Global stability in n-dimensional discrete Lotka-Volterra predator-prey models
© Choo; licensee Springer. 2014
Received: 31 October 2013
Accepted: 16 December 2013
Published: 9 January 2014
There are few theoretical works on global stability of Euler difference schemes for two-dimensional Lotka-Volterra predator-prey models. Furthermore no attempt is made to show that the Euler schemes have positive solutions. In this paper, we consider Euler difference schemes for both the two-dimensional models and n-dimensional models that are a generalization of the two-dimensional models. It is first shown that the difference schemes have positive solutions and equilibrium points which are globally asymptotically stable in the two-dimensional cases. The approaches used in the two-dimensional models are extended to the n-dimensional models for obtaining the positivity and the global stability. Numerical examples are presented to verify the results.
MSC: 34A34, 39A10, 40A05.
where , for , , and for .
where x and y denote the population sizes of prey and predator, respectively.
There are a number of works on investigating nonstandard finite difference schemes for the Lotka-Volterra competition models (see  and the references given there), but relatively few theoretical papers are published on discretized models of equation (1.2). In particular, to my knowledge, Euler difference schemes for equation (1.2) have not theoretically been studied for the global stability of the equilibrium points except a recent paper . In Section 2, it is shown that the Euler difference scheme has positive solutions. In order to show the global asymptotic stability of the equilibrium point whose components are all positive, the paper  assumes that is globally stable. Without using the assumption, we show the global stability of all of the equilibrium points in Section 3. In addition, we also analyze the Euler difference scheme for equation (1.2) with replaced by . We are interested in extending the method used in the two-dimensional discrete models to the n-dimensional discrete models for equation (1.1). In Section 4, we demonstrate the positivity and the global stability in the n-dimensional discrete cases. Numerical examples are given to verify the results of this paper.
2 Two-dimensional predator-prey model
Theorem 2.1 Let Δt, , satisfy equation (2.4) with . If , then for all k with .
By the principle of mathematical induction, the proof is completed. □
From now on, we assume that and for denote the solutions of equation (2.1). For simplicity of notation, we write for all k instead of for all k with when there is no confusion.
- (a)Suppose . Since , we have
- (b)Suppose . This gives and , and then
- (c)Suppose . Since , we have
- (d)Suppose , which means that and . Then
Set , and use the notation for and a positive integer N to denote both for all k with and . Then equation (2.7) implies the following theorem.
3 Dynamics of the two-dimensional predator-prey models
which is equivalent to .
If , then for all k.
If , then for all k or for some .
If , then for some .
Proof (a) Since , the set is empty, and then Theorem 2.4 gives for all k. Thus equation (2.7) shows that is bounded and increasing, and is bounded and decreasing. Hence and . Finally since otherwise , which is a contradiction. Therefore by using equation (2.1) with both and .
(b) Theorem 2.4 and (a) in this theorem show that it suffices to show in the case for all k, where and by equation (2.7). Then since otherwise the last equation in equation (2.1) gives , and hence equation (3.1) implies , producing a contradiction to .
Note that for all k since for all k. Thus if , then , which contradicts to . Hence . Consequently, by using equation (2.1) with .
(c) Assume, to the contrary, that for all k. Then equation (2.7) implies that and have positive limits, and hence equation (3.1) gives , which is a contradiction. Therefore for some , which gives by (b) in this theorem. □
which implies that for all k.
The following lemma is used for showing that the equilibrium point θ of the nonlinear system is locally asymptotically stable.
In order to find conditions for when rotates finitely or infinitely many times around θ, we need the following theorem about a hyperbolic point: A point p of is called hyperbolic if all of the eigenvalues of have nonzero real parts.
Theorem 3.4 (Hartman-Grobman theorem for maps)
Let p be a hyperbolic fixed point of , where H is a continuously differentiable function defined on a neighborhood of for . Then there exist neighborhoods U of p, V of the hyperbolic fixed point 0 of , and a homeomorphism such that for all .
Theorem 3.4 (see  for the proof) states that the nonlinear system is topologically equivalent to the linearized system of the nonlinear system at p.
If , then rotates finitely many times around θ in the counterclockwise direction and finally stays in one of for with .
- (b)If the four inequalities , ,(3.4)
are satisfied, then rotates infinitely many times around θ in the counterclockwise direction with .
Proof (a) Since all of the eigenvalues of in equation (3.3) are positive numbers less than 1, the fixed point θ of is hyperbolic and the solutions of the linearized system of equation (2.1) at θ rotate finitely many times around , converging to . Hence Theorem 3.4 with Theorem 2.4 gives the proof of (a) without showing .
