In most models of population dynamics, diffusion between two patches is assumed to be either continuous or discrete, but in the real natural ecosystem, impulsive diffusion provides a more suitable manner to model the actual dispersal (or migration) behavior for many ecological species. In addition, the species not only requires some time to disperse or migrate among the patches but also has some possibility of loss during dispersal. In view of these facts, a single species model with dissymmetric bidirectional impulsive diffusion and dispersal delay is formulated. Criteria on the permanence and extinction of species are established. Furthermore, the realistic conditions for the existence, uniqueness, and the global stability of the positive periodic solution are obtained. Finally, numerical simulations and
discussion are presented to illustrate our theoretical results.
1. Introduction
In the last few years, mathematicians and ecologists have been actively investigating the dispersal of populations, a ubiquitous phenomenon in population dynamics. Levin [1] showed that both spatial dispersal of populations and population dynamics are much affected by spatial heterogeneity. In real life, dispersal often occurs among patches in ecological environments; because of the ecological effects of human activities and industries, such as the location of manufacturing industries and the pollution of the atmosphere, soil, and rivers, reproduction- and population-based territories and other habitats have been broken into patches. Thus, realistic models should include dispersal processes that take into consideration the effects of spatial heterogeneity.
In recent years, increasing attention has been paid to the dynamics of a large number of mathematical models with diffusion, and many nice results have been obtained. The persistence and extinction for ordinary differential equation and delayed differential equation models were investigated in [2–6]. Global stability of equilibrium and periodic solution for diffusing model were studied in [7–12]. However, in all of above population dispersing systems, it is always assumed that the dispersal occurs at every time. For example, in [7], Beretta and Takeuchi proposed the following single-species diffusion Volterra models with continuous time delays:
(1)x˙i=xi(ei-aixi+γi∫-∞tFi(t-τ)xi(τ)dτ)+∑μ=1nDiμ(t)(xμ-xi),i∈N,
where N={1,…,n}n is the number of patches, and xi is the population density in the ith patch. The form of the dispersal established in this model is continuous; that is, the dispersal is always happening at any time.
Actually, real dispersal behavior is very complicated and is always influenced by environmental change and human activities. In many practical situations, it is often the case that maybe one of the species suffers a significant loss or increase in density for some reason at some transitory time slots. These short-term perturbations are often assumed to be in the form of impulses in the modeling process. For example, when winter comes, birds will migrate between patches in search for a better environment, whereas they do not diffuse in other seasons, and the excursion of foliage seeds occurs during a fixed period of time every year. Therefore, impulsive differential equations [13] provide a natural description of such system. With the developments and applications of impulsive differential equations, theories of impulsive differential equations have been introduced into population dynamics, and many important studies have been performed [14–20].
In [14], the authors studied the following autonomous single-species model with impulsively bidirectional diffusion:
(2)x˙1(t)=x1(t)[a1-b1x1(t)],x˙2(t)=x2(t)[a2-b2x2(t)],t≠nτ,Δx1(t)=d1(x2(t)-x1(t)),Δx2(t)=d2(x1(t)-x2(t)),t=nτ,
where ai, bi(i=1,2) are the intrinsic growth rate and density-dependent parameters of the population xi and di is the dispersal rate in the ith patch. Consider Δxi=xi(nτ+)-xi(nτ-), where xi(nτ+)=limt→nτ+xi(t) represents the density of population in the ith patch immediately after the nth diffusion pulse at time t=nτ, while xi(nτ-)=limt→nτ-xi(t)=xi(τ) represents the density of population in the ith patch before the nth diffusion pulse at time t=nτ (τ the period of dispersal between any two pulse events is a positive constant, n=1,2,…). It is assumed here that the net exchange from the jth patch to ith patch is proportional to the difference xj-xi of population densities. The dispersal behavior of populations between two patches occurs only at the impulsive instants nτ. Obviously, in this model, species x inhabits, respectively, two patches before the pulse appears, and when the time at the pulse comes, species x in two patches disperses from one patch to the other. The boundedness and global stability of positive periodic solution were obtained.
Time delay often appears in many control systems (such as aircraft, chemical, or process control systems) either in the state, the control input, or the measurements. In order to reflect the dynamical behaviors of models that depend on the past history of system, it is often necessary to incorporate time delays into systems [21]. There have been extensive theoretical works on delay differential equations in the past three decades. The research topics include global asymptotic stability of equilibria, existence of periodic solutions, complicated behavior, and chaos (e.g., [8, 22–24]).
