AMER. ZOOL., 21:831-844 (1981)
Predicting Populations1
GEORGE OSTER
Department of Entomology, University of California, Berkeley, California 94720
SYNOPSIS. Deterministic models are valuable for gaining insight into the mechanisms of
population dynamics. However, they are not generally useful for the practical problem of
estimating real populations. Herein we present a method for estimating the birth and death
rates as well as the age structure for a population using only "aggregated" measurements
such as total adult deaths and birthrates. The method applies to nonstationary populations with nonlinear birth and deathrates.
INTRODUCTION
or organize the data—for a specific experiment or data set. Examples of GT fill the
pages of the mathematical biology journals. Thousands of papers have been published on competition and predation theory, most of which are simply devoted to
examining the mathematical properties of
a set of differential equations that purport
to model interactions between real populations. Remarkably, for a long time most
ecologists took these equations quite seriously—as if there were hidden in them
some great, but subtle, truth about nature.
What was lost in the proliferation of paper
was that the subtlety was mostly mathematical, and the truth they contained
mostly allegorical. Indeed, one of the most
pernicious consequences of this flood of
theorizing-sans-data was that it lent an
aura of respectibility to allegorical models.
This trend ultimately led to such excesses
as Catastrophe Theory, the ultimate allegorical model: rich in mathematical subtlety and virtually devoid of empirical content! (I recently overheard one eminent
biologist snipe that catastrophe theory
turned out to be about as useful to biologists as the Book of Genesis is to geologists!)
Not that theories must always be used
exclusively to fit curves to data. Sometimes
a theory does not so much explain a surprising observation as to change our view
of what should be surprising. For example,
Robert May's analysis of the ubiquity of
certain log-normal species-abundance distributions: It turned out on closer examination that it would be surprising if they
1
From the Symposium on Theoretical Ecology pre- were not log-normal. Thus, among other
sented at the Annual Meeting of the American Society of Zoologists, 27-30 December 1980, at Seattle, things, theories can protect the empiricist
from fortuitous numerology.
Washington.
Beginning around the time of Robert
MacArthur there began such a population
explosion of theorists that it seemed for a
while as if no biology department was complete without its resident theoretical ecologist. Recently, however, the market seems
to have crashed, and biology departments
no longer feel incomplete without an indigenous theoretician. The reason for this
bearishness on theory and theoreticians is
easy to see. It is not that theoretical ecology
has not produced anything of value, but
rather that it promised too much. Everyone, it seemed, developed inflated hopes
of quantifying ecology; with hard work we
would eventually understand ecology like
physicists understand physics! If theorists
had been a bit more modest in their
claims—and experimentalists had been a
bit more cynical about the powers of mathematics—perhaps there would be no need
for this symposium on the role of theory
in ecology.
For the purposes of my discussion here
I would like to make a distinction between
two kinds of theory. At one extreme there
is what I will call GENERAL THEORY
(GT). These are models which address
some general phenomenon, but are not
designed to explain any particular set of
data. At the other extreme is SPECIAL
THEORY (ST)—or, as-it is called in physics, "in-house" theory. These models are
designed to answer a particular question—
831
GEORGE OSTER
832
1 0000
0.
. 50 . . 100. . 150. . 200 . . 250. . . 300. . 350. . 400
ADULT COMPETITION FOR 0.4 GRAMS OF LIVER PER DAY
NICHOLSON EXPERIMENT I
3000
2000
1000-
UJ
Q
ID
I—
H
100 -
50.20
10
5
4
PERIOD CDAYS>
3
POWER SPECTRUM DENSITY OF NICHOLSON'S EXP.
I
FIG. 1. a. Nicholson's "L97 cage I" blowfly experiment (cf., Nicholson, 1960) showing the population dynamics
resulting from competition between adult flies for limited resources, b. Fourier analysis of la showing the
amounts of each harmonic frequency present in the data. The major frequency near 30 days is evident, but
the abundance of other frequencies attests to the complexity of the dynamics (Brillinger et al, 1980).
