]. subjects will show the response
1.
' n.
\I .1
,J
:L
1
\
n. - r.
1
..,
1.
hi
The
t'stimai.:(~
and thi s is
of
'y",.
1
'n (x.)
1.
is taken as the percentage actually observed
The variance of Yi is
96
T)(x.) (1- T)(x.)
~
~
n.
~
y.(1-y.)
~
~
n.
~
'.. :I(,.s, in this ,case the. E:xperimental design is
SUdl
that an estimate
<If the "ariance is automatically furnished.
In many instances in bioassay when the response is not qualitative
b'lt: quanti tative,
sigmoid
etC)
,~urve
the response as a function of the stimuli follow a
(someti.mes a transformation is needed).
eXj}E:,dment analyzed by Finney (1964)
bubsl'.al1:e is s;:Jught.
For instance, in
the vitcnnin D content of a
The substance is fed to rats and after a period
the rats are killed and a characteristic bone is analyzed to determine
che antirach:U:ic properties of the substance.
The bones are given
one i',f thirteen ratings which can be called 0, 1, ... , 12.
d·;.v:l.sil,'·, by 1'2 these lie between U and one.
On
When the responses are
p i.e ::t:8d against !:he logirithm of the dose a sigmoid (;urve results.
Ie th2 e:-:W:i-JE::riynent cited here, there are at least
i ", E:a~h
t.h e
Se"VeIl
subjects
l::r2a':ntc:!Jr: group so that it is possible to get an estimate of
·'i ar.iafi,~·e
at. each puinL
Finney (1964.) also suggests that iin
sie:,n,id dat:a the 'variance uf the depe:ldent: variable might folL)w a
tjuad~Cdc:i.L 1.~el.atiui.1., i.~.,
97
If t"his f s assumed an analysis of variance on the data after an
<:i'c::::s in
tansfocmation
t
:a
will pruvide an estimate of 00'
thus giving
eS"imatEs uf the individual varianceso
Anothic''t' pllssibilir,y is to assume homoscedast.icityo
l:(J
This proves
be a r"asu"'at,Ie assumption if the dat.a is not far from the point of
infle,:!'1on in the sigmoid curve.
val'ian~e
Indeed if a quadxatic furm of the
is assdmed then there is very little change i.n the variances
over the interval from u
5" 3.,3
Autuc..atalysis I Enzyme Reaction and Haemolysis
Thest' are a few examples of the application uf the equation in
such pi aces as biochemistry and irrrrnunologyo
produce 1.s
tOf'Tp,ed
In autucatalysis, a
f:com a reaction and then acts as a catalyst
0
The
rate of the reaction is proportional to the product of the densities
of the OJ'lginal subst.anl:e and the product.
to one
0
These two densities add
Thu,s, if ui s the density of the pt'oduct and 1., u is the
density lIf the original substance,
u = ~u(l-u).
In
I.er[~al n
enzym.E: reactions such as the hydrol.ysis of succose by
In',,'8r'.:a6t:', t'h€: rate. ,:,f (he reaction is pruportional t'o t:hE cuncentration,
again l't:,su.'i[in,g
:U~,
the l.:Jgistic equatiun as a modeL
Finaliy the
pEt',enr:etge uf ,,,11 slysed ty a C'oncentrat1.UD vf haem.olysis follows the
All ,:,f these
F~anrples
result i.n extremely al:cu:care fitso
resulting errur ':,iarianCE,s are quite small and thus
rangE:. f.r:u;n
11=
0'3 i:u u= 0'/
I
eSIJ'E::('icdJ~,,-
The
over the
there is very Ii,tele diffe:rem.t=:, whether
98
homoscedastL:ity is assumed or whether the variances are allowed to fall
off toward the asymtotes.
5.4
Results of Monte Carlo Simulation of Data
The cwo methods of estimati.ng
tesred by l'1on1:e Carlo simulations.
S presented so far in Chapter 5 were
Preliminary tests showed that the
second method of simultaneously estimating
unacceptable when
I
S
and
I
were completely
represented a cubic spline function.
Howe\;'er, the results of the test of the sequential method are
given i.n Table 5.1.
Preliminary runs with no error i.n the data to
determine the spacing of the knots necessary to give acceptable results
were made.