Takeuchi et al. in [25] studied the following population model with time delays that introduced the dispersal time for individuals to move from one patch to other patches:
(3)x˙i(t)=xi(t)[ai-bixi(t)]+∑j=1n[εijdijxj(t-τij(t))-dji(t)xi(t)],hhhhhhhhhhhhhhhhhhhhhi=1,2,…,n,
where bi, i=1,…,n, are positive constants, dij, i, j=1,…,n, are nonnegative constants, εij=eγijτij satisfy 0<εij≤1, ai, i=1,…,n, are constants, and some of them may be negative. In this paper, the authors took account of dispersal delay; however, they assumed that the dispersal is continuous.
It is well known that the application of impulsive delay differential equations to population dynamics has been an interesting topic since it is reasonable and correct in modelling the evolution of population, such as pest management [26].
However, in all of the impulsive dispersal models studied up till now, there are few papers considering the dispersal delay, which is really a pity. Actually, in the real world, the migration between patches is usually not immediate; that is, dispersal processes often involve time delay. For example, elks move from higher to lower elevations to escape cold in winter, and ungulates migrate annually among grazing areas to follow spatiotemporal changes in rainfall. Obviously, this kind of dispersal delay between patches extensively exists in the real world. Therefore, it is a very basilic problem to research this kind of population dynamic systems.
Moreover, in the above impulsive dispersal models, it is assumed that the dispersal occurs between homogeneous habitat patches; that is, the dispersal rate between any two patches is equal or symmetrical [1, 11], which is really too idealized for a real ecosystem. Actually, in the real world, due to the heterogeneity of the spatiotemporal distributions in nature, movement between fragments of patches is usually not the same rate in both directions. In addition, once the individuals leave their present habitat, they may not successfully reach a new one, due to predation, harvesting, or other reasons, so that there are traveling losses. Thus, the dispersal rates among these patches are not always the same. Rather, in real ecological situations they are different (or dissymmetrical) [27, 28].
Therefore, it is our basilic goal to investigate a single species model with dissymmetric impulse dispersal and dispersal delay. Motivated by the calculation hereinbefore, in this paper, we extend system (2) with dispersal delay and dispersal loss and consider it
(4)dx1(t)dt=x1(t)[a1-b1x1(t)],dx2(t)dt=x2(t)[a2-b2x2(t)],t≠nτ,Δx1(t)=d2x2(t-τ0)-D1x1(t),Δx2(t)=d1x1(t-τ0)-D2x2(t),t=nτ,
where Di(i=1,2) is the rate of population xi emigrating from the ith patch and di is the rate of population xi immigrating from the ith patch. Here we assume 0≤di≤Di≤1, which means that there possibly exists mortality during migration between two patches. τ0≤τ stands for the time delay; that is, a period of time of species x dispersing between patches.
The organization of this paper is as follows. In Section 2, as preliminaries, the definition of permanence and some useful lemmas are introduced. From discrete dynamic system theory, we establish the stroboscopic map of system (4), by which we can obtain the dynamical behaviors of it. In Section 3, the results of permanence and extinction for the system are presented. The existence and the uniqueness of the positive periodic solution for system (4) are established in Section 4. In Section 5, using the discrete dynamic system theory in [29], we can get the global stability of the positive periodic solution for the system. Finally, we give a brief discussion and our theoretical results are conformed by numerical simulations.
2. Preliminaries
In this section we introduce a definition and some notations and state some results which will be useful in subsequent sections.
For any fixed t∈R+=[0,∞), let ϕ(s):[t-τ0,t]→R2 be a piecewise continuous function such that ϕ(s) is continuous in s≠nτ, ϕ(nτ-)=lims→nτ-ϕ(s)=ϕ(nτ) and ϕ(nτ+)=lims→nτ+ϕ(s) exist. For any fixed t∈R+, let PC[t-τ0,t] denote the Banach space of all such piecewise continuous functions ϕ(θ):[t-τ0,t]→R2 with the norm ∥ϕ∥=supt-τ0≤θ≤t|ϕ(θ)|. Further, let PC+[t-τ0,t] = {ϕ=(ϕ1,ϕ2)∈PC[t-τ0,t]:ϕi(θ)≥0 for all θ∈[t-τ0,t] and ϕi(t)>0 for i=1,2}.
