# Permanence and Global Stability of Positive Periodic Solutions of a Discrete Competitive System

Authors: Wenjie Qin; Zhijun Liu; Yiping Chen
Journal: Discrete Dynamics in Nature and Society
 (2009)

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## Abstract

We consider the dynamic behaviors of a discrete
competitive system. A good understanding of the permanence, existence,
and global stability of positive periodic solutions is gained. Numerical
simulations are also presented to substantiate the analytical results.


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## Body

## 1. Introduction

In biomathematics, one of the most challenging aspects of mathematical biology is competition modeling. Although the mathematical idea is simple [1], this type of modeling is so difficult to carry out in any generality since there are so many ways for a population to compete; many classical competitive models have been established to describe the relationships between species and the outer environment, and the connections between different species. Lotka-Volterra competitive model of two species communities is probably the best known model of mathematical ecology. In 1934, Gause [2] found out the competitive exclusion theory, which states that two species that compete for the exact same resources cannot stably coexist. One of the two competitors will always have an ever so slight advantage over the other that leads to extinction of the second competitor in the long run. Since then the competitive model has increasingly won attention as an important and fundament model in biomathematics. The dynamic relationship between species and their competitors has long been one of the dominant theses in both ecology and mathematical ecology. As a consequence, many excellent results concerned with permanence, extinction, stability and hopf bifurcations, and existence and global stability of positive periodic solutions of Lotka-Volterra competitive system are obtained (see [3–16]).Although much progress has been seen for Lotka-Volterra competitive systems, such systems are not well studied in the sense that most results are continuous time versions related. Many authors [17–21] have argued that the discrete-time models governed by difference equations are more appropriate than the continuous ones when populations have a short life expectancy, nonoverlapping generations in the real word. Discrete-time models can provide efficient computational models of continuous models for numerical simulations. So it is reasonable to study discrete-time competitive systems governed by difference equations.In this paper, we will consider the dynamic behavior of a discrete-time competitive system. Let us first introduce its continuous time version which is motivated in [22]
(1.1)ẋ1(t)=x1(t)[r1(t)-a1(t)x1(t)-c2(t)x2(t)1+x2(t)],ẋ2(t)=x2(t)[r2(t)-a2(t)x2(t)-c1(t)x1(t)1+x1(t)],
where x1(t),x2(t) are the population densities of two competing species; r1(t),r2(t) are the intrinsic growth rates of species; a1(t),a2(t) are the rates of intraspecific competition of the first and second species, respectively; c1(t),c2(t) are the rates of interspecific competition of the first and second species, respectively. All the coefficients above are continuous and bounded above and below by positive constants.Following the same idea and method in [21], one can easily derive the discrete analogue of system (1.1), which takes the form of
(1.2)x1(n+1)=x1(n)exp[r1(n)-a1(n)x1(n)-c2(n)x2(n)1+x2(n)],x2(n+1)=x2(n)exp[r2(n)-a2(n)x2(n)-c1(n)x1(n)1+x1(n)],n=0,1,2….
The exponential form of system (1.2) is more biologically reasonable than that directly derived by replacing the differential by difference in system (1.1) because this exponential form can assure xi(n+1)>0 if xi(0)>0(i=1,2). Here xi(n) represent the densities of species xi at the nth generation, ri(n) are the intrinsic growth rates of species xi at the nth generation, ai(n) measure the intraspecific effects of the nth generation of species xi on own population, and ci(n) stand for the interspecific effects of the nth generation of species xi on species xj(i,j=1,2;i≠j).It is well known that, compared to the continuous time systems, the discrete-time ones are more difficult to deal with. The principle aim of this paper is to explore the permanence, existence, and global stability of positive periodic   solutions of system (1.2). To the best of our knowledge, no work has been done for system (1.2).For the sake of simplicity and convenience in the following discussion, the notations below will be used through this paper:(1.3)fU=supn∈ℕf(n),fL=infn∈ℕf(n),
where {f(n)} is a bounded sequence and ℕ is the set of nonnegative integer numbers.For biological reasons, in system (1.2) we only consider the solution {x1(n),x2(n)} with the initial value {x1(0),x2(0)}>0.The organization of this paper is as follows. In the next section, we establish the permanence of system (1.2). In Section 3, we obtain sufficient conditions which ensure the existence and global stability of positive periodic solutions of system (1.2). Numerical simulations are present to illustrate the feasibility of our main results in final section.