For obtaining the limit of consider the case for and all sufficiently large k. Then equation (2.7) yields the result that is increasing with the upper bound less than , and hence . Consequently , since otherwise , which is a contradiction. Therefore we obtain equation (3.1).
In the case for and all sufficiently large k, it follows from equation (2.7) that and are both increasing and bounded, which gives equation (3.1).
Similarly, the other two cases for can be proved by using equation (3.1), equation (2.7), and the method of proof by contradiction.
for a positive constant .
This is a contradiction, since and is bounded by Theorem 2.1. □
with and . Then , , and . Since and , the condition in Theorem 3.5(a) is satisfied. Figure 1(b) shows the dynamics in Theorem 3.5(a) with the limit θ.
with and . Then , , and . Since and , the conditions in Theorem 3.5(b) are satisfied. Figure 1(c) shows the dynamics in Theorem 3.5(b) with the limit θ.
Replace in Section 2 with . For example, is replaced with . Then Theorem 2.1, equation (2.7), and Theorem 2.4 remain true. In the case , the set is empty, and hence we can prove the following theorem, which corresponds to Theorem 3.1.
If , then for all k.
If , then for all k or for some .
If , then for some .
Remark 3.9 The Jacobian matrix of equation (3.10) at ϑ equals as defined in equation (3.2), and then Theorem 3.5 remains true if , , and θ in Theorem 3.5 are replaced with , , and ϑ, respectively. Therefore only the two equilibrium points and ϑ of equation (3.10) are globally asymptotically stable.
with the three initial conditions , , and . Then the conditions in Theorem 3.8 are satisfied. Figure 2(a) shows that the three trajectories converge to as in Theorem 3.8. Replacing the values in E with and letting , we have for and , which imply for all k.
with the four initial conditions , , , and . Then and . Since and , the condition in Theorem 3.5(a) is satisfied. Figure 2(b) shows that solutions rotate finitely many times around the limit ϑ.
with the four initial conditions , , , and . Then and . Since and , the conditions in Theorem 3.5(b) are satisfied. Figure 2(c) shows that the spiral trajectories rotate infinitely many times around the limit ϑ.
4 n-Dimensional predator-prey models
with and for .
which is the generalization of equation (2.4).
Theorem 4.1 Let Δt and satisfy equations (4.4)-(4.6). If , then for all k.
and otherwise, we have with .
and hence mathematical induction completes the proof. □
where is the transpose of matrix .
Then equation (4.7) is the generalization of the condition in both Lemma 3.3 and Theorem 3.5.
where , and are the real and imaginary parts of , respectively. Hence equation (4.10) together with equation (4.11) is the generalization of Lemma 3.3.
Remark 4.2 The condition equation (4.9) with implies that θ of equation (4.1) is asymptotically stable by using Theorem 3.4 with the following two facts: first, θ is a hyperbolic point since all of the eigenvalues of are of the form , which is nonzero. Second, equation (4.9) is equivalent to the fact that the linearized system of equation (4.1) is asymptotically stable (see Theorem 3.3.20 in ).
If we assume equation (4.9) with , then θ is asymptotically stable (see Remark 4.2). Hence the following lemma can be proved.
which are the generalization of equations (3.4) and (3.5), respectively.
Theorem 4.6 Let the assumptions in Lemma 4.4 hold and let equations (4.15), (4.17), and (4.18) be satisfied. If , then .
This is a contradiction, since and is lower bounded for all k by using the boundedness of the solutions for . □
Let , , and . Then equations (4.4), (4.5), and (4.6) are satisfied.
The condition (4.7) is satisfied and .
The conditions (4.10), (4.11), and (4.12) are satisfied, since with the three roots , −0.330, and .
The values satisfy equation (4.15), since the three equations () have solutions , , and , respectively.
The inequalities (4.17) and (4.18) are satisfied, since , , and .
Set , , , , and for the three difference schemes. The points in (a), (b), and (c) are , , and , respectively. Hence the conditions in Theorem 4.6 are satisfied. Consequently, the solutions of the three difference schemes are positive and the equilibrium point θ is globally asymptotically stable, which are demonstrated in Figure 3(b), (c), and (d).
Remark 4.9 In the n-dimensional cases, we only consider the equilibrium point which components are all positive. Thus a future study is to investigate dynamics on the other equilibrium points with some zero components.
This work was supported by the 2013 Research Fund of University of Ulsan.
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