# Global Dynamics of Discrete Competitive Models with Large Intrinsic Growth Rates

Authors: Chunqing Wu; Jing-an Cui
Journal: Discrete Dynamics in Nature and Society
 (2009)

---

## Abstract

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.


---

## Body

## 1. Introduction

In this paper, we further consider the global dynamics of discrete Lotka-Volterra model(1.1)x1(n+1)=x1(n)exp(r1-a11x1(n)-a12x2(n)),x2(n+1)=x2(n)exp(r2-a21x1(n)-a22x2(n)),
with positive initial conditions x1(0),x2(0)>0. Here xi(n)(i=1,2) is the density of population i at nth generation, ri(i=1,2) is the intrinsic growth rate of population i. aij(i,j=1,2) represents the intensity of intraspecific competition or interspecific action of the two species. It is assumed that ri and aij(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 ri≤1(i=1,2). And from [3] we know that (1.1) can be globally attractive under the assumption that ri<1+ln2(i=1,2).It is well known that for the single-species Logistic model(1.2)x(n+1)=x(n)exp[r-ax(n)],
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 ri=r≥3.13(i=1,2). Therefore, questions can be proposed naturally: how to investigate the global dynamics of (1.1) when 1+ln2<ri<3.13(i=1,2)? Can model (1.1) be also globally asymptotically stable when ri>1(i=1,2)? Can model (1.1) be globally attractive when ri>1+ln2(i=1,2)?Our aim of this paper is to obtain some global dynamics of (1.1) when the intrinsic growth rate ri(i=1,2) is large (ri≥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 ri>1+ln2(i=1,2) and (1.1) can also be globally asymptotically stable when ri>1(i=1,2).The paper is organized as follows. We give some preliminaries in Section2. 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.


## 2. Preliminaries

A pair of sequences of real positive numbers{x1(n),x2(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 x1(0)>0,x2(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
(2.1)G={(x,y)∣x≥0,y≥0}.If a solution of (1.1) is a pair of real constants {x1,x2}, then it is an equilibrium of (1.1).Lemma 2.1.
Assume that(2.2)D≜a11a22-a12a21≠0,
then system (1.1) has four equilibria.Proof.
Solving the following scalar equation system:(2.3)x1=x1exp(r1-a11x1-a12x2),x2=x2exp(r2-a21x1-a22x2).
We obtain that the four equilibria of system (1.1) are
(2.4)(0,0),(r1a11,0),(0,r2a22),(x1*,x2*),
respectively. Here and the following, we denote
(2.5)x1*≜D1D,x2*≜D2D,D1≜r1a22-r2a12,D2≜r2a11-r1a21.The equilibria(0,0),(r1/a11,0) and (0,r2/a22) are the so-called “boundary equilibrium.” If we further assume that
(2.6)D1>0,D2>0,
which implies that D>0, then (x1*,x2*) is the unique positive equilibrium of (1.1).Lemma 2.2.
Denotef(x,y)=xexp(r1-a11x-a12y), r1≥1, then the maximum M̂1 of f(x,y) in the domain
(2.7)G1≜{(x,y)∣x≥0,y≥0,r1-a11x-a12y≤0,r2-a21x-a22y≤0}
is (1)
ifa22≤D1ora22/D≥r1/a21,thenM̂1=max{(r2/a21)exp(r1-a11(r2/a21)),D1/D},(2)
ifD1/D<a22/D<r1/a21,thenM̂1=(a22/D)exp((D1/a22)-1).Denote g(x,y)=yexp(r2-a21x-a22y), r2≥1, then the maximum M̂2 of g(x,y) in domain G1 is(1)
ifa11≤D2ora11/D≥r2/a12,thenM̂2=max{(r1/a12)exp(r2-a22(r1/a12)),D2/D},(2)
ifD2/D<a11/D<r2/a12,thenM̂2=(a11/D)exp((D2/a11)-1).Proof.
For any fixedx(ory), let y→+∞(orx→+∞) we get f(x,y)→0. Note that
(2.8)limx→+∞,y→+∞f(x,y)=0,
therefore, the maximum of f(x,y) in domain G1 exists. Direct computation gives M̂1, we omit the details. Similarly, M̂2 exists and its value can be obtained directly.Lemma 2.3.
(1) IfF: R+=[0,+∞)→R  is monotonously increasing, then for each positive sequence {y(n)}n=1∞,
(2.9)lim supn→∞F(y(n))=F(lim supn→∞y(n)),lim infn→∞F(y(n))=F(lim infn→∞y(n)).
If F: R+=[0,+∞)→Ris monotonously decreasing, then for each positive sequence {y(n)}n=1∞,
(2.10)lim supn→∞F(y(n))=F(lim infn→∞y(n)),lim infn→∞F(y(n))=F(lim supn→∞y(n)).
(2) For any positive sequences{y(n)}n=1∞,{z(n)}n=1∞ one has
(2.11)lim supn→∞[y(n)z(n)]≤lim supn→∞y(n)lim supn→∞z(n),lim infn→∞[y(n)z(n)]≥lim infn→∞y(n)lim infn→∞z(n).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
(2.12)m≤lim infn→∞xi(n)≤lim supn→∞xi(n)≤M,i=1,2.Definition 2.5.
System (1.1) is strongly persistent if each positive solution of (1.1) satisfies
(2.13)lim infn→∞xi(n)>0,i=1,2.Definition 2.6.
The solution{x1(n),x2(n)} of system (1.1) with initial values x1(0)>0, x2(0)>0 is said to be stable if for any ε>0, there is a δ>0 such that if max{|x1(0)-x̅1(0)|,|x2(0)-x̅2(0)|}<δ, we have |x1(n)-x̅1(n)|+|x2(n)-x̅2(n)|<ε for all positive integers n, where {x̅1(n),x̅2(n)} is the solution of (1.1) with initial values x̅1(0)>0,x̅2(0)>0.Definition 2.7.
Suppose that{x1*,x2*} is the positive equilibrium solution of (1.1). If for each positive solution {x̅1(n),x̅2(n)} of system (1.1), we have max{|x1*-x̅1(n)|,|x2*-x̅2(n)|}→0 as n→∞, we say (1.1) is globally attractive or the equilibrium {x1*,x2*} of (1.1) is globally  attractive.Definition 2.8.
The positive equilibrium solution{x1*,x2*} of (1.1) or system (1.1) is said to be globally asymptotically stable if this equilibrium is stable and globally attractive.
The following lemma can be found in [10].Lemma 2.9.
Consider the following difference system:(2.14)xi(n+1)=xi(n)exp(ri(n)-∑j=1naij(n)xj(n)),i=1,2,…,l.
Assume that (i)
there exist positive constantν and positive constants  ci such that
(2.15)ciaii(n)>∑j=1,j≠ilcj|aji(n)|+ν,i=1,2,…,l,
for all large  n;(ii)
system (2.14) is strongly persistent;(iii)
for any positive solution{x1(n),x2(n),…,xl(k)} of system (2.14),
(2.16)aii(n)xi(n)≤1,i=1,2,…,l,
for all largen.
Then system (2.14) is globally asymptotically stable.


