THEORETICAL POPULATION BIOLOGY 9, 202-221 (1976)
Discrete
Time
Models
for Two-Species
Competition
M. P. HASSELL AND H. N. COMINS
Department
Imperial
of Zoology and Applied Entomology,
College, London S. W.7., England
Received
December
10, 1974
A discrete (difference) single age-class model for two-species competition is
presented and its stability properties discussed, It resembles the Lotka-Volterra
model in having linear zero growth isoclines, and thus, also in its general
requirements for the coexistence of competing species. It differs in allowing the
populations to show damped oscillations, stable cycles or even apparent “chaos”
if competition is sufficiently severe. A similar two age-class model is discussed
where there is both intra- and interspecific competition in one of the developmental stages, but only intraspecific competition in the other. Even this slight
increase in complexity leads to markedly different properties. The zero growth
curves become nonlinear and up to three equilibria
between two competing
species are possible.
1. INTRODUCTION
The history of modelling
competitive
interactions
commenced with the
pioneering studies of Lotka (1925) and Volterra (1926). They independently
proposed a linear two-species competition model often expressed in the form:
dXjdt = rX(K - X - aY)/K,
dY/dt = r’Y(K’ - Y - /3X)/K’,
(1)
where r and T‘ are the intrinsic rates of increase, K and K’ are the equilibrium
densities (carrying capacities) and oi and p are the competition
coefficients
defining the equivalences between the two species. The model permits either
the extinction of one or other of the competing species or their stable coexistence.
The populations can only approach their equilibrium
positions monotonically
and never exhibit oscillatory behaviour. It has stimulated much valuable research,
both experimentally
(e.g., Gause 1934 and Crombie 1945, 1946) and theoretically
(e.g., Levins 1968, MacArthur
1972, May and MacArthur
1972, May 1973).
This model and most of its more sophisticated analogues (e.g., Ayala et al.
1973) ignore any time lags in the density dependent feedback mechanism. This
is done to avoid the intractable mathematics that often results when time delays
are introduced into differential equations. It is an important omission as it is
202
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All rights
0 1976 by Academic
Press, Inc.
of reproduction
in any form reserved.
DISCRETE TIME COMPETITION
MODELS
203
now clear that time delays have a great effect on the stability properties of models
(May 1973, McMurtrie
1975). The same problems are inherent in predator-prey
models where time delays are also an essential feature. But here the development
from the Lotka-Volterra
predator-prey
models has followed a different path.
Several workers have based their models on difference rather than differential
equations (e.g., Nicholson and Bailey 1935, Watt 1959, Griffiths and Holling
1969, Hassell and May 1973, 1974). Such models are particularly
appropriate
whenever generations are relatively discrete since they implicitly
include time
lags of one generation length.
In the first part of this paper, we discuss a general two-species competition
model based on difference equations and show how this introduces a rich
diversity of stability behaviour. We then proceed to a more specific case with
two age-classes in each competing population and find that this introduces
further variety in the global stability properties.
2. DISCRETE COMPETITION
MODEL
The principal component of a two species competition model is the function
describing the intraspecific and interspecific density dependent feedback. In the
Lotka-Volterra
model (Eqs. (1)) this follows directly from the logistic model
of single species population growth. In the discrete models to be discussed below,
the dependence is upon the appropriate single species difference model. The
properties of such models (including a difference analogue of the logistic) have
recently been explored by May (1974a, b; 1975). He showed them to exhibit
a variety of stability behaviour, with the populations either damping monotonically or oscillatorily, showing stable limit cycle behaviour or even irregular
fluctuations (“chaos”). A two species model based on the “difference logistic”
showed a similar range of stability behaviour (May 1974b). Further difference
equation growth models have been discussed by May et al. (1974) and Hassell
(1975). Hassell assumed that the density dependence was of the form
X n+l = Xx,(1
+ a-Wb,
(2)
where h is the finite rate of increase (h = er) and a and b are constants defining
the form of the feedback relationship. This has the advantage of describing a
wide range of intraspecific competitive interactions between the extremes of
scramble and contest (Nicholson 1954). Equation (2) is readily extended to a two
species interaction:
X
n+l = ~X,U + 4X, + c.~Y,)l-~,
X’Y,[l + a’(Y, + B&Xb’,
Y n+l =
where 01and /I are as defined for Eqs. (1).