It was found that with knots at -1.0, 0.0) and 1.0 and
measurements taken at -1.0, .5, 0.0, .5 and 1.0, the resulting estimate
of
S was
exact for
S
= 1.0, and about five per cent off for
Thus runs wi th er:cor were made for
S
= 1.0 and 1.5.
S
= 2.0.
The resu 1 ts are in
Table 5.1.
In addition to the value of
S
there are other sources of variability.
In order to assess the effects of some of these the inflections point
was moved tLl the left and right of the knot in the center.
seems robust with respect to this possible source of t:couble.
error variances were used, '12
with
(j2
=
. CJ2,
=
.01 and
(12
=
.015.
The model
Two
Runs were attempted
however, the resulting data were so variable that they
were no longer monotonic thus precluding any reasonable estimate of
100
6.
S
ESTIMATING
IN d (~:-:Il2 =
S(1
- 'n)
Tn thi.s chapter, t:he logistic differential equation is put into
a different form and this form is used to estimate S.
6.1
A Different Form of the Logistic Differential Equation
The I.ngistic equatiun is defined to be
(6.1)
T f the in.it-La1 (ooditiuo is such that any part of the curve lies between
1l= ()
andTl
= 1,
then the whu1e solution curve lies withi.n these limits.
In addition as x ..... co,
1l -., 1
on the value 0 and 6. 1
\'~an
~
1l < 1,
-7
O.
Thus, u never takes
be written
(6.2)
Now since 0 <
1l
and as x .... -co,
S(l - 1l) .
=
..enCn) is always defined and
d(..en 1])
dt
= 1:
1l
11
1l
E.1]
dt
Substituting this into (6.2) gives
(6.3)
(..en
1l)
S(l - 1l) .
This is the versi0n of the logistic that will be of concern in this
chap ter.
Tn the usual case, obsen,-ations are made on Tj at times x. giving
1.
the \lEctor of observations v
"'-
So
are the -en(y ) '" £i.
i
~hat
= "1
v ,
... , y ).
n
Since the y" are available
1
1£ (6.3) is to be made into a linear model,
is needed is the derivative of In(Tj) at each x.'
1
But the methods
already used to estimate Tj. from.y can be used to estimate ..e. from ..e.
1
1
101
6.2
The Variance of ..enen)
The method t<J be used in this chapter to estimate
a
bias~d
estimater.
of ..en(y).
S will
result in
The bias will be a function of the error structure
For perfectly general error structure tn(y} will not exist
but a gaud approximati0o over the range of interest hE:re can be obtained.
In cases «(,nsidert::d here negative y will not occur in practice
Let y
~
~
+
0
B where n is the true value and y is the observed value
and € is a random error with expectation zero.
..en(y) = ..en(n + €)
=
..enn + ..en(l +
Then
tnT](l+ sin)
e/n) .
Taking the first term of the Taylor's series fur tn(l+
e/n), the
following approximation results:
The approximate variance would be
where
O'
2
'T] is the error variance of the observations y whose true
value is z.
'I'he other important quantit.y needed in the calculation
of che bias is
CO'i1
(y, tn(y)
Here
..e
:=
..eny and A
:=
..enn.
A Mvnte Cad,o t.est of che approximation was cundueted.
resui ts are given iq Table 6.1.
apprLYximatiun
Ser",i(-;S
qui te we 11.
The
Over the area of interests, the
Table 6.1
Results of Monte Carlo Test of the Approximations to Variance of log(y).
Given in Parenthesis.
SD = .01
Var= .0001
SD = . 05
Var
= • 0025
Estimated Values Arc
SD = . 075
SD = .01
SD = .05
SD=.CH
Var= .0056
Var = .0001
Val' = .025
VCi:C"'" .0001
SD=.05
SD"'.075
Var=. 0025 'v- ar=. 0056
.0099
.38
.82
.0009
.023
.058
.0001
.0024-
.0055
(.0010)
(.25)
(.56)
(.0009)
(.022)
( .051)
(.0001)
(.0025)
(.0056)
.0024
.0058
y = .1
~.