I plead as guilty as anyone in yielding to
the seduction of GT and mathematical elegance. But I perceive a recent trend
back to ST models; it is probably time for
theorists to return to the less glamourous
service of particular experiments. General
principles of ecology will suggest them-
selves, I believe, only after many examples
and special cases have been worked out.
These remarks serve my purposes in two
ways: they have fulfilled my obligation to
the symposium organizers to pontificate
on the general role and relevance of theory in ecology, and they provide an excuse
833
PREDICTING POPULATIONS
Actual data
Simulated data
300
Time (day)
FIG. 2. A replicate of Nicholson's blowfly competition experiments performed by Y. C. Wu (1978). The data
were fitted to a deterministic "von Foerster-like equation that included age structure, density dependent birth
and death rates as well as functional dependence of fecundity on nutritional history (Oster, 1977a, b).
for the rest of my talk, which is definitely
ST, but which has, I think, general practical applications.
DETERMINISTIC MODELS
When I first got into the ecological modeling business I was impressed (as was
everyone else) by A. J. Nicholson's extraordinarily detailed studies of the demography of laboratory-reared blowfly populations (Nicholson, 1957, 1960). These
experiments were run under various conditions of competition for food between
adults or larvae. In almost all cases the
population began to oscillate in a more or
less regular fashion; these oscillations persisted for many generations. Figure la
shows an example of one of Nicholson's
data sets. The major oscillatory period of
about 45 days is easy to understand, being
merely a manifestation of the time delay
between birth and adulthood (cf., May,
1975). However, a lot more is going on in
the population dynamics, as can be seen by
calculating the frequency spectrum of the
data as shown in Figure lb. Because Nicholson had taken such pains to gather information on the density dependence of
the vital rates this seemed to be an ideal
system on which to test population models;
and in particular, those involving age
structure. Therefore, I and my colleagues
spent a great deal of effort in constructing
models of this system. These models
ranged from simple difference equations
to systems of nonlinear partial differentialintegral equations; an account of our efforts can be found in Oster (1976, 1977a,
b, 1978), Ipaktchi (1977) and Wu (1978).
We soon found that, although Nicholson
had been quite perceptive in deciding
which data he should record, nevertheless,
the models required information that he
had not thought to gather. Therefore, we
were inspired to replicate some of these
experiments ourselves in order to fill in the
834
GEORGE OSTER
10000
100
300
200.
400
500
600.
TIME CDAYS5
SIMULATION OF NICHOLSON EXPERIMENT
I
3008
50
20
10
5
4
3
PERIOD CDAYS)
POWER SPECTRUM DENSITY OF SIMULATION
10000
=> 2000
o
0
100.
200.
300.
TIME CDAYS)
400
500
SIMULATION OF NICHOLSON EXPERIMENT I
FIG. 3. Two simulations of the model of Figure 2 with identical parameter values, but with slightly different
initial conditions. The population trajectories diverge from one another at an exponential rate due to the
presence of a "strange attractor" (cf., Guckenheimer ft ai, 1976). A power spectrum of the simulation reveals
as rich a frequency structure as the experiments themselves!
835
PREDICTING POPULATIONS
missing data (cf., Wu, 1978). After a lot of
hard work we managed to fit some of these
experiments passably well. One (of the
best) example(s) is shown in Figure 2.
maturation delay
aUuhs Nit)
However, there was—er—a fly in the ointment. The behavior of the models was frequently as complicated looking as the data
itself! This can be seen in the model sim300
I
\
ulation and its accompanying frequency
\
1
\
\
spectrum in Figure 3a, b, c. Indeed, theses
1
1
258
\
^ ~~
models seemed to point to a disasterous
l\
1
conclusion, which can be seen by compar\
200.
ing the population trajectories shown in
\
{
\
Figure 3a, b. These are numerical simum 150.
lations of the identical population model
\ 4
(a complicated set of differential equaiee
tions), the only difference being a tiny difz
50
ference in the initial population size. If
Figure 3a is superimposed upon 3b, you
-<—
will see that this small difference in initial
10
15
28.