(3)
204
HASSELL
AND
COMINS
We shall here show a general recipe for determining
of all models of the form
X
the stability
properties
n+1= -G[w&2 + aYnl)-“>
(4)
Yn+l= Y,[g(Y, + BX’J-“‘,
and in particular Eqs. (3), which can be rewritten as a special case of this. The
details of the stability analysis are presented in Appendix I. We concentrate here
on the conclusions that emerge.
The basic properties of a competition model may be discussed under three
headings:
(a) The form of the zero isoclines for the two species that separate the
phase plane regions of positive and negative growth.
(b)
The conditions
(c)
The
dynamical
First, we rewrite
for coexistence
properties
Eqs. (3) to simplify
x
of the competing
of the model
the stability
species.
when the species coexist.
analysis:
n+1= Xn[fl + Y(Xn+ ~YTw,
Y n+l =
Y# + r’(Yn + PxL)l-b’>
(5)
where f? = h--ljb and y = Ba and similarly for 0’ and y’.
By plotting the zero growth isoclines of species X and Y, we see that Eqs.
(4) and (5) have some properties in common with the Lotka-Volterra
model.
In the first place, the zero isoclines are always linear as in the Lotka-Volterra
model. Further to this, the conditions for the coexistence of the two species
are not dissimilar. Coexistence requires that the linear isoclines intersect as
shown in Fig. 1 and that the isocline for species X has a steeper negative slope
than that for species Y, namely,
(1 -
6)/a > 1 - 8
and
(I - q/p
> 1- e
(64
If these conditions are not satisfied, one or other of the species will move to
extinction. It will be apparent that an equilibrium
also occurs when
(1 - q/c%< 1 - B’
and
(1 -
el)p < i - e,
but in this case the equilibrium
is an unstable one where the initial
determine which species survives.
(6b)
conditions
DISCRETE
TIME
COMPETITION
MODELS
205
FIG. 1. The conditions for a competitive equilibrium between species Y and X from
Eqs. (5). The lines represent the zero growth isoclines of the two species that separate the
regions of positive and negative growth. The relation of the intercepts on the axes provide
the necessary and sufficient condition (6a) for coexistence of the two species.
The essential differences from the Lotka-Volterra
model lie in the stability
properties of the coexisting populations. Instead of only a monotonic approach
to the equilibrium
being possible, models of the general form of ,Eqs. (4) permit
monotonic or oscillatory damping as well as stable limit cycles and irregular
fluctuations as discussed by May (1974b). Th is range of behaviour is illustrated
in Fig. 2. The local stability properties of any model of the form of Eqs. (4)
may be deduced from the diagram in Fig. 3. The coordinates X, Y of a potential
equilibrium
point are given by the intersection of the zero isoclines as shown
in Fig. (1). The local stability properties can then be found by plotting the
point (Xjb,
Y 2 b’) on Fig. (3) wheref and 2 are the derivatives of the functions
f and g evaluated at (X, Y). In the particular model we are considering (Eq. (5))
this becomes (X y b, Y y’ b’).
The lines in the top part of Fig. 3 give the stability boundaries for different
values of the “feedback” parameter c+. Note that the stability boundaries are
only shown for a/3 < I, since ~$2> 1 is inconsistent with Eq. (6a) and thus
ensures the extinction of one of the species. All points within the relevant
boundary correspond to stable equilibria. If the point falls in the shaded part of
the diagram, the equilibrium
is stable and is approached monotonically,
while
for points within the stability boundary but not in the shaded region, the
approach to equilibrium
is oscillatory. Thus, for point P in Fig. 3, there is an
oscillatory approach to the equilibrium if + < i but stable limit cycle behaviour
if c@ > I.