.0026
.075
.23
.0004
.010
.02.3
(.0025)
(.06.3)
(.14)
(.0004)
(.099)
(.022)
(.00001)
(.0025)
(.0056 )
.0011
.030
.077
.0002
.0059
.014
.00010
.0025
.0056
(.0011)
(.028)
(,062)
(,0002,)
(.0058)
(.013)
(.0025)
(.0056)
.00061
.016
.0.38
.0001
.0037
.0085
.0025
.0056
:.00063)
(.016 )
(.035)
(.0002)
(,0038)
( .O08~)
(.0025)
{.0056)
.00042
.011
.024
.0001
.0027
.0059
.0024
.0060
(.00040)
r. 010)
".022)
COO01)
(. 002.5)
(.0056 )
(. Oe25)
(. 0056)
.000097
= .2
v::: .3
Y='o
y
=
(.0001)
.000097
~
,,5
(.0001)
.00010
(.0001)
~_t,
'.~-'
1'-'
Table 6.1
Continued
SD
= .01
Var == .0001
SD = .05
Var= .0025
SD
lIar
= .075
= .0056
SD == .01
SD
= .05
Var = .0001
Var==.02S
SD== .01
\7 ar
=
<
0001
.00027
.0071
.017
.00006
.0017
.0039
(.00028)
(.0069)
(.016)
(.OOOO?)
(.0017)
( .0037)
.0020
.0053
.012
.00004
.0011
.0025
(.00020)
(. 0051)
(. all)
(.00004)
(.0011)
(.0024)
.00016
.0042
.0089
.000026
.00066
.0014
(.00016)
(.0039)
(.0087)
(.000025)
(.00062)
(. 0014)
(.0001)
.00012
,0030
.0071
.OOOOll
.00027
.00068
. 0001,J
(.00012)
(',0031)
(.0069)
(.000011)
(.OO028)
(.00062)
.OOOll
SD=.05
SD=.075
Var;.0025 Var~.OOSb
.0025
.0056
(.0025)
(.0056)
.0025
.0056
(.0025)
(.0056)
0025
.0059
y == .6
(. 0001)
.000099
y == ,7
(.0001)
.000098
y == .8
y
(.0025)
(.0056)
.0025
.0037
(.0025)
(,0056)
= .9
(. (001)
.-'
C
104
6.>3
F':1t:Ung the Spline and
d (.tn]) -
dx
Let V
-
.t
=
=
-
~ (l - 11)
Estimat:ing~
Separately
(y , " ' , y ) be a vector of observations and let
n
1
.t ) be the positions of the knots and let r be the
(.t , ... ,
1
1
unknown vector of parameters specifying the spline.
spline is the une spe,cified by
r
Then the desired
in t.he model
-.t
wr-
~
with the matrix W being the same as the matrix constructed in Section
3.2.
Letting
r"
be the estimate obtained as the LS estimator in the
pree-eeding equation and letting Wi be the derivative of W (as before),
then
i.s an estimator of the dHivative of .tn(l1) at the points x..
1.
Using
(6,3), i t follows that: approximately
D.t
=
~(l-11)
l'he least squares estimator of
~
(6.4 )
where
~
and
1:
=
~
=
~C
-
~€,
say.
is
T ~l T
(~~) ~ D.&
are observations on
a biased E,sCimator.
~.£
-
C and
A,
As in Chapter 5, this is
However, the structu:re of the bias is some what
simpler due to the fact that z in (6.4) is linear in the error and,
the covaxiance between .z and .t is simpler than it was between corre-
sponding quantities in Section 5.1.
Appendix.
For the
exp€c~ted
value of
II
~
see
105
6.4
F} t!ing the Spline and Estimating
S Simultaneously
As in the manner first introduced in Section 4.4, the parameters
of the spline can be Estimated simultaneously with
lee t be the vector uf logs of the observations.
such that
wr
is the an approximation to
1:.
1:..
WI and, therE;fore, an approximation to
and
w/r
S.
Let
To this end,
r,
Wand Wi be
the derivative of
Then, as in Section 4.4, the
model is
,
For "observations"
where z.
~
1 - y..
1
1:.,
e
:,j
S~
will be used since
r
Then,
t
e
Ri
S~
IW I
I
II
I
I
I
I
I Wi II
L J
or
As in Section 4·,4 and 5.2,
J:/~
is a discrete version of the
Sobolev norm
and solving the resulting normal equations gives
106
Again, this is biased and the approximate expected value given by the
now familiar method is in the Appendix. The variance of
6.5
1\
S
is in Appendix.