## 3. Main Results

In this section, we will obtain the permanence, global attractivity, and global asymptotical stability of system (1.1) when ri≥1(i=1,2).Lemma 3.1.
For every positive solution{x1(n),x2(n)} of system (1.1) with initial values x1(0)>0,x2(0)>0, one has
(3.1)lim supn→∞x1(n)≤s1,lim supn→∞x2(n)≤s2,
where
(3.2)s1=1a11exp(r1-1),s2=1a22exp(r2-1).Proof.
Note that(3.3)x1(n+1)=x1(n)exp(r1-a11x1(n)-a12x2(n))=x1(n)exp(r1-a11x1(n))exp(-a12x2(n)),exp(-a12x2(n))≤1
for all n, therefore
(3.4)x1(n+1)≤x1(n)exp(r1-a11x1(n))≤1a11exp(r1-1).
Here we used
(3.5)maxx≥0xexp(r-ax)=1aexp(r-1)
for a>0. Then
(3.6)lim supn→∞x1(n)≤1a11exp(r1-1).
The proof of
(3.7)lim supn→∞x2(n)≤1a22exp(r2-1)
is similar.Lemma 3.2.
Assume that{x1(n),x2(n)} is the solution of (1.1) with initial values x1(0)>0,x2(0)>0  and
(3.8)1a11exp(r1-1)<r2a21,1a22exp(r2-1)<r1a12,
then
(3.9)lim infn→∞x1(n)≥t1>0,lim infn→∞x2(n)≥t2>0,
where
(3.10)t1=r1a11(1-a12r1s2)exp(r1-a12s2-a11r1s1),t2=r2a22(1-a21r2s1)exp(r2-a21s1-a22r2s2),
and s1,s2 are the same as in Lemma 3.1.Proof.
The proof of this lemma is similar to that of [3, Proposition 2].Note thatt1>0,t2>0, therefore, system (1.1) is permanent from Lemmas 3.1 and 3.2 under the assumption (3.8).Theorem 3.3.
Assume that (3.8) is satisfied then system (1.1) with initial values x1(0)>0,x2(0)>0  is permanent.Theorem 3.4.
Assume that (2.6), and (3.8) hold. The coefficients of (1.1) satisfy ri≥1(i=1,2)  and (1)
a22≤D1ora22/D≥r1/a21,(2)
a11≤D2ora11/D≥r2/a12.Further, assume that
(3.11)M̂1≤D1D≠1a11,M̂2≤D2D≠1a22,
where M̂1  and M̂2  are defined in  Lemma 2.2. Then the unique positive equilibrium (x1*,x2*) of (1.1) is globally attractive.Proof.
If we denote(3.12)l1=lim infn→∞x1(n),l2=lim infn→∞x2(n),L1=lim supn→∞x1(n),L2=lim supn→∞x2(n)
for any positive solution {x1(n),x2(n)} of system (1.1) with initial conditions x1(0)>0,x2(0)>0, we have
(3.13)0<l1≤L1<+∞,0<l2≤L2<+∞
from Theorem 3.3 and Definition 2.4. Moreover,
(3.14)l1≥l1exp(r1-a11L1-a12L2),(3.15)l2≥l2exp(r2-a21L1-a22L2),(3.16)L1≤L1exp(r1-a11l1-a12l2),(3.17)L2≤L2exp(r2-a21l1-a22l2)
from (2) of Lemma 2.3.
Note (3.13), the inequalities (3.14)–(3.17) can be written as follows:
(3.18)r1-a11l1-a12l2≥0,(3.19)r2-a21l1-a22l2≥0,(3.20)r1-a11L1-a12L2≤0,(3.21)r2-a21L1-a22L2≤0.
From (3.18)–(3.21), it is clear that (l1,l2) lies in the domain
(3.22)G2={(x,y)∣x≥0,y≥0,r1-a11x-a12y≥0,r2-a21x-a22y≥0},
while (L1,L2) lies in the domain  G1  (see (2.7). Therefore, from (3.11) and Lemma 2.2, the maximum of f(x,y)=xexp(r1-a11x-a12y) in domain G1 is D1/D, the maximum of g(x,y)=yexp(r2-a21x-a22y) in domain G1 is D2/D. Then
(3.23)L1≤D1D,L2≤D2D.
But in domain G1, only the point (x1*,x2*)=(D1/D,D2/D) satisfies these two inequalities, then
(3.24)L1=D1D=x1*,L2=D2D=x2*.
At this point, we claim that(3.25)l1=D1D=x1*,l2=D2D=x2*.