Figure 3 is particularly valuable for following changes in stability properties
as the parameters 6 and b’ in Eq. (5) are varied. Since the equilibrium
point
(X, Y) does not depend on these parameters, the coordinates (X y b, Y y’ b’)
of the corresponding
point on Fig. 3 are simply proportional
to b and b’,
206
HASSELL
AND
COMINS
(b)
‘:.:
10
20
30
GENERATIONS
FIG. 2. Numerical simulations from the model (5). The parameters used in (a), (b),
and (c) correspond to points A, B, and C in Fig. 3 and show the effects of gradually
increasing the level of competition in species Y. Figure 2(d) shows the apparently chaotic
behaviour occurring when competition
is further increased (note compressed scale).
Species X is omitted here since its fluctuations are not visible on this scale.
respectively.
For example, varying b’ alone would move the monotonically
stable point, A in Fig. 3, through oscillatory stability (B) to the point C, in the
locally unstable region. Finally, with further increases in b’, we move into a
region of apparently chaotic behaviour. This progression, illustrated in Fig. 2
(a)--(d), is similar to that observed by May (1974b, 1975) and reinforces his
conclusion that such behaviour is a general property of such discrete time models.
A feature of the model that is not evident from the stability diagram is that the
stability properties do not depend on the parameters y and y’. This can be shown
DISCRETE TIME COMPETITION
MODELS
207
2
1
Xib
FIG. 3.
position of
determines
area exhibit
A general stability diagram for models of the general form in Eq. (4). The
the competitive equilibrium
for a specific model relative to the ~$3boundary
whether there is a stable point, or local instability. Points within the shaded
monotonic stability. For further discussion, see text.
by appropriate scaling of the parameters in Eqs. (5). A corresponding lack of
dependence of stability on the parameter a was found by Hassell (1975) for the
single species model given by Eq. (2).
3. A Two
AGE-CLASS
COMPETITION
MODEL
In the previous section we explored the general properties of a single ageclass difference model for two-species competition. We now consider how these
properties may alter if two species compete at more than one age-class. To do
this we take a simple two age-class model of the form
t+l = xt exp[--a& + olyt)L
Yt+I = Yt exp[--c(yt+ P41,
x~+~= Xt exp[r- a’(& + cJYt)I,
Y~+~= Yt exp[s- c’(Yt + F&)1,
X
(7)
where x and y are the larval or juvenile stages, and X and Y the adults. The
generations and stages are discrete and nonoverlapping
so that adults and larvae
are not present at the same time. The time interval from t to t + 1 represents the
ki3/9/2-6
208
HASSELL
AND
COMINS
duration of adult or juvenile stages. The parameters r and s are the intrinsic
rates of increase of the two species and the terms a, a’, c, and c’ define “threshold
densities” (l/a, l/a’, etc.) above which competition becomes severe (as in Eqs. (2)
and (3)).
To reduce parameter proliferation,
the model presented here is not exactly
derived from the single age-class competition model discussed in the previous
section (Eqs. (3) and (5)). This was an extension of the single species model in
Eq. (2) where the density dependence is characterized by two parameters, a
and 6. In this section we adopt instead the model proposed by May (1974a),
namely,
X n+l = XX, exp(-SJ,
(8)
where /\ is the finite rate of increase and 4 is a constant defining a threshold
density as above. This model may be obtained directly from Eq. (2) for the
special case where b - 00 (with ab kept constant) and exhibits the same range
of stability properties. Unfortunately,
even for this simplified two age-class
model, it is not possible to present the full stability properties analytically.
We shall here concentrate on some interesting properties of the model, whilst
leaving the mathematical details for Appendix II.
Firstly, we note that the two age-class model reverts to the single age-class
case when all the intraspecific and interspecific competition is confined to one
of the developmental
stages, larvae for example. In this case, a’ = c’ = CL’=
B’ = 0, and thus,
x,,-1 = x, exp[r - a(x, + ciyn)].
yn-l-lL=yn exp[s- c(ynf P41,
(9)
where rz to n + 1 is a generation interval equivalent to t to t + 2 in Eq. (7).
This model has the same properties as discussed in the previous section.