Error Structure and Estimations of Bias
and Variance
Like the estimators derived in Chapter 5, the estimates given here
are biased and the bias and variance of the estimators depend on the
error structure of che variable y.
In this chapter though, the error
structure of 1- is of moderate concern but as was shown in Sect.ion 6.1,
the structure of
~
is determined by that of
y.
In Section 5.3, in the discussion of uses of the logistic essentially
two error structures for y were mentioned.
and the other was the quadratic.
variance bet.ween
1:.
One was the constant variance
Therefore, the variance of
and 1. will be determined for the two cases
and
Case I
In Section 6.1, it was shown that
~
and co-
107
Thus,
The covariance Cov(£',z) is needed in the calculationso
z
-'='
l'~y,
Cavce, z)
-Cov(.t,y)
0
Then
-Cov(£"y)
= -e(.t,u) + e(£)e(y)
-e[ (A + ~)(T] + e) ] + AT]
2
e
= -e(··-)
T]
since e(e)
= 0 and e(A) = A and eCn) = T]o
But
so
Cov(£,z)
Case II
Since
then
=
For the covariance,
Cov(£,z)
0'2 (l ~
-.::..0
T])
_
Recalling that
108
Unfortunately, the variance
is unbounded as i) ..... 0 in Case II.
However, in practice i) is never zero but nearly always between .1 and
.9.
Over this range, the variance is finite.
6.6
Results of Monte Carlo Simulation of Data
Exactly the same designs as were described in Section 5.4 were
simulated to test the model.
The averages of the esti.mates remain quite
good, but the variance of the estimator does seem high.
It seems to
increase with the error variance and is probably due to a breakdown
in the estimate
1n(y) - 1n(i) + eli) .
109
Table 602
Munte Carlo Values of
II
a and
6.2 with Inflection at x
s=
0'2
.01
II
a/a = -.1
1.01
a
V(~)
0044
II
a/a
a
=
0
II
V(a)
1\
S
a/a
=
.1
=
.986
.060
= 1.01
II
V(S)
Variance of
a for
Model of Section
= a/a.
1.0
S = 1.5
0'2 =
II
a
v(~)
II
S
= 1.02
=
0'2
II
a
v(~)
II
a
II
II
S
.066
= 1.02
.067
V(a)
II
.015
= 1.02
V(S) =
V(S)
II
S
001
1.53
.073
= 1.53
.067
V(a)
0'2 =
II
s
v(~)
II
S
.015
= 1.55
=
.110
1.55
II
.072
= 1. 53
V(a)
II
S
.110
= 1.54
II
II
II
.044
II
.071
V(S)
.109
110
7.
SUMMARY AND OVE:RVIEW OF OPEN
Four meth.ods of estimating the parameter
PROBLE.-~S
S
in the logistic equation
I) = SI)(l - I))
or equivdlently
dUn ])
dx
have been described.
The two methods that fitted a cubic spline to
observations on the dependent variable then differentiated the spline
tD obtain an estimate of the deri.vative proved satisfactory in Monte
Carlo tests.
The two methods t.hat involved minimizi.ng a discrete
version of a Sobolev norm and simultaneously estimating the parameters
of the cubic spline and
S
proved completely unacceptable.
This is
surprising in light of the success that Bell and van der Vaart (1973)
enjoyed using piecewise polynomials with fewer continuity conditions
at the knots
and a similar differential equation.
Preliminary
computer runs show that the method they used gives results that are
just as poor as using the method described here.
Work should be undertaken to determine the cause of the poor
performance of the simultaneous method.
It is possible this method
only works well on certain class of differential equations.
I f the methods discussed here are to be useful in a practical
se.nse, a way of dealing with the bias and with the variance of the
Estimator must be discovered.
The approx.imate estimates gi.ven here
are diffit:ult to compute and did not prove sat.isfactory in the Monte
Carlo tEsts often giving negative values for r.he variance"
111
The method could be used to estimate the parameters in the VolterraLotka equation.
These equations can be written
X
Q'x2 +Sxy
y_yy2+ pxy
These can be modified in a manner similar to that in Chapter 6 to
give
(.tnx) = Q'x+ S Y
( .tny )
=
y y +Px •
l'he method of Chapter 6 could be applied to these equations.
112
8.
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Charles
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App lied Mathematics 4: ,49-99.