Note (3.11), we must consider the following four cases to prove claim (3.25):Case (i):
D1/D<1/a11,D2/D<1/a22,Case (ii):
D1/D<1/a11,D2/D>1/a22,Case (iii):
D1/D>1/a11,D2/D<1/a22,Case (iv):
D1/D>1/a11,D2/D>1/a22.
It is easy to verify that the functionh(x)=xexp(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)x1(n+1)=x1(n)exp(r1-a11x1(n))exp(-a12x2(n)),x2(n+1)=x2(n)exp(r2-a22x2(n))exp(-a21x1(n)).
Note that
(3.27)L1=x1*=D1D<1a11,L2=x2*=D2D<1a22,
we have x1(n)<1/a11,x2(n)<1/a22 for n sufficiently large. Then
(3.28)l1≥l1exp(r1-a11l1-a12L2),l2≥l2exp(r2-a21L1-a22l2).
That is
(3.29)r1-a11l1-a12L2≤0,(3.30)r2-a21L1-a22l2≤0.
The inequalities (3.24), (3.29) together with (3.30) imply that
(3.31)l1≥D1D=x1*,l2≥D2D=x2*.
From (3.13) and (3.24), we get
(3.32)l1=D1D=x1*,l2=D2D=x2*.Case (ii)
Similarly, we have(3.33)l1≥l1exp(r1-a11l1-a12L2),(3.34)l2≥L2exp(r2-a21L1-a22L2).
From (3.13), (3.24), and (3.33), we get l1=D1/D=x1*. And from (3.13), (3.24), and (3.34), l2=L2=D2/D=x2* follows.
The proof of Case (iii) is similar to that of Case (ii).Case (iv)
We have(3.35)l1≥L1exp(r1-a11L1-a12L2),l2≥L2exp(r2-a21L1-a22L2).
Therefore,
(3.36)l1=D1D=x1*,l2=D2D=x2*
are consequent from (3.13), (3.24), and (3.35).
The proof of claim (3.25) is completed. Note (3.24) and (3.25),
(3.37)limn→∞x1(n)=x1*,limn→∞x2(n)=x2*
for any positive solution {x1(n),x2(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 Theorem3.4 are satisfied, moreover,
(3.38)D1D<1a11,D2D<1a22,
then the unique positive equilibrium of system (1.1) is globally asymptotically stable.Proof.
From Theorem3.3, system (1.1) is strongly persistent. That is, condition (ii) of Lemma 2.9 is satisfied.
Di>0,i,j=1,2 implies that r1a22-r2a12>0,r2a11-r1a21>0. Set c1=r2,c2=r1, it is clear that c1a11>c2a21,c2a22>c1a12. Thus, condition (i) of Lemma 2.9 is satisfied.
Let{x1(n),x2(n)} be any positive solution of system (1.1). We show below that
(3.39)a11x1(n+1)≤1,a22x2(n+1)≤1
for all large n. By Theorem 3.4, we know that (x1*,x2*)=(D1/D,D2/D) is globally attractive. That is
(3.40)limn→∞x1(n)=D1D,limn→∞x2(n)=D2D.
From (3.38) we first select ε>0, such that
(3.41)D1D+ε<1a11,D2D+ε<1a22.
Further from (3.40), we know that there exists N1 and N2, such that
(3.42)x1(n+1)<D1D+ε,forn≥N1,x2(n+1)<D2D+ε,forn≥N2,
respectively. Then denote N=max{N1,N2}, we get
(3.43)x1(n+1)<D1D+ε<1a11,x2(n+1)<D2D+ε<1a22
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 ri≥1(i=1,2) and
(3.44)D1D<a22D<r1a21,D2D<a11D<r2a12,(3.45)a22Dexp(D1a22-1)<1a11,a11Dexp(D2a11-1)<1a22,
then the positive equilibrium of system (1.1) is globally asymptotically stable.Proof.
From the proof of Theorem3.4, we know that (L1,L2) lies in domain G1. Therefore, we obtain
(3.46)lim supn→∞x1(n)≤a22Dexp(D1a22-1)<1a11,lim supn→∞x2(n)≤a11Dexp(D2a11-1)<1a22
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.