25
°0
5
conditions grows continuously until the
BIFURCATION DIAGRAM OF THE DIFFERENTIAL
two trajectories are completely out of
DELAY EQUATION
phase. It was impossible, in principle, to
use such deterministic models to project FIG. 4. Bifurcation diagram of the simplest model
population behavior into the future, since which includes age structure effects and Nicholsondensity dependent birthrates. Various combithe slightest error in estimating the initial type
nations of birth and death rate parameters yield perpopulation rendered prediction impossi- iodic orbits of increasing length. (The dashed lines
ble. This kind of behavior turned out to be are approximate bifurcation boundaries computed
typical not only of these complicated from a linearized analysis.) (Oster and Ipaktchi, 1978)
models but of much simpler models which
contained time delays and/or nonlinearities. An explanation of this phenomenon problems of estimating the model paramcan be found in May and Oster (1976); it eters from data, estimating the measureturns out that it is practically impossible to
distinguish the chaotic behavior that non- ment error bounds, and estimating the
linear models can generate from true ran- "goodness-of-fit" of the model to the data.
dom processes {cf., Guckenheimer et at, For these tasks we must relinquish our es1976). The best one can do in most cases thetic commitment to deterministic modis to work out the generic behavior of the eling and turn to some recent ideas in time
model; for example, Fig. 4 shows the bi- series analysis.
furcation behavior of an age-structured
STOCHASTIC MODELS
model for the adult population of NicholIf
deterministic
models are of no use in
son's flies showing the parameter values
predicting
the
fate
of Nicholson's blowwhich generate periodic orbits of various
flies,
then
how
shall
we
proceed? Let us go
lengths.
back to the experiments and see what variSo the situation seemed to be this: how- ables Nicholson actually measured. Figure
ever useful deterministic models were for 5a shows a schematic flow chart of the flies'
gaining a theoretical understanding of life history. Nicholson could not measure
population dynamics, it seemed that they the entire age structure every day; what he
were not of much use for the practical did measure every two days was total adult
problem of actually forecasting population deaths, the number of pupae emerging,
behavior. Moreover, there remained the and the dates when the emerging flies had
t
UL
836
GEORGE OSTER
Births = Bit)
(eggs laid)
Adults = Nit)
£55$
Pupae
Larvae
(iz-zf hrs) (5-iodays) (6- 0 days)
aae
emergences
*E(t)
Adult deaths =
egg +• pupal + larval
deaths
FIG. 5. a. Schematic of the life cycle of a blowfly showing the aggregate measurements taken by Nicholson,
b. The time series actually measured by Nicholson in his experiments (cf., Brillinger et al., 1980). Because of
the large differences between the maximum and minimum population sizes it is customary to plot the square
root of the data so as to make the variability more uniform.
been laid as eggs. From these measurements he could calculate the total adult
population, N(t), for each time interval
since
N(t + 1) = N(t) - D(t) + E(t)
(1)
E(t) = B(t - T)
(2)
where D(t) is the adult deathrate, E(t) is
the emergence rate, B(t) is the birthrate
(i.e., eggs laid/2 days), and T is the measured time between birth and eclosion.
Figure 5b shows the time series for N(t),
D(t), E(t), and B(t) taken from one of Nicholson's data sets.
Now these were very aggregated measurements, and it is not clear that they are
sufficient to predict the population's future course. For that one needs to know
not only the age structure—which is constantly changing—but the birth and death
rates, which are complicated functions of
the population's density and history. Ordinary life-table analysis will not work here
because the age structure is not stationary
and the vital rates are nonlinear. Fortunately, it turns out that something can be
done even for so complicated a population
history as this. Below I will describe a
methodology my colleagues David Brillinger, Peter Guttorp, John Guckenheimer
and I developed to deal with such situations. My description here will be sketchy
and heuristic; readers interested in the
whole story, with all the gory mathematical
details, should consult Brillinger et al.
(1980, and 1981) and Guttorp (1980).