114
Sch,,,U:z, He
19>H.L '.l'hs Stcmdard ErriH' uf a FurE:2asr from a CU:r.'"'ve.
J. ArneI'. StaU,sUcal Assoc" 52:.567~5n.
S6~r~SL,
D.
1965.
Error Bounds Ear Interpolation and Differentiation
SIAM Series B, Numerical Analysis
by the tse of Spline Functions.
2. ' !.; ~U -~,!:, 7 •
TS~Jkus,
C" Po, and S. W. Hinkley.
1973.
A Sr:~)\.:hastic Bivariate Ecology
ModeJ fur (;'Jmp2cing Species 16: 191~208.
\lerhulst, P. F
18~5
de la p.lpdlariDoo
0
0
Recherckes Mathematiques sur la 10i d'accrocssemtn
Academic de Brurelles 18: 1,,,'38.
Wadswurth, S. 1%7. Standard Methods.
Baltimore, Maryland.
Williarns and Wilkins Co.,
\,rhitE, A., Phi Up Handler, and E. L. Smith.
Bio(;hemi sery
McGraw,"Hill, New York.
1970.
Pr.inciples of
0
Wilks, So S.
1962.
New Yurk"
Weld, Svanteo
16: 1<~11.
1974.
Mathematical Statistics.
John Wiley and Sons,
SpLine Functions in Data Analysis.
Technametrics
115
9.
APPENDIX
l'hl'uughdU t: the following derivatiuns the followi ng t.wo rules will
be uSE.d:
[{,lJ
Ie 1:
If
and e, arE' normal random variables tbat are uncorro1ated
0:0
J
1
unless i
and if they have means of 0 and variances of (j~ and (j~ then
1
J
'" j
where
6.
f
0
1J
i = j
1
~, 0
"
i
This result is obvious.
Rule 2:
If
€
,
€n are normal random variables with mean 0 and
, •• ,
1
. " c,
variances 0'2,
1
j
0'2
n
and e. is independent of
= g.
unless e.
g,
1
J
J
1
then
if n is odd
e(e
1
... e)
n
=0 .
A long case-ex.haust.:i.ve proof is a-va:Uable but a short: argument will
give the rul", for n =:3 (it is obvious for n = 1) .
e.
Consider
€
,
1
e ,
2
If they are different then by independence
3
e(s e e: )
1
If €
,- €
1
Z
f €
1
1f
e:= e:
1
:2
.,
0:
3
1
2
3
e(e: e: ) e(e: )
then e (E:
€
. 1
1.
:2
3
:2
3
eC€e:e:)
o .
eCe ) e(e: ) e(e )
:i.8 independent: of e:
t:hE:o €
3
3
:2
3
and €
1
so
2
e (e.,
.l
€_ )
0= 0
.,
e: ) in the third central ITlL"lmen t
a 3
random veft'iabLe with mean O.
That is O.
0
f a normal
116
9.1
The Expected Value of
1\
~
1\
The express j ons for (3 in Sect:ion 5.1 and .5.2 both have the form
y
and
'=
]+ £
with
be normal and have variance O'~.
€.
1
Then
1
Z.= V. (l
1"1
-, y .) = T]. (l - T] .)
1
1
1
+ €.1 (l
.. 2T]. - €.) .
1
1
This can be written as a vector
where
CL
= T]. (l ~ T].) and e. = €.1 (1 - 2'l'l.
- €.).
ILL
1,
1
1
l'he e:Kpt'ession for
1
1\
S
O'~.
Thus e(e.)
1
1
can now be written
(.h + ~) 'rAe] + of)
(£. + ~) TB (£ + ~)
If
(£T A ])
--~-
is added and subtral:.ted and the numerator expanded this gives
(h TB .£)
13
s;,TA
,n
.~TB
s;,
U'. is stl]} impossible to obtain an expression fur
presence of quot.ients of random vari abIes.
1\
e (13)
because of the
Hut-Ievet' , an approximation
that wiLt give products of random variables can be used.