The interesting effects of multiple age-classes emerge when competition
is
also allowed in the other age-class, the adults in this case. We shall illustrate
this for the case where intraspecific and interspecific competition
is allowed
between larvae, but only intraspecific
interactions
between the adults (i.e.,
only a’ = p’ = 0). Equation (7) thus becomes
X til = xt exp[--a@,
+ vt)l,
I’tyt,-l= Yt exp[--c(39+ P41,
5t+l = Xt exp[r - a’X,],
ytel
= Y, exp[s -
(10)
c’Yt].
The addition of this single component (i.e., a’ and c’ not equal to zero),
leading from Eq. (9) to (IO), has very important effects on the dynamical properties of the model. The zero isoclines or growth curves (adult to adult or
DISCRETE
TIME
COMPETITION
MODELS
209
larvae to larvae) need no longer be linear and there now can be more than one
equilibrium
point in phase space (cf. Fig. 1). The various growth curves may
be almost linear, concave, or convex and there can be up to three equilibrium
points between the competing species as shown in Figs. 4(a)-(c). In each case the
graphs are drawn for both the adults and larvae. From this we see, interestingly,
that the adult and larval growth curves tend to be inverted, in some cases almost
mirror images as in Fig. 4(a). Figures 4(b), ( c ) a1so show how multiple equilibria
can arise. There are two equilibria in Fig. 4b, one unstable and one stable
(or possibly a region of stable limit cycles). Disturbance from the unstable
equilibrium
may either lead to the extinction of one of the species or movement
to the stable equilibrium
position. Figure 4(c) has an additional equilibrium
point or stable limit cycle region. Here, neither population can become extinct,
(4
(b)
POPULATION
OF
X
FIG. 4. Typical zero growth curves from Eqs. (IO) that yield one or more equilibria.
In each case (a)-(c) the larval growth curves are to the left and the corresponding adult
ones to the right of the figure. S = stable equilibrium;
U = unstable. Broken lines
indicate the growth curves for species X; solid lines species Y.
210
HASSELL
AND
COMINS
but the system may move from one equilibrium
to the other if sufficiently
strongly disturbed. This is illustrated in Fig. 5. An approximate condition for
the existence of three equilibria is derived in Appendix II, namely, that
cm/c > s/(r -
1)
and
cp/a > r/(s -
1).
This requires that in the larval stages an increase in either
has an effect on the other species sufficiently greater than
This is analogous to the condition for competitive exclusion
models. However, due to the self-regulation imposed in the
species can force the other to extinction.
10
20
30
40
(11)
one of the species
the effect on itself.
in single age-class
adult stage, neither
CBU
GENERATIONS
FIG. 5. A numerical simulation of Eqs. (10) where the zero growth curves have the
form in Fig. 4(c). The parameters have the following
values: r = s = 3; iy = /3 = 2;
a = c = 0.2; a’ = c’ = 1. PI and P, represent perturbations
of different strengths.
In using the zero growth curves to investigate the range of stability behaviour
of the model, we may consider either the adult population growth curves or the
corresponding larval ones since there is a one to one relationship between the
respective equilibrium
points and their stability properties. This is obviously
necessary since both graphs refer to the one system. In the case where both ageclasses interact intraspecifically
and interspecifically,
the zero growth curves
are mutually dependent. In the present model, however, the adults only complete
intraspecifically
so that the larval growth curves depend on disjunct sets of
parameters: {r, a, a’, a) for species X and {s, c, c’, j3> for species Y. Therefore,
it is possible to alter one of the larval growth curves at will without affecting
the other.
DISCRETE
TIME
COMPETITION
211
MODELS
The general form of the larval growth curves is derived in Appendix II. It
depends upon the sets of parameters mentioned above, except that the parameters 01and p merely change the scale on the y and x axes, respectively, without
altering the shape of the curve. This shape is always convex (except possibly
below the turnover point). The condition that the curve shall have a turnover
point (as in Fig. 4(b), (c)) is:
a/a’ < (r - 1) exp[-(r
- l)].
(12)
The existence of the turnover arises from the form of the adult survival curve
defined in Eq. (10) and shown in Fig. 6. Let us suppose that for a sufficiently
low larval population of species Y the larvae of species X suffer little competition.