In order to illustrate the procedure we
will deal with the deathrate series only,
D(t) ( = the number of adult flies which
died during the 2-day period, t). This time
series is plotted in Figure 6a above a plot
of N(t), the total number of adults alive
during period t (Fig. 6b). A clue as to the
density dependence of deathrate on population size can be gotten by cross plotting
these two time series to obtain a scatter
diagram of deathrate versus population
size; this is shown in Figure 6c. Contrary
to Nicholson's intended design, the death
rate clearly increases with population size,
albeit with considerable variation. Thus it
is necessary to write a stochastic version of
the population balance equation as follows.
By analogy with the deterministic Leslie
matrix method we define the following
quantities.
N(t) = the population vector whose entry in row i gives the number of adults in
age class i at time t.
E(t) = (E (t), 0, 0, . . . , 0) = emergence
837
PREDICTING POPULATIONS
VfOJLT PDPULflTIDN-
125
100
75
50
25
100
200
100
300
400
500
600
700
500
600
700
VDEflTHS
75
50
25
Q
0
100
200
300
100
DEMERGING FLIES
100
?S
50
25
Q
0
100
200
300
400
500
600
700
0
100
200
300
400
500
600
700
FIG. 5. Continued.
838
GEORGE OSTER
58.
108.
150.
200.
250.
300.
358.
400.
50.
188.
150.
200. 258.
TIME CDAYS5
388.
358.
488. 8.
10008
.2
.4
.6
DEATH RATE
.8
I.
VARIATION OF THE DEATH RATE IN EXP. I
FIG. 6. The deathrate and adult population size time series cross plotted as a scatter diagram to detect the
presence of density dependence.
vector giving the recruitment of adults
from pupae.
P(N) = the survival matrix: the entry in
row i + 1 and column i gives the fraction
of age class i which survives to age i + 1,
all other entries being zero.
If the population dynamics were deterministic, we could write the balance equations as:
E(t), N(0) = No
(4)
and the total adult population is N(t) = 1
N(t), where 1 is the unit vector whose
entries are all 1. Thus the population trajectory can be projected into the future if
the emergence series, E(t), and the survival
matrix, P(N), is known. The deathrate series is then given simply by
N(t
= P(N(t))-N(t)
D(t) = N(t - 1) - N(t) + E(t). (5)
In order to have a closed system of equations we also need to have an expression
for the emergences as a function of the
population history, H:
E(t + 1) = f(H)
(6)
where H = (N(t), N(t - 1), . . .). (Equation
6 is not to be confused with equation 2
which is simply Nicholson's measured
emergence series.)
The stochastic version of the above model is written as:
N(t + 1)= P(N(t))-N(t)
+ E(t + 1) + e(t + 1)
(7)
where P(N) is stochastic and e(t) is an error
variate.
In what follows I will illustrate the method by focussing on the relationship between the deathrate series, D(t), and the
adult population series, N(t).
First we must assume a particular model
for the survival matrix. Let q(x,H) = probability of dying between ages x and x + 1
839
PREDICTING POPULATIONS
diath
tnodni
Data = {N(t)/ D(t)\
FIG. 7. The iterative algorithm for simultaneously estimating the deathrate parameters and the age structure
vector. At each stage the current estimates are compared with the data by minimizing the function S; thus
the longer the recorded history of the population, the better will be the forecasted trajectory.
conditional on having survived to age x
and on the population history, H.
In what follows we will assume that
the population history, H, consists of
(N(t),N(t - 1)) only and the functional
form of q(x,H) is
q(x,H)= 1 - ( 1 - a x ) ( l -)8N(t))
•(l-yN(t-l))
(8)
where (a,/3,y) are the parameters to be
estimated from the data. (We emphasize
that the method is in no way restricted to
this particular model, which assumes that
the density, age and history effects act independently; but such a model is a reasonable first approximation.) Then the model
for the adult population dynamics is:
N(x,t) - N(x + l,t +1) = q(x,H) N(x,t)
N(0,t) = E(t)
(9)
The problem now comes down to estimating the age structure vector, N(t), and
the deathrate parameters (a,/3,y) simultaneously. The algorithm for doing this is
shown in Figure 7. The method consists
of: (1) making an initial guess for the parameters and the population vector, (2)
DEATH PROBABILITIES
LflRGE
20
HGE
FIG. 8. The computed deathrate probability, q(x,N),
as a function of age, x, and for various population
sizes. (Brillinger et ai, 1980)
840
GEORGE OSTER
DBSERUED RND PREDICTED CDflSHED: DEflTHS
110
88
66
44
22
Q
100
200
300
400
500
600
700
500
600
700
WEIGHTED RESIDURLS.