Note
11'7
1
1
e
+e
T
Ii 8
1
\\,'tlt-.re rhe rW0 middle terms can be combined since Eo is symetric .in both
(,dBc' 6
Now this last expression can be written as
T
2r B e
(£'['B -') _.{
... 1 + \;, ,-
T -1
+ -e B -e}
«(rBQ
whf rte· the approximation is gained by using the first two terms of the
bi llGmial expansion
f or
0
Now this can be used to get an approximate expression
/')
e \S),
Lose.r:tlng the approximation into the expression for
S gives
CAn
~Tb f.
!'h,,~
sF.o,'id and::.hird
indj' idual
at a
ri,TIt_
terms are stochastic and thesf:':v'1t:ain f:i fteen
addi tLve e"X:p:t'essionso
All s':.nnrnations will be
The e.xpectari.:..qs wiU b", derived one
OVtoet'
all indi:i2,o p"E;;oenL
118
\u:Tlt:
';1'0
rat:()!:' of the first two terms is
! [I [hi: st-(ond term in brackets the product of 1 and the second
roC un
1n
Sq'ld('t'
bracket.s gives
R
EA. " e(e: . e: . (l - 211.) - e:~ e:.) - I: cr~ A.,~ C + EA. • (l ~ 211.) cr~
1
l\:dt:~
w'hHt2
J
1.
J
~
1.
J
1
.1. j
J
1 1.
1
1
1 and '2 have been used on the lase summation,
['1 the first curly brackEts the prodU'.:t of the firsT term in
sqJar~
t',Cd'
,rhi,; hct-,
r
ktt's and the second term in the second squdretcackets give
~.pt: .LEd
value
119
In the first curly brackets there are two terms of the form
Their sum expectation is
(l ~. 21l'k) <~ (l - 211 0) (l ,- 2T].) E:
J .k
1,
+
Y1
(1-2'Y1.,)e;0E:
III
Jk' - (l-2 II J.)(l-2'Y1'k)E:o
'I
1
+
(l - 2T]k) E: . E:. - E:. E: 0E:'k) ]]
1 Jl J
All terms have 4, 5 or 6 E:' s.
O.
- (I - 2,110) (l - 2T]k) E:o
J
1,
+ (1-2'Y1·)e:·E:
II
J 1 k
By Rule 2 those with 5 have expectation
The other four are
e (E: i E: j E:~)
e(E:lE:jE: k ) = O. 0'20'~
ill
61jO'22
10' j
e( E:~ E:JE:~)
:3
-
eCE:iejE: k )
2
°ijO'iO'k
Jm
J
222
O'iO'jO'k
Thus
The product of the last: terms in square brackets is
The expected value is
R
5
= E B o oBkme[ e. E:, E:'kE:
1J
1
J .
ill
0- 2T] • ., eo )(1- 2T]. ,. E: ,)(1- 2T]k ~ e;'k) (l
1
1
J
J
'
,c,
21\ ., E: )].
m
m
120
:'h,;: J. il.dt.L.', 10 square brackets gives 16 tenos of IAr1;,,:h eight: Ldnt:aLn an
udd nUillh,r .'. f ;:;; s.,
b,,' Rule 2 these a:r12 zero
/
I'he rE:'[fidlnln,g eight: are
0
J, k - m
i
or
k,
ifl
\ i
o
;Z)
e
:3)
e( E; " g~ 0;.,
~)
e( g, is,2€k€
2
E;i €j
2
)
cL CT~ O'~ CT~
1,k 1, J 1,
)
'
fL (J~(J~0':3
€j€k€;n}
'2-
6 "kCT.O'.O'
;::):3:3
6)
e (€ 1. €~J €~K E:!1i, )
6.I,m0'.0'
:1 J'O'k
J)
e( €~ €~ €
6
1
1
'))
ef €i2
1.
S",
8
8
,I K m
.J ,'m
J k
lIO 1
;::)
)
E;
:-
se g i ';e
:\ )
L.
f'
F
.',
I j " ."
KI"
0'2
rn
;::)
;::)
:3;::)
2
kr
0'.0' 'O'k
. TIL
J '
III
e( E:~1 E?J €~k •.2 )
~
J
JJ1m
ill
I'hl;:
;::)
°ijO'iO'kO'rn
22
," 0' ]"(J"k,
1
0'~0'~0'20'2
:l
J k m
In
j
k
or
=
\)W
;::) 2
8)
E;k€m'
j
In,
121
6
f'
<:"['
'-
1;
'
(J8(J~(J;)
",.