This leads to abundant adults of X, which are then severely reduced by intraspecific competition (A in Fig. 6). Evidently, for values of X, greater than m in
Fig. 6, the larval population at t + 1 is actually increased by a decrease in the
number of adults at t. Thus, a moderate amount of interspecific competition
can increase the larval equilibrium population while decreasing that of the adult.
Vandermeer (1973) has postulated the existence of this type of zero growth curve
in his graphical analysis of two species interactions. These would be obtained
when one species has an augmentative effect on the other at certain densities.
FIG. 6. The
Xt exp(r - ax,)).
adult survival curve for species X from
Points A and m are discussed in the text.
Eqs.
(10) (i.e.,
xtil
=
The analytic conditions for stability of a particular equilibrium point are also
given in Appendix II. Although further scaling in Eq. (10) is possible, there
remain six independent parameters, all of which affect stability. It is not possible,
therefore, to produce a general synthesis as was done in Fig. 3 for the single
age-class model. The equations can, however, be used to investigate further the
properties of equilibrium points discovered by graphical analysis of the isoclines.
212
HASSELL
AND
COMINS
Since the parameters 01and /3 have the sole effect of compressing the respective
isoclines in one direction, it is easy to observe graphically their effects when
varied. For example, the triple intersection shown in Fig. 4c will clearly only
occur for very restricted ranges of 01and /3. For the same reasons, we would
expect that single intersections, as in Fig. 4(a) can occur for relatively many
parameter choices. This is illustrated in Fig. 7 where we have plotted the
number of intersections of the two isoclines as a function of the parameters 01
and /3. The remaining parameters are fixed at the values used for Fig. 4(c).
4-
0
1
(unstable)
B
1
2
0
(stable)
2
4
Frc. 7. Graph showing the dependence of the number of potential competitive
equilibria on the interspecific
competition
coefficients, OLand /I Other parameters are
fixed at the values used for Fig. 4(c).
4.
DISCUSSION
There have been several major studies on interspecific
competition under
laboratory conditions: for example Park (1948, 1954) Crombie (1945) (1946)
Birch (1953) and Birch et ~2. (1951) using stored product insects; Gause (1934)
using Paramecium species, Frank (1957) using Duphniu species and Ayala
(1969) using Drosophila species. The most frequent result has been the elimination of one of the species. Coexistence has been achieved, but generally only
where there was an effective refuge for one of the species (Gause 1934, Crombie
1946), or under carefully controlled conditions such as particular temperature
regimes (Ayala 1969). Experiments
such as these have lent weight to the
“principle
of competitive exclusion” (see Hardin (1960) for its origins). This
states that the coexistence of two species competing for limited resources is only
possible if each species inhibits its own growth rate more than it affects the
growth rate of the other species. This is likely to occur when one species has
some partial or complete refuge in space or time or some differences in feeding
habit or in territoriality.
DISCRETE
TIME
COMPETITION
MODELS
213
The conditions for coexistence at a stable equilibrium
have been precisely
defined in terms of the Lotka-Volterra
model (Eq. (1)) by Gause and Witt
(1935); that
ci<K/K’
and
p < K’IK.
(13)
These are merely the necessary conditions for the zero growth isoclines to
intersect, as they do in Fig. 1, with that for species X having the greater slope
at the equilibrium
point. The analogous conditions from the discrete model
(Eq. (5)) are given in Eq. (6a). We believe that this difference model has two
significant advantages over its differential counterparts, both hinging on the
single-species growth model (Eq. (2)) u p on which it is based. In the first place,
the two-parameter
growth model describes a wider range of data than existing
single parameter characterizations
(Hassell 1975). More importantly,
the
implicit inclusion of time lags due to the difference format has a great effect on
the stability properties (May 1974a, b; 1975; May et al. 1974), allowing the
kinds of oscillatory behaviour often seen in laboratory, single-species interactions
(Fujii 1968, Crombie 1945, Utida 1967, Nicholson 1954).