Motivated by the biological background of system (4), in this paper we always assume that all solutions of system (4) satisfy the following initial conditions:
(5)xi(θ)=ϕi(θ)∀θ∈[-τ0,0],i=1,2,
where ϕ=(ϕ1,ϕ2)∈PC+[-τ0,0]. For any ϕ∈PC+[-τ0,0], by the fundamental theory of impulsive functional differential equations [30, 31], system (4) has a unique solution x(t,ϕ)=(x1(t,ϕ),x2(t,ϕ)) satisfying the initial conditions (5).
Definition 1.
System (4) is said to be permanent, if there are positive constants mi and Mi such that
(6)mi≤liminft→∞xi(t)≤limsupt→∞xi(t)≤Mi,i=1,2,
for any positive solution x(t)=(x1(t),x2(t)) of system (4).
We consider the following scalar impulsive differential equation:
(7)x˙(t)=f(t,x(t)),t≠nτ,x(nτ+)=I(x(nτ)),t=nτ,
where t∈R+, nτ>0, is an impulsive time sequence, x∈R, f(t,x):R+×R→R is continuous, and I(x):R→R is a nondecreasing function. We have the following comparison theorem [30, 31] for (7).
Lemma 2.
Let x(t) be a solution of system (7) defined on [t0,T] and satisfy
(8)u˙(t)≤(≥)f(t,u(t)),t≠nτ,u(nτ+)≤(≥)I(u(nτ)),t=nτ.
If u(t0)≤(≥)x(t0), then u(t)≤(≥)x(t) for all t∈[t0,T].
Lemma 3 (see [<xref ref-type="bibr" rid="B32">32</xref>]).
Consider the following nonlinear impulsive system:
(9)x˙(t)=x(t)(a(t)-b(t)x(t)),t≠tk,x(tk+)=(1-θk)x(tk),t=tk,k∈N,x(0+)=x0,
where a(t) and b(t) are bounded and continuous ω-periodic functions defined on R+, b(t)≥0 for all t∈R+ and impulsive coefficients θk∈(0,1),θk=θk+q, q is a fixed positive integer, and tk+q=tk+ω. If ∫0ωa(s)ds+ln∏i=1q(1-θi)>0, then there exists a unique positive periodic solution of system (9), which is globally asymptotically stable.
Remark 4.
If system (9) degenerates into the following autonomous impulsive differential equation:
(10)x˙(t)=x(t)(a-bx(t)),t≠tk,x(tk+)=(1-θ)x(tk),t=tk,k∈N,x(0+)=x0,
where a, b, θ are positive constants, tk+1=tk+ω. As a consequence of Lemma 3, we have the following result: if 1-θ-e-aω>0, then system (10) has a unique positive periodic solution x*(t), which is globally asymptotically stable. In fact, here ∫0ωads+ln(1-θ)>0≡1-θ-e-aω>0.
Lemma 5 (see [<xref ref-type="bibr" rid="B29">29</xref>]).
Let F:R+n→R+n be continuous, C1 in int (R+n), and suppose DF(0) exists with limx→0+DF(z)=DF(0). In addition, assume
DF(x)>0, if x>0;
DF(y)<DF(x), if 0<x<y.
If F(0)=0, let λ=ρ(DF(0)). If λ≤1, then, for every x≥0,Fn(x)→0 as n→∞; if λ>1, then either Fn(x)→∞ as n→∞ for every x>0 or there exists a unique nonzero fixed point q of F. In the latter case, q>0 and, for every x>0, Fn(x)→q as n→∞.
If F(0)≠0, then either Fn(x)→∞ as n→∞ for every x≥0 or there exists a unique fixed point q of F. In the latter case, q>0 and, for every x>0, Fn(x)→q as n→∞.
Next, we analyze system (4). Integrating and solving the first two equations of system (4) between pulses, we have
(11)xi(t)=[biai+(1xi(nτ+)-biai)e-ai(t-nτ)]-1,hhhhhhhhhnτ<t≤(n+1)τ,i=1,2.
Similarly, considering the last two equations of system (4), we obtain the following stroboscopic map:
(12)x1n+1=(1-D1)x1nh1+c1x1n+d2x2nh2ea2τ0+l2x2n,x2n+1=(1-D2)x2nh2+c2x2n+d1x1nh1ea1τ0+l1x1n;
here xin+1=xi[(n+1)τ+], ci=(bi/ai)(1-e-aiτ)>0, li=(bi/ai)(1-e-ai(τ-τ0))>0, 0<hi=e-aiτ<1.