,,,.I
1
.1,
'k',
7
.,
;'h e 8\11[\"f
ch""", ;;;;;Igh '
fe,ems givps (hE: ,,<,Kp,c,_t:l-,d
,:)f E
'CdIUt"
'1'
2{:,-l<
i;;
and
~"A ~
has \.',I\:J't:~t:ed ·i?dlue.
() t'
... ,
k
E' J. A
L'L: C,
i
vl,"'m
E,.. A
:I"
''<'Ii
eO.,. 8
H',' I{"
1."
~
,'"
TI
'jj
t'
,,,"b
~,.
122
R
The pruduct of the second tfrm in the fir" t bracket and the third
in the second gives
Its expectation is
By Rule 2 the first and last term in brackets have zero expectation.
The remainder
g~ves
or
The product of the third term in the first brackets and the first:
term in the second brackets give
'1'.
~-b ,.§.
,k
T
A ,~
Its expectation is
Again t:here are terms with 3, 4 or 5 ens.
l'hos.:: with 4 give
123
/ t:htc second
'1'
e
~,"'B
,A.
,
,,,
j
~
I
..
!
"Ii
1
.
"
"
t~ L E: • ~ ,
)'
I
12. "
J k
4>
"
'. r,
'I,
(1 . LTi1, " '" I' 'J (1 ~.
.)
eli \d (:
rp
S
H
,,
..
"
c;;
.. r
<•
<..
;
(.,
y:
- 1;-
'l.
.i I..
.'"
Ip
e"}) €, e A 8
h.
'~I, h .. A.
I <'
J.
I,
f~l t:;
.
~,
e ,8)
e.. (1 - 21l,o1 " ;';,) (1. ]
.:<. ,I,
or
c::"~ :
F.'
r
p., A
•
~i I .
J,
"
',.'1
:,'
('1
A..
;'11.'..1.
A,
\, 2 2 G
,n en,'L )(J.C1.(JL.
.1
'
1'1
.~.
,K,
A
.
1\
h
(":
".'
", <'
-
:'~
~ >cd
"1
1\
, 'J
8,'
!< •
•
124
Thus the approximate expected value of
"S is
1
+
9.2
I'
£B",
Approximate Expected Variance
The formula
will be used to get an expression for the variance of
expression is already available for
the formula for
"Sa
e (S).
"
"S.
An approximate
Because of the complexity of
it will be necessary to resort to the approximation
1
1
---------- = - - -
('" + ~) TB ('" +~)
",TB '"
This corresponds to using only the first term of the binomial expansion
used in Section 8.1 or equivalently assuming the errors in'" can be
ignored.
Thus
Using the approximation this becomes
125
S2
~
2
__1 __
(1:l'B ,b)2
+
T
T
T
2~ A ](1: A.§. + ~ A.§.)
There are ten terms in the curly brackets taking their expectations one
at a time as was done in Section 8.1.
T
1
T
2
= e (1:TA
])2
=e(~TA]
=
(1:TA ])2
T]ji\nAij~e[€i€k(l-
!;;,TA ]) = E
2T]i
~
€i)(l- 2T]k - €k)]
By Rule 2, this is
T
=E T].'T1 A.. A. (1- 2T].)20'~+E T].T1 A.. A 0'~O'k2
T
= e (1:TA€- 1:TA -€)
2
3
J'm 1J 1m
1.
1.
J'm 1.J-Km
e(e J. em)
= E C. CkA .. A_
1.
l.J-Km
1.
= E C. C A. . A O'~
1.'.
k 1.J kln J
u
T
T =e(e Aeel'Ae)=e[E e.e.e € A.. A 0-2T].-€.)(l-2.T]'k-ek)]
4
-1. J k m 1.J-Km
1.
1
There are ter'ms with 4,5 and 6 els and Rule 2 gives
T = E A.. A
l.l.--k k
4
(l- 2T].)(1 - 2T]k)0'~0'2k+ E A~ . (1 - 2T]. )20'~0'~
1.
'.
1,
1.J
1
1. J
+ E A.. A.. (1 - 2T].) 0 - 2T] .)O'~O'~ - 2E A~. (1- 2n. )20':
1J
+E
J1.
1.
222
A1J
.• --kJ
A .0' 1..(1'O'k
•
]
]
1
J
1.1
1.