It is unfortunate that there is little available data that can be used to validate
a specific, two-species competition model, much less than is at hand for predatorprey models. The most useful data is that from which phase plane diagrams
may be constructed (as in Fig. 1) to show the zero growth lines and also the
trajectories of the two species from different experiments. Such results have
been obtained by Crombie (1945, 1946) and Gilpin and Ayala (1973). Crombie’s
classical series of experiments using stored product insects provide some support
for the predictions of the Lotka-Volterra
model. Coexistence or extinction were
obtained as expected and the populations followed roughly the expected trajectories. Gilpin and Ayala’s Drosophila data, however, indicates markedly concave
zero isoclines to explain the observed equilibrium
(Fig. 8).
The form of the zero isoclines for Drosophila in Fig. 8 is not dissimilar to
that in Fig. 4(a), where the larvae have been “sampled.”
These results were
obtained from the two-age class model described in the previous section,
allowing intraspecific and interspecific competition
between larvae and only
intraspecific competition between adults. The introduction
of such a minimal
age class structures has interesting consequences, affecting the shape of the
zero isoclines and the number of possible equilibrium
points. The model
suggests several questions in relation to Gilpin and Ayala’s data.
(1) Do the nonlinear growth curves solely result from competitive
actions at more than one age class ?
inter-
(2) Would the Drosophila isoclines have been convex rather than concave
(as in Fig. 4(a)) if the insects had been sampled in a different stage of their life
cycle ?
214
HASSELL
AND
COMINS
(3) Would there have been oscillatory behaviour about the equilibria
if the level of competition had been increased (e.g., by reduction in food supply) ?
(4) Would an increased intensity of interspecific
competition
more than one stable equilibrium
as shown in Fig. 4(b), (c)?
D, PSEUDOOBSCURA
lead to
(P)
FIG. 8. The zero growth isoclines for two species of Drosophila as determined from
competition
experiments. The solid circles show the equilibrium
populations in single
and mixed species culture as determined from independent
experiments (after Gilpin
and Ayala 1973).
The results in this paper suggest that a single age-class competition model
remains a useful tool in studying the simplest of laboratory systems or in answering very general questions (MacArthur
1972, May and MacArthur
1972).
Natural interactions usually involve competition (intra or inter) at more than
one age-class. This can lead to complex behaviour such as nonlinear zero
growth curves and multiple equilibria, which would not be predicted from a
single age-class model, but are observed in the simplest of two age-class models.
Naturally,
these effects become more complicated as further age classes are
allowed to interact. However, except in the most detailed of studies, it should
be possible to compound some of the age-classes, especially where consecutive
ones have very similar interactions.
APPENDIX
I
We derive here the local stability properties of the general
competition model defined in Eq. (4) of the first section:
X n+1=
Y n+1=
zaLf(-G + ~Yn)l-b,
YnMYn + P-L)l-“‘.
two-species
(Al.l)
DISCRETE
TIME
In the vicinity of an equilibrium
to a linear equation of the form
COMPETITION
point (X*,
xnll
215
MODELS
Y*) Eqs. (Al.l)
can be reduced
(A1.2)
= AX”,
where the vector X, is
(Al .3)
and the elements of A are the partial derivatives of the components
respect to those of X, . For this model we have
i4 =
1 - x*bf(x*
( -Y*b’&(Y*
+ au”)
+/3x*)
-x*bolf(X*
1 - Y*b’g(Y*
+ aY”)
+/Ix*)
of X,,,
1.
with
(Al .4)
The equilibrium
conditions f(X* + olY*) = 1, g(Y* + /3X*) = 1 have been
used to reduce the matrix elements to their simplest possible form. The
derivatives of the functions f and g are indicated by dots. Let us define
7) = x*qx*
+ my*),
f = Y*b’g(Y*
+ /3X”).
(A1.5)
Then the eigenvalues h, of A are the roots of the characteristic
AZ + A(71+ t - 2) + 776(1 - 4)
- (7 + 5 -
equation:
1) = 0.
(A1.6)
The equilibrium point is locally stable if both the eigenvalues of A have modulus less than unity. If they are given by a quadratic equation X2 + pih + p, = 0
the equivalent requirements are
IPOI < 1,
IP*l<J +A.