## 2. Permanence

In this section, we will establish sufficient conditions for the permanence of system (1.2).Definition 2.1.
System (1.2) is said to be permanent if there exist positive constants mi and Mi such that each positive solution {x1(n),x2(n)} of system (1.2) satisfies
(2.1)mi≤lim infn→+∞xi(n)≤lim supn→+∞xi(n)≤Mi,i=1,2.Proposition 2.2.
Any positive solution{x1(n),x2(n)} of system  (1.2) satisfies
(2.2)lim supn→+∞xi(n)≤Mi=def[exp(riU-1)]aiL,i=1,2.Proof.
To prove Proposition2.2, we consider Case 1 and Case 2.
Case 1.
Assume that there exists ann0∈ℕ such that x1(n0+1)≥x1(n0), from the first equation of system (1.2), it follows that
(2.3)r1(n0)-a1(n0)x1(n0)-c2(n0)x2(n0)1+x2(n0)≥0,
which implies,
(2.4)x1(n0)≤r1(n0)a1(n0)≤r1Ua1L.
Then
(2.5)x1(n0+1)=x1(n0)exp[r1(n0)-a1(n0)x1(n0)-c2(n0)x2(n0)1+x2(n0)]≤x1(n0)exp[r1U-a1Lx1(n0)]≤[exp(r1U-1)]a1L=defM1,
where we use the fact that maxx∈ℝ[xexp(a-bx)]=[exp(a-1)]/bfora,b>0, and ℝ is the set of all real numbers. Hence x1(n0)≤M1.
We claim thatx1(n)≤M1 for all n≥n0. By way of contradiction, assume that there exists a p0>n0 such that x1(p0)>M1, then p0≥n0+2. Let
(2.6)p̂0=min{p0:p0≥n0+2,x1(p0)>M1},
that is, x1(p̂0)>M1 and p̂0≥n0+2, then x1(p̂0)>M1≥x1(p̂0-1). It is easy to obtain that x1(p̂0)≤M1 from the above argument, which is a contradiction. Therefore, x1(n)≤M1 for all n≥n0, then limsupn→+∞x1(n)≤M1. This proves the claim.
Case 2.
Suppose thatx1(n+1)<x1(n) for all n∈ℕ. In particular, limn→+∞x1(n) exists, denoted by x̅1, we will prove x̅1≤r1U/a1L by way of contradiction as follows. Assume that x̅1>r1U/a1L, taking limit in the first equation of system (1.2), which leads to
(2.7)limn→+∞[r1(n)-a1(n)x1(n)-c2(n)x2(n)1+x2(n)]=0,
however,
(2.8)limn→+∞[r1(n)-a1(n)x1(n)-c2(n)x2(n)1+x2(n)]≤limn→+∞[r1(n)-a1(n)x1(n)]≤r1U-a1Lx̅1<0,
which is a contradiction. This proves the claim. By the fact that minx∈ℝ+{[exp(x-1)]/x}=1, we obtain that x̅1≤r1U/a1L≤r1U/a1L·exp(r1U-1)/r1U=defM1. Therefore,
(2.9)lim supn→+∞x1(n)≤M1=[exp(r1U-1)]a1L.
Analogously,
(2.10)lim supn→+∞x2(n)≤M2=[exp(r2U-1)]a2L.
This completes the proof of Proposition 2.2.Proposition 2.3.
Suppose that system (1.2) satisfies the following assumptions:
(2.11)r1L>c2U,r2L>c1U.
Then any positive solution {x1(n),x2(n)} of system  (1.2) satisfies
(2.12)lim infn→+∞xi(n)≥mi=defriL-cjUaiUexp(riL-aiUMi-cjU),i≠j;i,j=1,2.Proof.
