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Global stability in n-dimensional discrete Lotka-Volterra predator-prey models
Advances in Difference Equations volume 2014, Article number: 11 (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.
Consider the n-dimensional system
where , for , , and for .
The system equation (1.1) can be seen as a generalization of the two-dimensional Lotka-Volterra predator-prey model
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
In this section, we consider the Euler difference scheme for equation (1.2)
where Δt is a time step size, , for , and
Note that if and are positive constants such that
Let Δt satisfy . Take positive constants and such that
Theorem 2.1 Let Δt, , satisfy equation (2.4) with . If , then for all k with .
it follows from equation (2.3) that
If , then , and otherwise,
by the condition . Hence equation (2.3) implies that
Similarly if , then , and otherwise,
by the condition . Thus equation (2.3) gives
Finally we obtain, if , then
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.
Remark 2.2 Theorem 2.1 gives for all k
Hence it follows from equation (2.3) that for every fixed
Let and . Since f and g are decreasing and increasing, respectively, it follows from equation (2.4) that for all k
Set , and let for denote the four areas
Remark 2.3 Let for some k. The following can be obtained by using equation (2.5), equation (2.6), and the definitions of and .
Suppose . Since , we have
Suppose . This gives and , and then
Suppose . Since , we have
Suppose , which means that and . Then
Therefore in , , , and moves to
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.
Theorem 2.4 Let the assumptions of Theorem 2.1 hold. Suppose .
3 Dynamics of the two-dimensional predator-prey models
In this section, we first consider dynamics of the Euler difference scheme for equation (1.2) and next for equation (1.1) with and . Let and . For calculating the limits of and , we use the inequalities
which is equivalent to .
Theorem 3.1 Let the assumptions of Theorem 2.1 hold. Suppose . Then satisfies the following dynamics with the limit .
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. □
Example 3.2 Consider the discrete system
with and . Then equation (2.4) and the conditions and in Theorem 3.1 are satisfied. For the four initial conditions
which implies that for all k.
In order to show the global asymptotic stability of the equilibrium point , the linearized system of equation (2.1) at θ is used: Consider the Jacobian matrix of at θ
Letting with the identity matrix I, we have , , and the eigenvalues of
The following lemma is used for showing that the equilibrium point θ of the nonlinear system is locally asymptotically stable.
Lemma 3.3 Suppose . Let . Then all of the eigenvalues of have magnitude less than 1 if one of the following is true.
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.
Theorem 3.5 Let the assumptions of Theorem 2.1 hold and let . Suppose and .
If , then rotates finitely many times around θ in the counterclockwise direction and finally stays in one of for with .
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.
(b) Since all of the eigenvalues of in equation (3.3) have positive real parts and magnitude less than 1 by , the fixed point θ is hyperbolic and the solutions of the linearized system of equation (2.1) at θ rotate infinitely many times around , converging to . Hence (b) is proved by using Theorem 3.4 and Theorem 2.4. It remains to show that . Consider a function defined by
for all the solutions . Letting , , and , we have and by equation (2.1). Then the Mean Value Theorem gives for some α, β with
where equation (3.4) gives
Then equation (3.7) becomes
where , , and
by equation (3.8). Hence equation (3.9) together with equation (3.5) becomes
for a positive constant .
Now assume, to the contrary, that does not converge to θ. Since θ is locally asymptotically stable by the linearization method (see ), the assumption implies that has a positive lower bound. Then there exists a positive constant such that for all k with
This is a contradiction, since and is bounded by Theorem 2.1. □
Example 3.6 Consider the discrete system
and the four initial conditions
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.
Example 3.7 Consider the discrete system
and the four initial conditions
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.
In the remainder of this section we consider the Euler difference scheme for equation (1.1) with and
which has the three nonzero equilibrium points
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.
Theorem 3.8 Let the assumptions of Theorem 2.1 hold with replaced by and let . Suppose . Then the solution of equation (3.10) satisfies the following dynamics with the limit .
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.