Remark 6.
System (12) is a difference system, which means that densities of population in two patches have values at the previous pulse. We are, in other words, stroboscopically sampling at its pulsing period. The dynamical behavior of system (12), coupled with (11), determines the dynamical behavior of system (4). In the following sections, we will focus our attention on system (12) and investigate various aspects of its dynamical behavior.
To write system (12) as a map, we define the map :R+2→R+2:
(13)F1(x1,x2)=(1-D1)x1h1+c1x1+d2x2h2ea2τ0+l2x2,F2(x1,x2)=(1-D2)x2h2+c2x2+d1x1h1ea1τ0+l1x1.
The set of all iterations of the map F is equivalent to the set of all density sequences generated by system (12); F(x) is the map evaluated at the point x=(x1,x2)∈R+2. Consequently, in system (12), Fn describes the population densities in the time nτ.
On the positivity of solutions of system (4) we have the following result.
Lemma 7.
The solution x(t,t0,ϕ) of system (4) with initial condition (5) is positive, that is, x(t,t0,ϕ)>0 on the interval of the existence.
The proof of Lemma 7 is simple; we hence omit it here.
3. Permanence and Extinction
In this section, we present conditions to ensure that system (12) is permanent and extinct which will imply the permanence and extinction of system (4). The permanence plays an important role in mathematical ecology since the criterion of permanence for ecological systems is a condition ensuring the long-term survival of all species. So, we firstly prove system (12) is permanent.
Theorem 8.
Suppose
(H1)1-Di-hi>0,i=1,2,
hold; then system (12) is permanent.
Proof.
Let xi(t)∈PC′ be the solution of system (4) satisfying the initial conditions (5). From the first equation of system (12), we have
(14)x1n+1=1-D1h1(x1n)-1+c1+d2h2ea2τ0(x2n)-1+l2<1-D1c1+d2l2.
Similarly, we have
(15)x2n+1=1-D2h2(x2n)-1+c2+d1h1ea1τ0(x1n)-1+l1<1-D2c2+d1l1.
Hence, by (14) and (15) we know that system (12) has an ultimately upper bound.
Next, we prove that all the solutions of system (4) are ultimately below bounded. Since di≥0(i=1,2), from the third equation of system (4), we have
(16)x1(nτ+)=x1(nτ-)+Δx1(nτ)=(1-D1)x1(nτ)+d2x2(nτ-τ0)≥(1-D1)x1(nτ).
Similarly,
(17)x2(nτ+)≥(1-D2)x2(nτ).
Thus, system (4) becomes
(18)x˙1(t)=x1(t)[a1-b1x1(t)],x˙2(t)=x2(t)[a2-b2x2(t)],t≠nτ,x1(nτ+)≥(1-D1)x1(nτ),x2(nτ+)≥(1-D2)x2(nτ),t=nτ.
From (18), we find that there is no relation between x1(t) and x2(t). Therefore, we will discuss them, respectively;
(19)x˙1(t)=x1(t)[a1-b1x1(t)],t≠nτ,x1(nτ+)≥(1-D1)x1(nτ),t=nτ,x˙2(t)=x2(t)[a2-b2x2(t)],t≠nτ,x2(nτ+)≥(1-D2)x2(nτ),t=nτ.
If (H1) holds, from Remark 4, we can obtain that the auxiliary system
(20)u˙1(t)=u1(t)[a1-b1u1(t)],t≠nτ,u1(nτ+)=(1-D1)u1(nτ),t=nτ,
has a unique positive periodic solution u1*(t)=x1*(t) which is globally asymptotically stable.
Let u1(t) be the solution of system (20) with initial value u1(0+)=x1(0+). By Lemma 2, we have
(21)x1(t)≥u1(t),∀t≥0.
Hence, for any ε>0 sufficiently small, there exists a T1>0 such that
(22)x1(t)≥x1*(t)-ε≜m1,fort≥T1.
Similarly, if (H1) holds, for above ε>0, there exists a T2>0 such that
(23)x2(t)≥x2*(t)-ε≜m2,fort≥T2.
Denote m=min{m1,m2} and T=max{T1,T2}; then we have x1(t)>m and x2(t)>m, t≥T. Finally, we can determine that there exist constants αi, βi(0<αi<βi)(i=1,2), such that α1≤liminfn→∞x1n≤limsupn→∞x1n≤β1, and α2≤liminfn→∞x2n≤limsupn→∞x2n≤β2. The proof of Theorem 8 is completed.