By Proposition2.2, since limsupn→+∞x1(n)≤M1 for each ε>0, then there exists an n*∈ℕ such that x1(n)≤M1+ε for n≥n*. In order to prove Proposition 2.3, there are two cases to be considered as follows.
Case 1.
Assume that there exists ann0≥n* such that x1(n0+1)≤x1(n0), by the first equation of system (1.2), it derives that
(2.13)x1(n0+1)=x1(n0)exp[r1(n0)-a1(n0)x1(n0)-c2(n0)x2(n0)1+x2(n0)]≥x1(n0)exp[r1L-a1Ux(n0)-c2U],
therefore,
(2.14)r1L-a1Ux(n0)-c2U≤0.
It follows from the inequality (2.11) that
(2.15)x1(n0)≥(r1L-c2U)a1U>0.
By (2.13) and (2.15) we have
(2.16)x1(n0+1)=x1(n0)exp[r1(n0)-a1(n0)x1(n0)-c2(n0)x2(n0)1+x2(n0)]≥r1L-c2Ua1Uexp[r1L-a1U(M1+ε)-c2U]>0.
Hence x1(n0)≥x1ε, where x1ε=def((r1L-c2U)/a1U)exp[r1L-a1U(M1+ε)-c2U].
In the following we will provex1(n)≥x1ε for all n≥n0. By way of contradiction, assume that there exists a p0>n0 such that x1(p0)<x1ε, then p0≥n0+2. Let
(2.17)p̃0=min{p0:p0≥n0+2,x1(p0)<x1ε},
that is, x1(p̃0)<x1ε and p̃0≥n0+2, then x1(p̃0)<x1ε≤x1(p̃0-1), the above argument produces that x1(p̃0)≥x1ε, which is a contradiction. Therefore, x1(n)≥x1εforalln≥n0, since ε can be sufficiently small, it gives that
(2.18)x1(n)≥r1L-c2Ua1Uexp(r1L-a1UM1-c2U)=defm1>0,
then liminfn→+∞x1(n)≥m1. This proves the claim.
Case 2.
Assume thatx1(n+1)>x1(n) for a sufficiently large n≥n*. In this case, limn→+∞x1(n) exists, denoted by x̲1. For the sake of proving x1≥(r1L-c2U)/a1U, by way of contradiction, assume that x̲1<(r1L-c2U)/a1U , taking limit in the first equation of system (1.2), it follows that
(2.19)limn→+∞[r1(n)-a1(n)x1(n)-c2(n)x2(n)1+x2(n)]=0,
however,
(2.20)limn→+∞[r1(n)-a1(n)x1(n)-c2(n)x2(n)1+x2(n)]≥limn→+∞[r1(n)-a1(n)x1(n)-c2(n)]≥r1L-a1Ux̲1-c2U>0,
which is a contradiction. It implies that x̲1≥(r1L-c2U)/a1U. By the fact that minx∈ℝ+{[exp(x-1)]/x}=1, we obtain that M1=[exp(r1U-1)]/a1L≥r1U/a1L≥r1L/a1U. Thus x̲1>(r1L-c1U)/a1U≥x1ε. Therefore, liminfn→+∞x1(n)≥x1ε. Since ε can be sufficiently small, we have
(2.21)lim infn→+∞x1(n)≥m1=defr1L-c2Ua1Uexp(r1L-a1UM1-c2U)>0.
Analogously, by the second inequality in (2.11), we can obtain that
(2.22)lim infn→+∞x2(n)≥m2=defr2L-c1Ua2Uexp(r2L-a2UM2-c1U)>0.
This completes the proof of Proposition 2.3.Now, we are in a position to state Theorem2.4 whose proof is a direct consequence of Propositions 2.2 and 2.3.Theorem 2.4.
If the inequalities in (2.11) hold, then system (1.2) is permanent.