## 4. 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+ln2≐1.6931, and this result can be obtained by Theorem 3.4.Example 4.1.
Consider the following case of system (1.1):
(4.1)r1=1.95,r2=1.8,a11=0.5,a12=0.1,a22=0.5,a21=0.09,
then
(4.2)D1=0.7950,D2=0.7245,D1D=x1*=3.2988,D2D=x2*=3.0062,1a11exp(r1-1)=5.1714,1a22exp(r2-1)=4.4511,r2a21=20.0000,r1a12=19.5000,1a11=1a22=2.0000,r2a21exp(r1-a11r2a21)=0.0064,r1a12exp(r2-a22r1a12)=0.0069.
We see that the conditions of Theorem3.4 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(ri-1)-1<1(i=1,2) when ri>1, that is, ri<1+ln2≐1.6931(i=1,2). In Example 4.1, ri>1+ln2(i=1,2).
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.Figure 1
Solutions of system (1.1) with initial values (x1(0),x2(0))=(10,9), (1,10), (0.01,0.09), (9,0.1), and r1=1.95,r2=1.8.Example 4.2.
Consider the following case of system (1.1):
(4.3)r1=1.1,r2=1.2,a11=0.05,a12=0.02,a22=0.06,a21=0.015,
then
(4.4)D1=0.0420,D2=0.0435,x1*=15.5556,x2*=16.1111,1a11exp(r1-1)=22.1034,1a22exp(r2-1)=20.3567,r2a21=80.0000,r1a12=55.0000,r2a21exp(r1-a11r2a21)=4.4019,r1a12exp(r2-a22r1a12)=6.7351,a22Dexp(D1a22-1)=16.4626,a11Dexp(D2a11-1)=16.2610,1a11=20.0000,1a22=16.6667.
It is clear that the conditions of Theorem3.6 are satisfied. Thus by Theorem 3.6 the positive equilibrium of system (1.1) is globally asymptotically stable (see Figure 2).
Example4.2 shows that our results improve [12, Theorem 3] by providing estimates for the smallness of  r1,r2. The work in [10, Theorem 2] states that if D1>0,D2>0,ri≤1(i=1,2), then the positive equilibrium (x1*,x2*) 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 ri>1(i=1,2).Figure 2
Solutions of system (1.1) with initial values(x1(0),x2(0))=(50,45),(1,47),(0.1,0.09),(48,0.1), and r1=1.1,r2=1.2.


## 5. 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 than1+ln2 ([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+ln2.For the global stability of the system, the following condition in Theorem3.5:
(5.1)D1D<1a11,D2D<1a22,
can be reduced to the following by direct computation:
(5.2)r1<1+a12a22,r2<1+a21a11.
And the above inequalities imply that ri(i=1,2) can be greater than 1 while the system is globally asymptotically stable.



---

*Source: 710353-2009-08-06.xml*