Global Dynamics of Discrete Competitive Models with Large Intrinsic Growth Rates

The global dynamics of discrete competitive model of Lotka-Volterra type with two species is considered. Earlier works have shown that the unique positive equilibrium is globally attractive under the assumption that the intrinsic growth rates of the two competitive species are less than 1 ln 2, and further the unique positive equilibrium is globally asymptotically stable under the stronger condition that the intrinsic growth rates of the two competitive species are less than or equal to 1. We prove that the system can also be globally asymptotically stable when the intrinsic growth rates of the two competitive species are greater than 1 and globally attractive when the intrinsic growth rates of the two competitive species are greater than 1 ln 2.


Introduction
In this paper, we further consider the global dynamics of discrete Lotka-Volterra model with positive initial conditions x 1 0 , x 2 0 > 0.Here x i n i 1, 2 is the density of population i at nth generation, r i i 1, 2 is the intrinsic growth rate of population i. a ij i, j 1, 2 represents the intensity of intraspecific competition or interspecific action of the two species.It is assumed that r i and a ij i, j 1, 2 are positive constants throughout this paper.
The discrete Lotka-Volterra models have wide applications in applied mathematics.They were first established in biomathematical background and then have proved to be a rich source in analysis for the dynamical systems in different research fields such as physics, chemistry, and economy 1 .
Model 1.1 was first introduced in May 2 then was investigated by many authors 3-14 .The difference system 1.1 is autonomous, some of the works mentioned above are the nonautonomous case of 1.1 .Many results about the global dynamics of 1.1 such as permanence, global attractivity, global asymptotical stability have been obtained.For example, it is shown in 10 that 1.1 can be globally asymptotically stable when r i ≤ 1 i 1, 2 .And from 3 we know that 1.1 can be globally attractive under the assumption that r i < 1 ln 2 i 1, 2 .
It is well known that for the single-species Logistic model x n 1 x n exp r − ax n , 1.2 the positive equilibrium x * r/a is globally asymptotically stable if and only if r ≤ 2 and there exists periodic cycles when r > 2. When r ≥ 3.13, 1.2 exhibits chaotic behavior e.g., see 15 .That is, the global dynamics of 1.2 is very complex when the intrinsic growth rate r is large.It is clear that 1.1 is a coupling of two equations described by 1.2 .And it is proved in 16 that 1.1 also exhibits chaotic behavior when r i r ≥ 3.13 i 1, 2 .Therefore, questions can be proposed naturally: how to investigate the global dynamics of 1.1 when 1 ln 2 < r i < 3.13 i 1, 2 ?Can model 1.1 be also globally asymptotically stable when r i > 1 i 1, 2 ?Can model 1.1 be globally attractive when r i > 1 ln 2 i 1, 2 ?Our aim of this paper is to obtain some global dynamics of 1.1 when the intrinsic growth rate r i i 1, 2 is large r i ≥ 1, i 1, 2 and give answers to the above questions.First we obtain permanent result of 1.1 , then global attractivity of 1.1 is obtained through geometrical properties of 1.1 .Last, we obtain the global asymptotical stability of 1.1 by applying a theorem in 10 .After these theoretical results for 1.1 obtained, we give numerical examples to confirm these theoretical results and show that our theoretical results imply that 1.1 can be globally attractive when r i > 1 ln 2 i 1, 2 and 1.1 can also be globally asymptotically stable when r i > 1 i 1, 2 .
The paper is organized as follows.We give some preliminaries in Section 2. In Section 3, we discuss permanence, global attractivity, and global asymptotical stability of 1.1 theoretically.Numerical examples are given in Section 4 to show the feasibility of the assumptions of the main results and on the other hand, to show that our main results can be applied to larger intrinsic growth rates than earlier works.Brief conclusion is given in Section 5.