Example 3.10 Let . Consider the Euler difference scheme for equation (3.10) with .
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
In this section, we consider the Euler difference scheme for equation (1.1)
where denotes and
with and for .
Let denote . Note that if are positive constants such that for
Assume that there exist positive constants such that 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.
Proof It follows from equation (4.4) and the definition of that for
Then equation (4.3) with gives
If for , then
and otherwise, we have with .
Since by equation (4.6), it follows from equation (4.3) with that
where the last inequality is obtained by equation (4.5). Therefore we find that if , then
and hence mathematical induction completes the proof. □
From now on, we consider the global asymptotic stability of the equilibrium point of equation (4.1) whose components are all positive. Let if , and otherwise, . Assume that
where is the transpose of matrix .
which gives for
Then equation (4.7) is the generalization of the condition in both Lemma 3.3 and Theorem 3.5.
In order to use the linearized system of equation (4.1) at θ, consider the Jacobian matrix of at θ
Since the global stability of θ implies that θ is locally stable, we need to assume the condition for the local stability of θ:
Let for the identity matrix I and let be the characteristic polynomial of the matrix J. Note that are the eigenvalues of for the roots of . Thus equation (4.9) is equivalent to the condition
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 ).
Remark 4.3 Using Routh-Hurwitz criteria, we can find conditions for both and to satisfy equation (4.10). Let with real constants for . Define the n Hurwitz matrices for
Then equation (4.10) is equivalent to . Thus Routh-Hurwitz criteria for are
If we assume equation (4.9) with , then θ is asymptotically stable (see Remark 4.2). Hence the following lemma can be proved.
Lemma 4.4 Let , , Δt satisfy equations (4.4)-(4.7), (4.10), and (4.11) with . If , then there exists a positive constant C such that
We need to find constants that play the same role as and in equations (3.6), (3.7), and (3.9): If and , then
Hence we choose positive constants such that the matrices
are positive definite. Note that all of the eigenvalues of are positive if and only if
which is equivalent to
Hence consider the quadratic equation in the variable t
which has two different solutions denoted by and with for . Therefore in order that the eigenvalues of are positive, assume that there exist positive constants such that for
Lemma 4.5 Suppose that there exist positive constants () satisfying equation (4.15). Let be the minimum of the two positive eigenvalues of for in equation (4.13). Then for all
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 .
Proof Consider a function
for all the solutions of equation (4.1). The Mean Value Theorem with and equation (4.8) shows that there exist constants such that
where for , for , and
by equation (4.17). Then equation (4.19) with (4.21) becomes
It follows from equations (4.20) and (4.13) that
Hence substituting equations (4.24), (4.25), and Lemma 4.5 into equation (4.23) yields
Using equations (4.26), (4.22), and (4.18), we obtain , and then equation (4.27) becomes
for some positive constant . Now assume, to the contrary, that does not converge to θ as k goes to infinity. Then combining equation (4.28) with Lemma 4.4, we see that there exists a positive constant such that
This is a contradiction, since and is lower bounded for all k by using the boundedness of the solutions for . □
Example 4.7 Consider the three-dimensional scheme
which is equation (4.1) with , , , and . The initial condition is .
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 .
Hence the conditions in Theorem 4.6 are satisfied. Therefore the solutions of equation (4.29) are positive and the equilibrium point θ is globally asymptotically stable, which are demonstrated in Figure 3(a).
Example 4.8 Let . Consider the following three difference schemes for equation (1.1) with , , and .
, , and with the initial condition (see Figure 3(b)).
, , and with the initial condition (see Figure 3(c)).
, , and with the initial condition (see Figure 3(d)).
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.
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This work was supported by the 2013 Research Fund of University of Ulsan.
The author declares that he has no competing interests.
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Choo, S. Global stability in n-dimensional discrete Lotka-Volterra predator-prey models. Adv Differ Equ 2014, 11 (2014). https://doi.org/10.1186/1687-1847-2014-11
- Euler discrete schemes
- global stability
- predator-prey models