Substituting
(Al .7)
from Eq. (Al -6) we obtain:
I71LTl
+ 5- - 1)l- < 1,+
I + t- - 432 - < (17
75(1
7
I
@)
(7
5-
(A1.8)
2).
Note that these conditions depend only on the product B$ and not on 01 or
fi separately. For particular values of c$ these conditions yield the stability
boundaries shown in Fig. 3.
Finally we notice that the eigenvalues of A are real. The discriminant
of
Eq. (A1.6) is
(7 + 6 -
a2 - 4?((1 - 43) + 4(77 + E - 1)
= (7 -
0) + 47?&3 > 0,
(Al .9)
216
HASSELL
AND
COMINS
so that the roots of the quadratic are both real. An oscillatory approach to
equilibrium
therefore occurs if and only if the larger of the two eigenvalues is
negative. Equivalently
their sum must be negative. From Eq. (A1.6) therefore,
we obtain
(A1.lO)
7$-G-2>&
as the condition
for an oscillatory
approach to equilibrium.
II
APPENDIX
The two age-class model defined by Eqs. (10) 1s
. sufficiently complicated that
the equilibrium
conditions cannot be derived analytically.
The conditions for
zero growth of the various populations, however, can be found.
At first sight, it would appear that the isoclines must be drawn in four dimensions, since four populations are involved. Since there is no interaction between
adults and larvae at the same time step, the model has the formal property that
two completely independent sets of animals can exist; one having the adult
stages at t = 1, 3, 5, 7 ,... and the others being in the adult phase at t =
2,4, 6, 8,... . This means that the first population is in equilibrium
if the adult
populations at t = 3 are the same as those at t = 1: the larval populations at
these times are irrelevant. An equivalent condition is that the larval populations
at t = 4 must be the same as those at t = 2. The equilibrium
conditions for
the second set of animals are completely independent of these, being always
those one time step ahead.
In practice, two such interleaving
populations would be almost impossible
to obtain, because of the synchronisation
required between the lengths of the
two life stages. The realistic use of the model thus requires that only one interleaving set can have nonzero populations.
Let us first consider the adult populations at time t + 2 as a function of those
at time t. If these are unchanged then the system is in equilibrium.
The condition
for a constant adult X population is
A-t = xt,,
= xttl exp[--a(kl + w+dl
= Xi exp[r - a’X, - ax, exp(r which immediately
(A2.1)
a’X,)
- aolYt exp(s - c’Y,)],
reduces to
r - u’X, - ax, exp(r - a’X,)
- ao~Y, exp(s - c’YJ
= 0.
(A2.2)
The adult zero growth curves in Fig. 4 were obtained by plotting contours of
the function on the left-hand side of this equation and of the corresponding
DISCRETE
TIME
COMPETITION
217
MODELS
log-growth-rate
function for adult Y’s, Note that the function in Eq. (A2.2)
depends on all the parameters except c and p. Thus, the X and Y zero growth
curves are not independent and are not readily amenable to graphical analysis
techniques.
An alternative set of equilibrium
conditions is obtained by requiring the
larval populations to be unchanged after one generation (two time steps). The
resulting equilibrium
points must necessarily correspond exactly with those
found by drawing the adult growth curves, since both refer to the same system.
For a constant larval X population we derive
-u(xt + my,)+ Y- a’~,exp[---a&+ g~)l = 0,
and similarly
(A2.3)
for a constant Y population
-4x
+ Pxd + s -
C’Y, exp[--c(y,
+ /%)I
= 0.
Note that Eqs. (A2.3) and (A2.4) d ep en d on disjoint sets of parameters,
resulting zero growth curves are mutually independent.
Equation (A2.3) can be expressed parametrically
in the form
x =
(I
-
u)e”/u’,
(A2.4)
so the
(A2.5)
y = (u/a - x)/a.
Note that increasing a has the sole effect of compressing the curve in the y
direction. Some typical examples of these growth curves are given in Fig. 4.