Preliminaries
A pair of sequences of real positive numbers {x 1 n , x 2 n } ∞ n 1 that satisfies 1.1 is a positive solution of 1.1 .It is clear that the solutions of system 1.1 with initial values x 1 0 > 0, x 2 0 > 0 are positive ones, which is accordant with the biological background of 1.1 .That is, we only need to investigate the dynamics of system 1.1 in the plane domain Proof.Solving the following scalar equation system: We obtain that the four equilibria of system 1.1 are respectively.Here and the following, we denote

2.5
The equilibria 0, 0 , r 1 /a 11 , 0 and 0, r 2 /a 22 are the so-called "boundary equilibrium."If we further assume that which implies that D > 0, then x * 1 , x * 2 is the unique positive equilibrium of 1.1 .
Lemma 2.2.Denote f x, y x exp r 1 − a 11 x − a 12 y , r 1 ≥ 1, then the maximum M 1 of f x, y in the domain Proof.For any fixed x or y , let y → ∞ or x → ∞ we get f x, y → 0. Note that lim therefore, the maximum of f x, y in domain G 1 exists.Direct computation gives M 1 , we omit the details.Similarly, M 2 exists and its value can be obtained directly.
If F: R 0, ∞ → Ris monotonously decreasing, then for each positive sequence

2.11
Proof.One can refer to 17 for the proof of this lemma.
Next we give some definitions that will be used in this paper.
Definition 2.4.System 1.1 is permanent if there exist positive constants m and M such that The following lemma can be found in 10 .

2.14
Assume that i there exist positive constant ν and positive constants c i such that for all large n; ii system 2.14 is strongly persistent; iii for any positive solution {x 1 n , x 2 n , . . ., x l k } of system 2.14 , for all large n.
Then system 2.14 is globally asymptotically stable.

Main Results
In this section, we will obtain the permanence, global attractivity, and global asymptotical stability of system 1.1 when r i ≥ 1 i 1, 2 .
Lemma 3.1.For every positive solution {x 1 n , x 2 n } of system 1.1 with initial values x 1 0 > 0, Proof.Note that for all n, therefore Here we used max The proof of lim sup where

3.10
and s 1 , s 2 are the same as in Lemma 3.1.
Proof.The proof of this lemma is similar to that of 3, Proposition 2 .

3.24
At this point, we claim that It is easy to verify that the function h x x exp r − ax , a > 0 is monotonously increasing when 0 < x < 1/a and monotonously decreasing when x > 1/a.With this fact and Lemma 2.3, the proof of the claim is given as below.

Case (i)
We rearrange the two equations of 1.1 as

3.26
Note that we have x 1 n < 1/a 11 , x 2 n < 1/a 22 for n sufficiently large.Then

3.28
That is The inequalities 3.24 , 3.29 together with 3.30 imply that

3.31
From 3.13 and 3.24 , we get

Case (ii)
Similarly, we have

3.34
From 3.13 , 3.24 , and 3.33 , we get l 1 D 1 /D x * 1 .And from 3.13 , 3.24 , and 3.34 , l 2 L 2 D 2 /D x * 2 follows.The proof of Case iii is similar to that of Case ii .

Case (iv)
We have
The proof of claim 3.25 is completed.Note 3.24 and 3.25 , for any positive solution {x 1 n , x 2 n } of system 1.1 .That is, 1.1 is globally attractive according to Definition 2.7.
Theorem 3.5.Assume that the assumptions of Theorem 3.4 are satisfied, moreover, then the unique positive equilibrium of system 1.1 is globally asymptotically stable.
Proof.From Theorem 3.3, system 1.1 is strongly persistent.That is, condition ii of Lemma 2.9 is satisfied.
Thus, condition i of Lemma 2.9 is satisfied.
Let {x 1 n , x 2 n } be any positive solution of system 1.1 .We show below that for all large n.By Theorem 3.4, we know that

3.40
From 3.38 we first select ε > 0, such that Further from 3.40 , we know that there exists N 1 and N 2 , such that 3.42 respectively.Then denote N max{N 1 , N 2 }, we get for n ≥ N from 3.41 .That is, 3.39 is true for all sufficiently large n.Therefore, condition iii of Lemma 2.9 is satisfied.The proof is completed by applying Lemma 2.9.
Theorem 3.6.Assume that 2.6 , and 3.8 hold, the coefficients of 1.1 satisfy r i ≥ 1 i 1, 2 and then the positive equilibrium of system 1.1 is globally asymptotically stable.
Proof.From the proof of Theorem 3.4, we know that L 1 , L 2 lies in domain G 1 .Therefore, we obtain lim sup from Lemma 2.2.That is, condition iii of Lemma 2.9 is satisfied.Conditions i and ii of Lemma 2.9 are also satisfied.Then the positive equilibrium of system 1.1 is globally asymptotically stable by applying Lemma 2.9.