1
126
where the fi rs t: four terms come from the 4, e term and the last from the
6 e term.
The fifr.h thru seventh terms have one stochas tic factor
- I: T] A.. O'~
0
J
l'
T -=e(e-A e:)
'7
'--
-;
=
lJ
1
_
I: A.. eLe.e.(1- 2'Y1. - e.)]
lJ
1
Jill
1,
I:(1- 2'Y1.)A.. a~ .
111
11
1
The eighth term has expectation
I: C.T].A .. A_ (1- 2T].)a~
J J 1J-~
1 1
The ninth term gives
'1'9 =I:T1.A
.. A. e[e.eke
II
1
m(1-2'Y1.~e.)(1-·2'Y1k-ek)]
III
1
'I
'
J lJ~
Using Rule 2, this is
The tenth is
Thus
1
(£TA
]) a
J
(T +T +T +2-'A](1'_ +T +1' )+T +1' +1')
:<3
3
4
56?
8
9
10
.
127
9.3
1\
The Expectation of S in the Logarithmic Models
S obtained
The estimators of
where z.~ = l-y.~
and
£;~ =£n(y{).
~
in Chapter 6 are of the form
Let e.1 = e./,n
..
~ II~
Then using the first
two terms of the binomial expansion of
1
(£ +
T
e) B (£+ e )
gives
The first curly brackets contain five stochastic terms.
first two sections the expectations follow
T
a
T
= e (e
B
-
..§.)
=
E B .. (j~
~~
~
E C. CkB .. Bk .(j~
~
1
J
J
J
4E C.B
.. Bkrne(e.eke
)
~
~J.
J
m
o
As in the
128
~
T .
5
e(.€ IE
~ B o'''k".a
'S
:3 a + t.~ Ba•• a.a.
:3:3 + t.~ B •• B . . a.a.:3:3 2~
B:3•• C1.4
e:),z_-t.t.11 k:
1. k
1J 1 J
1.J J 1 1. J
J.J. ].
-
-
In the second curly brackets there are nine stochastic terms
T
6
=e(~TA~) =
T =
9
,T
e (2. b
B ~
0
'1'
£.
A -'=-)
2E
T
T
= e(e:- TB e: .h A e) =
T
=e(e:
B
-
12
13
T
€
e:
--
T
A,O
=
. . A . C1~ / T] .
Co1 'kBIJ-K]
J
]
0
a
+ E(B
.. A• . C1~C1~/'no) - 2E(B .. A.• C1~/T].) •
, 1J JJ. 1 J
J
11 11 1
1
Thus
(:;T A -A [ -1-'1
.-,1'
(T
.hB~_
~B.k_
1
+T
a
) -
T1
fB~
(1
+T +T
J
)
345
+---1
[ T +T +T - _1_ (T + T + T + T + T + T
}'
6
7
8
.£~.£
9
~
II
12
13
K
.('B ~
)J
129
1\
9, !'.,:.'tJe Variance of 8 in the Logarithmic Models
As in SEc!:iun 9.2., an approximate expression for the variance of
can be obtained by using an approximation to
1
(-,-+-~-);:;;-TB-(r + ~)
e(~2).
Assuming
1
=
.hIE .h
rhen
+
T.
T
2 (£ A~) (~ A~) .
There are nine stochastic terms
L;
A. A A.. A. a~
J m 1J 1m 1
+ L;(AijAjialaj)/llillj - ZL; A~ial/lll
T ::;
4
T5
e (2.£TA .-A§.T A ,2)
= e (2'"
·.k
T'
T
A _.A ,"
~ A .~)
-
::; 0
o
2L;
C.A.A
.. A. ka 2k,/ll k
1 J 1.Juk.
e
130
o
!~h' _ ~,~:
+
-~Cr,TA1 }y~
(or + T + T + T + T + T
1
:a
3
11
Bin\Jmial Approximation
th\:.'o
S"",'hClst:ic terms
L cr~B .. C.
=e
T
1
2
l'
·=e'c~.
3
A ,,~
,:= e I~ e -A
h
l'
6
+ T + l' + T
'l
. ~n,:._E,x:2.§'ct:ed Value of 8 with One T2r:m in the
'-1",
.~'herb'
45
(.
-
"
=e(e'A~
D
,~,:~.
...
n
1.1 J
8
9
~lJ