## 3. Existence and Global Stability of Positive Periodic Solutions

In this section, we suppose system (1.2) is a periodic system, and then we investigate the existence and global stability of positive periodic solutions of such system. To do this, assume that all the coefficients of system (1.2) are ω-periodic, in other words,
(3.1)ri(n+ω)=ri(n),ai(n+ω)=ai(n),ci(n+ω)=ci(n),i=1,2.Theorem 3.1.
If the inequalities in (2.11) hold, then system (1.2) has at least one strictly positive ω-periodic solution, denoted by {x1*(n),x2*(n)}.Proof.
We know thatK=[m1,M1]×[m2,M2] is an invariant set of system (1.2) from Propositions 2.2 and 2.3. Define the continuous mapping F on K(3.2)F{x1*(0),x2*(0)}={x1*(ω),x2*(ω)},for{x1*(0),x2*(0)}∈K.
Obviously, F depends continuously on {x1*(0),x2*(0)}, then F is continuous and maps the compact set K=[m1,M1]×[m2,M2] into itself. Therefore, F has a fixed point {x1*,x2*}. It is easy to see that the solution {x1*(n),x2*(n)} which passes through {x1*,x2*} is an ω-periodic solution of system (1.2). The proof is complete.Next, we derive sufficient conditions which guarantee that the positive periodic solution of system (1.2) is globally stable. We first give the definition of global stability.Definition 3.2.
A positive periodic solution{x1*(n),x2*(n)} of system (1.2) is globally stable if each other solution {x1(n),x2(n)} of system (1.2) with positive initial value defined for all n>0 satisfies
(3.3)limn→+∞|x1(n)-x1*(n)|=0,limn→+∞|x2(n)-x2*(n)|=0.Now, we give the main result in this section.Theorem 3.3.
In addition to (2.11), assume further that the following assumptions
(3.4)λ1=defmax{|1-a1Lm1|,|1-a1UM1|}+c2U<1,(3.5)λ2=defmax{|1-a2Lm2|,|1-a2UM2|}+c1U<1
hold. Then the positive periodic solution of system  (1.2) is globally stable.Proof.
Let{x1*(n),x2*(n)} be a positive periodic solution of system (1.2). We make the change of variables x1(n)=x1*(n)expu(n) and x2(n)=x2*(n)expv(n), then system (1.2) is rewritten as
(3.6)u(n+1)-u(n)=a1(n)x1*(n)[1-expu(n)]+c2(n){11+x2*(n)exp[v(n)]-11+x2*(n)},v(n+1)-v(n)=a2(n)x2*(n)[1-expv(n)]+c1(n){11+x1*(n)exp[u(n)]-11+x1*(n)}.
By the mean-value theorem, it derives that
(3.7)u(n+1)=u(n){1-a1(n)x1*(n)exp[θ1u(n)]}-c2(n)v(n)x2*(n)exp[θ2v(n)]{1+x2*(n)exp[θ2v(n)]}2,v(n+1)=v(n){1-a2(n)x2*(n)exp[θ3v(n)]}-c1(n)u(n)x1*(n)exp[θ4u(n)]{1+x1*(n)exp[θ4u(n)]}2,
where the constants θ1,θ2,θ3,θ4∈(0,1).
Now, by (3.4) and (3.5), we choose the constant ε sufficiently small such that
(3.8)λ1ε=max{|1-a1L(m1-ε)|,|1-a1U(M1+ε)|}+c2U<1,λ2ε=max{|1-a2L(m2-ε)|,|1-a2U(M2+ε)|}+c1U<1.
In view of  Propositions 2.2 and 2.3, there exists an n0∈N such that n≥n0, we have
(3.9)0<m1-ε≤x1*(n)≤M1+ε,0<m1-ε≤x1(n)≤M1+ε,0<m2-ε≤x2*(n)≤M2+ε,0<m2-ε≤x2(n)≤M2+ε.
Since θ1,θ2,θ3,θ4∈(0,1), both x1*(n)exp[θ1u(n)] and x1*(n)exp[θ4u(n)] are between x1(n) and x1*(n). Meanwhile, both x2*(n)exp[θ2v(n)] and x2*(n)exp[θ3v(n)] are between x2(n) and x2*(n). From the first equation of system (3.7), it follows that
(3.10)|u(n+1)|≤|u(n)||1-a1(n)x1*(n)exp[θ1u(n)]|+|c2(n)v(n)x2*(n)exp[θ2v(n)]1+x2*(n)exp[θ2v(n)]|≤max{|1-a1L(m1-ε)|,|1-a1U(M1+ε)|}|u(n)|+max{|c2(n)v(n)|}≤max{|1-a1L(m1-ε)|,|1-a1U(M1+ε)|}|u(n)|+c2U|v(n)|≤λ1εmax{|u(n)|,|v(n)|}.
Similarly, it follows from (3.5) that
(3.11)|v(n+1)|≤λ2εmax{|u(n)|,|v(n)|}.
Denote λε=max{λ1ε,λ2ε}, then λε<1. Therefore, when n≥n0,
(3.12)max{|u(n+1)|,|v(n+1)|}≤λεmax{|u(n)|,|v(n)|}≤(λε)n-n0max{|u(n0)|,|v(n0)|},
as a consequence, limn→+∞|x1(n)-x1*(n)|=0,limn→+∞|x2(n)-x2*(n)|=0. By using Definition 3.2, it follows that the positive periodic solution {x1*(n),x2*(n)} of system (1.2) is globally stable. This completes the proof.Remark 3.4.
Theorem3.3 shows that {x1*(n),x2*(n)} is the global attractor of all positive solutions of system (1.2), then {x1*(n),x2*(n)} is the unique positive periodic solution of system (1.2).