Next, we present condition to ensure that system (12) is extinct.
Theorem 9.
System (12) is extinct if
(24)[d1d2e(a1+a2)τ0-(1-D1)(1-D2)]e(a1+a2)τ+ea1τ(1-D1)+ea2τ(1-D2)≤1.
Proof.
Let us consider the system (13). Obviously, F(x1,x2) is continuous C1 in int(R+2), and F(0,0)=0. We obtain
(25)DF(x1,x2)=((1-D1)h1(h1+c1x1)2d2h2ea2τ0(h2ea2τ0+l2x2)2d1h1ea1τ0(h1ea1τ0+l1x1)2(1-D2)h2(h2+c2x2)2),DF(0,0)=(1-D1h1d2h2ea2τ0d1h1ea1τ01-D2h2).
Obviously, lim(x1,x2)→(0,0)DF(x1,x2)=DF(0,0); if x>0,DF(x)>0; if 0<x<y, DF(y)<DF(x). We have the characteristic equation of DF(0,0):
(26)λ2-(1-D1h1+1-D2h2)λ+(1-D1)(1-D2)h1h2-d1d2h1h2e(a1+a2)τ0=0.
Let λ=ρ(DF(0,0)); then we have
(27)λ=(+4d1d2h1h2e(a1+a2)τ0)1/21-D1h1+1-D2h2+((1-D1h1+1-D2h2)2-4(1-D1)(1-D2)h1h2+4d1d2h1h2e(a1+a2)τ0(1-D1h1+1-D2h2)2)1/2)×(2)-1=(+(1-D1h1-1-D2h2)2+4d1d2h1h2e(a1+a2)τ01-D1h1+1-D2h2+(1-D1h1-1-D2h2)2+4d1d2h1h2e(a1+a2)τ0)×(2)-1.
Assume λ>1; then by (27) we can obtain
(28)1-D1h1+1-D2h2+d1d2h1h2e(a1+a2)τ0-(1-D1)(1-D2)h1h2>1;
that is,
(29)[d1d2e(a1+a2)τ0-(1-D1)(1-D2)]e(a1+a2)τ+ea1τ(1-D1)+ea2τ(1-D2)>1,
which contradicts with (24). Therefore we have λ≤1. By Lemma 5, we can get Fn(x1,x2)→(0,0) as n→∞, which means that system (12) is extinct. This completes the proof.
4. Existence and Uniqueness of Positive Periodic Solution
In this part, we will prove the existence and uniqueness of the fixed points of system (12), which means that system (4) has a uniquely positive periodic solution.
Theorem 10.
If (H1) holds, then there exists a unique positive fixed point x*=(x1*,x2*) of system (12).
Proof.
Corresponding to (12), let us consider the following system:
(30)x1=(1-D1)x1h1+c1x1+d2x2h2ea2τ0+l2x2,x2=(1-D2)x2h2+c2x2+d1x1h1ea1τ0+l1x1.
From (30), we have
(31)x1-(1-D1)x1h1+c1x1>0,x2-(1-D2)x2h2+c2x2>0;
hence
(32)x1>1c1(1-D1-h1)=ξ,x2>1c2(1-D2-h2)=η.
From (30), we also obtain
(33)x2=h2ea2τ0[x1-((1-D1)x1)/(h1+c1x1)]d2-l2[x1-((1-D1)x1)/(h1+c1x1)],G(x1,x2)=d1x1h1ea1τ0+l1x1+(1-D2)x2h2+c2x2-x2.
Thus
(34)x2(ξ)=0,G(ξ)=d1ξh1ea1τ0+l1ξ>0.
Let x2→+∞; then f(x1)=x1-((1-D1)x1)/(h1+c1x1)→d2/l2. And f(x1) is an increasing function on the interval [ξ,+∞). Since f(ξ)=0, so there exists x-1>ξ such that f(x-1)=d2/l2. We can easily find that G(x-1)<0. By the zero theory of continuous function, there exists (x1*,x2*) such that
(35)ξ<x1*<x-1,G(x1*,x2*)=0,(36)x2*=h2ea2τ0[x1*-((1-D1)x1*)/(h1+c1x1*)]d2-l2[x1*-((1-D1)x1*)/(h1+c1x1*)].