-1
100
200
300
400
L0G10 SPECTRUM DF RESIDUflLS
5. Or
t.S
FIG. 9. a. The observed and computed deathrate series based on the deathrate model of equation (8). b.
The weighted residuals: (D(obs) — D(calc))/N(t - 1) between the model and the data. The substantial amount
of autocorrelation remaining indicates that there still remains unutilized information in the data. (Brillinger
etal, 1980)
calculating trial values for the deathrate
and the age structure, (3) estimating new
values for the deathrate parameters by
minimizing a certain function which compares the data with the model, and (4) using ihese new parameter values to recom-
pute the death rate and the population
vector. This procedure is iterated until the
trial values and the estimated values become the same. (That is, the system converges to a fixed point in (N(x,t),a,f3,y)
space.)
841
PREDICTING POPULATIONS
10Q
80
n
60
i
40
ICO
80
"•0
20
IOC
8C60-
i
4C-
IOC
9060-
100
200
300
WO
SOO
6O0
FIG. 10. The calculated age structure for 10 age classes. (Brillinger et ai, 1980)
700
842
GEORGE OSTER
100
80-
_. ec i
a
8C -
rt
6C-
X
20-
0
80
U> 6C
X
20
0
80-
n.
60-
l
2C
10Q
80 -
20-
k
AI
100
200
300
*0C
Fic. 10. Continued.
500
600
700
843
PREDICTING POPULATIONS
2300,
3S1
flUERflGE OF EGSS UIHEN PDPULflTIDN IN FDURTH QURRTILE
2700,
100
200
300
^00
flUERflGE fIGE
500
S00
700
361
flUERflGE OF EGGS UHEN PDPULflTIDN IN THIRD QURRTILE
3100,
ffe
361
flUERflGE OF EGGS UHEN PDPULflTIDN IN SECOND QURRTILE
2700
$55
300
tOO
500
600
AUERflGE FIDULT LIFE TIME
700
FIG. 11. The average adult age over the course of
the experiment. The average age is stabilizing and
decreasing toward the end of the experiment.
RESULTS
Nicholson claimed that his experimental
design precluded density dependent death
rates for the adult flies. The scatter diagram in Figure 6c belied that belief, although the deathrates should certainly be
age dependent. Figure 8 is a plot of the
deathrate computed from the model
(equation 8). As expected, q increases
monotonically with age and with density;
moreover, it indicates some dependence
on the phase, i.e., whether the population
is increasing or decreasing. A check on this
result can be obtained by using q(x,N) to
calculate the deathrate series and comparing this with the measured series. This
comparison is shown in Figure 9a; that the
fit is quite good can be seen by plotting the
weighted residuals ( = (actual deathrate measured deathrate)/N(t - 1)), as shown
in Figure 9b. (This plot also shows that the
product model for q leaves some autocorrelation in the residuals; thus age, density
and history effects are not independent
and/or a two-step history is not sufficient.)
3S1
ft)O
flUERflGE DF EGGS UHEN PDPULflTION IN FIRST QUflRTILE
Figure 5 3 3
FIG. 12. The evolution of the birthrate as a function
of density. As Nicholson suggested, there is a clear
trend toward increased fecundity at higher population densities. (From Guttorp, 1980)
Once the deathrate model parameters
have been estimated it becomes possible to
reconstruct the age structure of the adult
population, N(x,t). Figure 10 shows the
calculated age structure using this deathrate model. The point to be emphasized
here is that neither the population nor the
age structure is stationary.