Numerical Examples
In this section, we give two numerical examples to show the feasibility of the assumptions of the results.The first example also shows that system 1.1 can be globally attractive when the intrinsic growth rates of the two species are greater than 1 ln 2 .1.6931, and this result can be obtained by Theorem 3.4.
Example 4.1.Consider the following case of system 1.1 :

4.2
We see that the conditions of Theorem 3.4 are satisfied.Therefore, the positive equilibrium of system 1.1 is globally attractive see Figure 1 .But this result cannot be obtained by 3, Theorem 3 when consider the autonomous case of this theorem the model studied in 3 is nonautonomous .In fact, the condition of 3, Theorem 3 must satisfy exp The following example shows that system 1.1 can be globally asymptotically stable when the intrinsic growth rates of the two species are greater than 1, and this result can be obtained by Theorem 3.6.
Example 4.2.Consider the following case of system 1.1 :

4.4
It is clear that the conditions of Theorem 3.6 are satisfied.Thus by Theorem 3.6 the positive equilibrium of system 1.1 is globally asymptotically stable see Figure 2 .
Example 4.2 shows that our results improve 12, Theorem 3 by providing estimates for the smallness of r 1 , r 2 .The work in 10, Theorem 2 states that if D 1 > 0, D 2 > 0, r i ≤ 1 i 1, 2 , then the positive equilibrium x * 1 , x * 2 is globally asymptotically stable.Thus the global asymptotical stability of system 1.1 in the case of Example 4.2 cannot be obtained by 10, Theorem 2 because of r i > 1 i 1, 2 .

Conclusion
In this paper, we further discuss the global dynamics of a discrete autonomous competitive model of Lotka-Volterra type.Sufficient conditions are obtained to guarantee the permanence, global attractivity, and global asymptotical stability of the system.These conditions are expressed by the coefficients of the model and can be easily verified.Numerical examples are also given to show the feasibility of the conditions.
Earlier works have shown that the system of this type can be globally attractive when the intrinsic growth rates of the two species are less than 1 ln 2 3 , for single-species system see 18 .It is shown in 10 that the system can be globally asymptotically stable when the intrinsic growth rates of the two species are less than 1.In 16 , it is shown that the system can exhibit chaotic behavior when the intrinsic growth rates of the two species are equal and greater than 3.13.But the global dynamics of the system is not clear enough when the intrinsic growth rates of the two species are greater than 1 and less than 3.13.We obtain that the system can also be globally asymptotically stable when the intrinsic growth rates of the two competitive species are greater than 1 and globally attractive when the intrinsic growth rates of the two competitive species are greater than 1 ln 2.
For the global stability of the system, the following condition in Theorem 3.5: And the above inequalities imply that r i i 1, 2 can be greater than 1 while the system is globally asymptotically stable.

1
If a solution of 1.1 is a pair of real constants {x 1 , x 2 }, then it is an equilibrium of 1.1 .
for all positive integers n, where {x 1 n , x 2 n } is the solution of 1.1 with initial values x 1 0 > 0, x 2 0 > 0.Definition 2.8.The positive equilibrium solution {x * 1 , x * 2 } of 1.1 or system 1.1 is said to be globally asymptotically stable if this equilibrium is stable and globally attractive.
0.3.21From3.18-3.21, it is clear that l 1 , l 2 lies in the domainG 2 x, y | x ≥ 0, y ≥ 0, r 1 − a 11 x − a 12 y ≥ 0,r 2 − a 21 x − a 22 y ≥ 0 , 3.22 while L 1 , L 2 lies in the domain G 1 see 2.7 .Therefore, from 3.11 and Lemma 2.2, the maximum of f x, y x exp r 1 −a 11 x−a 12 y in domain G 1 is D 1 /D, the maximum of g x, y y exp r 2 − a 21 x − a 22 y in domain G 1 is D 2 /D.Then