Abstract
We consider n 2 populations of animals that are living in mutual predator – prey relations or are pairwise neutral to each other. We assume the temporal development of the population densities to be described by a system of differential equations which has an equilibrium state solution. We derive sufficient conditions for this equilibrium state to be stable by Lyapunov's method. The results supplement those published elsewhere.
Further we consider a modification of the Volterra – Lotka model which admits an asymptotically stable steady state solution. This model is discretized in two ways and we investigate how small the time step size has to be chosen in order to guarantee that the steady state solution is an attractive fixed point of the discretized model. This investigation is connected with the determination of the model parameters from given data.
1. Introduction
The oldest mathematical model to describe the predator – prey relation of two populations in a closed habitat is that of Volterra and Lotka. It consists of two differential equations of the form
Obviously, the above system has a steady state solution which is given by
This solution can be shown to be stable in the sense of Lyapunov (see, for instance Citation1).
In Section 2 we consider a generalization of the Volterra – Lotka model to the case of n 2 populations living in mutual predator – prey relations or being pairwise neutral to each other. This model is described by the system (2.1) and it is assumed that this system admits a steady state solution. In Section 2 we derive sufficient conditions for this steady state solution to be stable in the sense of Lyapunov. The results in this section supplement those in Citation2.
So far the predator – prey models are purely theoretical models.
The question now arises as to how these models can be adapted to measured data in order to find out whether these reflect predator – prey behaviour. Since the data are normally given at discrete points of time, it is reasonable to replace the time-continuous model by a time-discrete model that can be considered as an approximation of the time-continuous model. The simplest way of doing this is to replace the time derivatives [xdot](t) and [ydot](t) by difference quotients
In Section 3 we replace the Volterra – Lotka model by the system (3.1) where the additional coefficient c 22 is non-positive. If c 22 < 0 and the condition (3.3) is satisfied, then the system (3.1) has a steady state solution which is asymptotically stable (see Citation1) and the discrete system (3.4) has this solution as an attractive fixed point, if the step size is sufficiently small (also see Citation1). So the time-discrete system (3.4) can be considered as a good approximation of the system (3.1), if the time step size is sufficiently small. In addition, estimates for the step size are derived which guarantee that the fixed point of the time-discrete system (3.4) is attractive.
In Section 3 we also consider a second discretization of the system (3.1) which is given by (3.8) and also has the steady state solution of (3.1) as a fixed point which is attractive, if the time step size is sufficiently small. Again estimates for the step size are derived which guarantee the attractiveness of the fixed point of system (3.8).
In Section 4 we try to adapt the two systems (3.4) and (3.8) to realistic data which are taken from Citation3 with the aid of the least squares approximation method. It turns out that this is only possible for the second discretization of system (3.1) for which we can show that the sufficient conditions for the attractiveness of the fixed point are satisfied.
2 The model and sufficient conditions for stability of equilibrium states
In Citation2 we have investigated a general predator – prey model with respect to stability and asymptotic stability. In this model we consider n 2 populations Xi, i = 1, … , n, of animals that are living in mutual predator – prey relations or are pairwise neutral to each other. Let us denote by xi (t) the density of population Xi at time t. We assume the temporal development of these densities to be described by the following system of differential equations
We assume that there is an equilibrium state [xbar] = ([xbar]
1,…,[xbar]
n
) of
Equationequation (2.1) that is a solution of
Let us assume that, for every initial state x
0 ∊ IR
n
with for i = 1, … , n, there is exactly one solution x(t) = (x
1(t), … , xn
(t)) of
Equationequation (2.3)
with
In this section we want to derive sufficient conditions for the equilibrium state (with (2.2)) of
Equationequation (2.1)
to be stable, which is equivalent to (0, … , 0) ∊ IR
n
being a stable equilibrium state of Equationequation (2.4)
.
For this purpose we select a non-empty subset J of (1, … , n) and assume that
Now we consider the following special case: For some m∊{1, … , n−1} let J = {m + 1, … , n}. Further let
We end with the case n = 3, m = 2 in which, instead of (2.8), we assume that
3 Discretization of a modified Volterra – Lotka model
We consider a predator – prey model which is described by the following system of differential equations
If c 22 < 0, then (3.1) has the steady state solution
Now let h > 0 be a given step size. Then we discretize (3.1) by replacing [xdot](t) and [ydot](t) by the difference quotients
In the case c 22 = 0 (where (3.3) is satisfied) this fixed point, however, is repelling for every choice of h > 0 and hence not stable (see Citation1). If we replace the system (3.4) with c 22 = 0 by
In the case c 22 < 0 we have shown in Citation1 that the fixed point solution (3.5) of the system (3.4) is an attractor, if the step size h > 0 is chosen sufficiently small. The question is how small h has to be chosen. In order to answer this question we consider the Jacobi matrix of the right-hand side of (3.4) at the fixed point (3.5) which is given by
-
If this condition is satisfied, the fixed point (3.5) is an attractor.
-
-
If we replace the system (3.4) by
We again distinguish three cases:
-
(a + b)2 = 2b.
This implies b 2 < (a + b)2 = 2b, hence b < 2 or
which is equivalent to
2 or b < 2. Therefore the fixed point (3.5) is an attractor, if the condition (3.9) is satisfied.
-
(a + b)2 < 2b⇔1 – 2a > (1 – a – b)2. Then it follows that |λ1|2 = |λ2|2 = 1 – 2a < 1. If we put
which is only possible if
, in which case the last inequality is equivalent to
, then condition (3.10) is sufficient for the fixed point (3.5) to be an attractor. If
, case (2) cannot occur.
-
(a + b)2 > 2b. Then it follows that λ2 < λ1 <1 and −1 < λ2, if and only if
4 Determination of the model parameters from data
Let us assume that we are given data (x(i · h),y(i · h)) for some h > 0 and i = 0, … , N.
First we consider the system (3.4) with c 22 < 0. On using c 1 = −c 12[ybar] the first equation can be rewritten in the form
In order to determine c 2, c 21, c 22 we minimize
If we choose [ybar] = 0.75 the minimization of (4.1) leads to c 12 = 3.23. The minimization of (4.2), however, leads to c 2 = −1.62, c 21 = 3.6 and c 22 = 1.16 which violate the condition (3.2).
If we replace (4.2) by
Further we have α = 0.9, β = 0.969 which implies and
. Therefore condition (3.10) is satisfied which implies that (0.21, 0.75) is an attractive fixed point of (3.8).
The following table is also taken from Citation3:
If we choose [ybar] = .831, we obtain c 12 = 1.28 by minimizing (4.1). Minimizing (4.3) leads toFurther we obtain α = 0.82269, β = 0.2205115 which implies
In the first example the condition (a + b)2 <2b is satisfied which implies a 2 <2b which in turn is equivalent to
This can be considered as an explanation for the fact that the discretization via (3.4) does not lead to acceptible parameters.
This can be supported by the following remark: We have seen above that the condition (a + b)2 < 2b is equivalent to
References
- Krabs , W . 1998 . Dynamische Systeme: Steuerbarkeit und chaotisches Verhalten , Stuttgart : B.G. Teubner .
- Krabs , W . 2003 . A General Predator – Prey Model . Mathematical and Computer Modelling of Dynamical Systems , 9 : 387 – 401 .
- Krabs , W and Simon , R . “ Räuber-Beute-Verhalten in kleinräumigen Habitaten ” . Manuscript