The global stability properties of the model can be fairly thoroughly
investigated by graphical analysis of the larvae to larvae isoclines. Some general
properties of these curves will now be discussed. It must be remembered that
we are using a discrete time step model, so that the interpretation of the isoclines
is not quite the same as for a differential equation model. In particular, for
sufficiently large values of r and s, the two stable regions in cases such as Fig. 4(c)
can merge into a single limit cycle.
The condition for the existence of a turnover point (see Fig. 4(c) for example)
may be derived as follows. The derivative dx/dy is given by
dxjdu
-=
dyldu
(l/a’)(r
- u -
l)e”
dyF-’
WW
At the turnover point this expression is zero, so u = r - 1. For a turnover
point to exist this value of u must give a point in the positive quadrant:
i.e.,
ay = (Y a/a’ < (r -
1)/a -
e+l/a
1) exp(-(r
This requires that Y be greater than unity.
> 0,
-
1)).
(A2.7)
218
HASSELL
AND
COMINS
An approximate condition for the occurrence of three intersections between
a pair of zero isoclines is that the turnover point of the X isocline must occur
at a higher x value than that at which the Y isocline crosses the x-axis. The
corresponding condition with X and Y reversed also must be satisfied. From
Eq. (A2.5) the turnover occurs at x = exp(r - 1)/a’. Using the analogous
equation for the y zero isocline we find that the intersection with the x-axis
occurs at u = s, giving x = s/@. Thus, the condition becomes
s/f%2< e*-l/a’,
i.e.,
@/a’ > sel-‘.
Since both curves must have a turnover for three intersections to occur we can
assume that Eq. (A2.7) holds. Therefore,
exp(l - r) > a/[a’(r - I)] and
substituting in Eq. (A2.8), c/I/a’ > sa/[a’(r - l)]. Thus,
cppiu > s/(r Similarly,
the second condition
1).
(A2.9)
1).
(A2.10)
gives
m/c > r/(s -
The variation in the number of intersection points as (Y and B are varied is
shown graphically in Fig. 7, where the remaining parameters were fixed at the
values used in Fig. 4(c). This graph was produced by plotting the intersections
of a family of curves given by Eq. (A2.5) as (Ywas varied with the corresponding
family of y zero isoclines produced by varying /3. Note that within each family
the curves differ only in scale along one of the axes.
We now consider the local stability properties of the model. Because there is
no interaction between adults and larvae at the same time step many elements
of the stability matrix are zero. The analogous equations to Eqs. (A1.3) and
(A1.4) are
X ntl = AX, 1
where
(A2.11)
and A takes the form
! 0 0 al3al4
1
0a31
0a42
a23
00
a24
00
*
(A2.12)
DISCRETE TIME COMPETITION
The resulting characteristic
variable ,u = h2:
219
MODELS
equation
can be written
P2 + Plcl
+ PO =
as a quadratic
in the
(A2.13)
0.
This simplification
is a consequence of the formal separability of the model
(due to the noninteraction
of adults and larvae at a particular time step), which
was also observed in the discussion of the isoclines.
At an equilibrium point we obtain
p, = -[(l
- ex)(l
p. = [(I - ax)(l
- U’X) + (1 - cy)(l -
C’Y)],
- cy) - ac$?xy] (1 - a’X)(l
- C’Y),
(A2.14)
where X, Y, x, y are the equilibrium
populations of adults and larvae. For
stability we require 1h / < 1 for all eigenvalues of A. Since j h ( < 1 if and
only if / p 1 < 1, this reduces to the conditions (A1.7) on p, and p, . These
equations could theoretically be displayed in a diagram similar to Fig. 3. This
would be in four dimensions, with axes ax, a’X, cy, and c’Y. Since the stability
properties again depend only on the product $I, the stability boundaries would
be surfaces similar to the curves in Fig. 3.
ACKNOWLEDGMENTS
We are particularly
grateful to Professor Robert M. May for his encouragement and
comments on the manuscript. We would also like to thank Dr. P. Diamond for helpful
discussions and the Ford Foundation for support of one of us (H.N.C.) and for computing
facilities at the Imperial College Field Station at Silwood Park.
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