Having reconstructed the age structure
we can then address a number of other
demographic issues of interest. For instance, Nicholson asserted that a significant amount of selection took place over
the course of the experiment. In particular, he surmised that the flies were being
selected for egg-laying capacity at reduced
protein levels, a density adaptation. Figure
11 shows a plot of the average age of the
adult population over the course of the
experiment; it shows that the population
is apparently becoming younger towards
844
GEORGE OSTER
the end of the experiment. Thus it is possible that this observed change in egg-laying behavior can be traced in part to a lowering of the average age, younger flies
being generally more fecund. It turns out
that when the estimation method is applied to the birthrate time series (cf., Guttorp, 1980) the average egg-laying rate as
a function of density can be computed.
This is shown in Figure 11, which indicates
that, in addition to the above demographic
effect, there is indeed a trend towards
higher fecundity at high population densities, vindicating Nicholson's perception.
DISCUSSION
The forgoing section outlines a statistical
model for estimating density dependent
vital rates and age structures using only
aggregated measurements. The model is
not restricted to linear, stationary population dynamics, and can be regarded as an
extension of conventional life table analysis. Deterministic models are frequently
useless for predictive purposes because of
the ubiquity of chaotic dynamics. This
model, however, provides a practical
means of projecting measured population
trajectories into the future. Moreover, it
supplies error bounds on its predictions.
This is important, for example, if projections are being used for population management purposes (e.g., setting whaling
quotas or insect pesticide schedules). The
model can also be used to answer basic biological questions such as detecting longterm changes in the demographic properties resulting from systematic selective
forces acting on the population.
Finally, it seems to me that the process
which led to the development of the model
is an interesting example of how theory
itself evolves. One of the most practical
consequences of doing theoretical tinkering is that it stimulates you to think about
and do things in a way you might not have
done otherwise. Frequently, these other
things you did may or may not relate productively to your original research motives. But science progresses to a great extent by serendipity. Theory, providing you
are not too dogmatic, can actually facilitate
this process.
REFERENCES
Brillinger, D., J. Guckenheimer, P. Guttorp, and G.
Oster. 1980. Empirical modeling of population
time series data: The case of age and density dependent vital rates. In G. Oster (ed.), Lectures on
mathematics in the life sciences, Vol. 13, pp. 65-90.
American Mathematical Society, Providence, R.I.
Guckenheimer, J., G. Oster, and A. Ipaktchi. 1976.
Density dependent population models. J. Math.
Biol. 4:101-147.
Guttorp, P. 1980. Statistical modelling of population
processes. Ph.D. Diss., Univ. of Calif., Berkeley.
Ipaktchi, A. 1977. Dynamics of density dependent
populations. Ph.D. Diss., Univ. of Calif., Berkeley.
May, R. M. 1975. Stability and complexity in model
ecosystems, 2nd ed. Princeton University Press,
Princeton, N.J.
May, R. and G. Oster. 1976. Bifurcation and dynamic complexity in simple ecological models. Amer.
Natural. 110:573-599.
Nicholson, A. J. 1957. The self-adjustment of populations to change. Cold Spring Harbor Symp.
Quant. Biol. 22:153-173.
Nicholson, A. J. 1960. The role of population dynamics in natural selection. In I. S. Tax (ed.),
Evolution after Darwin, pp. 477-520. University
of Chicago, Press, Chicago.
Oster, G. 1976. Internal variables in population dynamics. In S. Levin (ed.), Mathematical problems
in the life sciences, Vol. 8. American Mathematical
Society, Providence, R.I.
Oster, G. 1977a. Lectures in population dynamics. In
R. DiPrima (ed.), Modern modeling of continuum
phenomena. American Mathematical Society,
Providence, R.I.
Oster, G. 19776. The dynamics of nonlinear models
with age structure. In S. Levin (ed.), Studies in
mathematical biology, Vol. 16, Part II. American
Mathematical Society, Providence, R.I.
Oster, G. and A. Ipaktchi. 1978. Population cycles.
In H. Eyring (ed.), Periodicities in chemistry and
biology, Vol. 4, pp. 111-132. Academic Press,
New York.
Wu, Y. C. 1978. An experimental and theoretical
study of population cycles of the blowfly Phaenicia sericata (Calliphoridae), in a laboratory ecosystem. Ph.D. Diss., Univ. of Calif., Berkeley.