## 4. Example and Numerical Simulation

In this paper, we have investigated the permanence and global stability of positive periodic solutions of a discrete competitive system. Each species is not isolated from its living environment, but competes with the other for the same resource. Sufficient conditions which guarantee the permanence, existence and global stability of positive periodic solutions are established, respectively. The theoretical results are confirmed by the following examples and their numerical results.To verify the sufficient conditions for permanence of system (1.2), we assume that r1(n)=0.46+0.01sinπn, r2(n)=0.88-0.02sinπn, a1(n)=0.75+0.01cosπn, a2(n)=0.52+0.01sinπn, c1(n)=0.28+0.02sinπn, c2(n)=0.13-0.01sinπn, and the initial condition (x1(0),x2(0))=(0.25,0.01). Clearly, (2.11) in Theorem 2.4 are satisfied, and hence system (1.2) is permanent (see Figure 1).Permanence of system (1.2) with initial value (0.25,0.01). (a) Time series of x1 for n∈[0,50]. (b) Time series of x2 for n∈[0,50].
(a)(b)Now, we further verify the sufficient conditions for the existence and global stability of positive periodic solutions of periodic system (1.2). Let us assume that all the coefficients of system (1.2) are periodic and listed in  Table 1.Table 1
The coefficient values whenn is under different conditions.
nr1(n)r2(n)a1(n)a2(n)c1(n)c2(n)Odd number0.761.581.251.800.030.02Even number1.260.981.651.980.020.01Besides, we choose the positive periodic solution with initial values(0.09,0.01), denoted by (x1*,x2*), and the positive solution with initial value (0.08,0.02) denoted by (x1,x2). By Theorem 3.3, a simple calculation shows that the assumptions in (3.4) and (3.5) hold. So from Theorem 3.3 we know that periodic system (1.2) has a positive 2-periodic solution which is globally stable. From  Figures 2(a), 2(b), and  2(c), we see that x1 with x1(0)=0.08 will tend to x1* with x1*(0)=0.09. Similarly, from  Figures 3(a), 3(b), and  3(c), we see that x2 with x2(0)=0.02 will tend to x2* with x2*(0)=0.01. Furthermore,  Figures 4(a) and 4(b) show the phase portrait of periodic system (1.2) with (x1*(0),x2*(0))=(0.09,0.01) for n∈[0,50] and (x1(0),x2(0))=(0.08,0.02) for n∈[0,50], respectively. From  Figure 4(c), we can see that periodic system (1.2) has a positive 2-periodic solution which is globally stable.(a) Time series ofx1* with x1*(0)=0.09 for n∈[0,50]. (b)  Time series of x1 with x1(0)=0.08 for n∈[0,50]. (c) Time series of x1* and x1 for n∈[0,50] in the same coordinate system.
(a)(b)(c)(a) Time series ofx2* with x2*(0)=0.01 for n∈[0,50]. (b)  Time series of x2 with x2(0)=0.02 for n∈[0,50]. (c) Time series of x2* and x2 for n∈[0,50] in the same coordinate system.
(a)(b)(c)(a) Phase portrait ofx1* and x2* with initial value (0.09,0.01) for n∈[0,50]. (b) Phase portrait of x1 and x2 with initial value (0.08,0.02) for n∈[0,50]. (c) Phase portraits of x1*,x2* and x1,x2 for n∈[30,50] in the same coordinate system.
(a)(b)(c)


## 5. Acknowledgement

The work is supported by the Innovation Term of Educational Department of Hubei Province in China (T200804), the National Science Foundation of Hubei Province in China (2008CDB068), and the Innovation Project of Hubei Institute for Nationalities for postgraduate students. We would like to thank the Editor Professor A. Vecchio and the referee for careful reading of the original manuscript and valuable comments and suggestions that greatly improved the presentation of this work.



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