Next, we will prove the uniqueness of the fixed point.
It follows from (33) that we obtain
(37)dx2dx1=d2h2ea2τ0[1-h1(1-D1)/(h1+c1x1)2]{d2-l2[x1-((1-D1)x1)/(h1+c1x1)]}2,dGdx1=d1h1ea1τ0(h1ea1τ0+l1x1)2+[h2(1-D2)(h2+c2x2)2-1]dx2dx1,dGdx1=1×({d2-l2[x1-(1-D1)x1h1+c1x1]}2×(h1ea1τ0+l1x1)2{d2-l2[x1-(1-D1)x1h1+c1x1]}2)-1*{d1h1ea1τ0{d2-l2[x1-(1-D1)x1h1+c1x1]}2+d2h2ea2τ0(h1ea1τ0+l1x1)2×[1-h1(1-D1)(h1+c1x1)2][h2(1-D2)(h2+c2x2)2-1]}.
Let
(38)φ(x)=d1h1ea1τ0{d2-l2[x1-(1-D1)x1h1+c1x1]}2+d2h2ea2τ0(h1ea1τ0+l1x1)2×[1-h1(1-D1)(h1+c1x1)2][h2(1-D2)(h2+c2x2)2-1];
then
(39)dφ(x)dx1=2d1h1ea1τ0{d2-l2[x1-(1-D1)x1h1+c1x1]}×{-l2[1-h1(1-D1)(h1+c1x1)2]}+2l1d2h2ea2τ0(h1ea1τ0+l1x1)×[1-h1(1-D1)(h1+c1x1)2][h2(1-D2)(h2+c2x2)2-1]+d2h2ea2τ0(h1ea1τ0+l1x1)2×2c1h1(1-D1)(h1+c1x1)3[h2(1-D2)(h2+c2x2)2-1]+d2h2ea2τ0(h1ea1τ0+l1x1)2×[1-h1(1-D1)(h1+c1x1)2][-2c2h2(1-D2)(h2+c2x2)3].
By (31), we have (1-D1)/(h1+c1x1)<1; since h1/(h1+c1x1)<1, so h1(1-D1)/(h1+c1x1)2<1. Similarly, we have h2(1-D2)/(h2+c2x2)2<1. Therefore, we obtain dφ(x)/dx<0, which implies that φ(x) is a decreasing function on the interval [ξ,+∞).
Since
(40)φ(ξ)=d1d22h1ea1τ0+d2h2ea2τ0×[h1ea1τ0+l1c1(1-D1-h1)]2×1-D1-h11-D11-D2-h2h2>0;(41)φ(x-1)=d1h1ea1τ0{d2-l2[x-1-(1-D1)x-1h1+c1x-1]}2+d2h2ea2τ0(h1ea1τ0+l1x-1)2×[1-h1(1-D1)(h1+c1x-1)2][h2(1-D2)(h2+c2x2(x-1))2-1]=0+d2h2ea2τ0(h1ea1τ0+l1x-1)2×[1-h1(1-D1)(h1+c1x-1)2](-1)<0,
using the zero theory of continuous function, there exists a unique point ξ1∈(ξ,x-1) such that φ(ξ1)=0. Besides,
(42)φ(x1)>0,∀x1∈(ξ,ξ1),φ(x1)<0,∀x1∈(ξ1,+∞);
thus
(43)dG(x1)dx1>0,∀x1∈(ξ,ξ1),dG(x1)dx1<0,∀x1∈(ξ1,+∞),
which, together with G(ξ)>0, leads to G(x1)>0,∀x1∈(ξ,ξ1). By G(ξ1)>0,G(x-1)<0, we have that there exists a unique point x1*∈(ξ1,x-1) such that G(x1*,x2*)=0. The proof is completed.
5. Global Stability
Now, we prove that the positive fixed points (x1*,x2*) of system (30) are globally stable by using Lemma 5, which means that the positive periodic solution of system (4) is globally stable.
Theorem 11.
If (H1) holds, then there exists a unique positive fixed point x*=(x1*,x2*) of the map F, and, for every x=(x1,x2)>0,Fn(x)→x* as n→∞.
Proof.
For any small ε1>0, ε2>0, we make the change of variable
(44)x1=u+ε1,x2=v+ε2.
By (13), we get the map F(u,v)=(F1(u,v),F2(u,v)); that is,
(45)u=(1-D1)(u+ε1)h1+c1(u+ε1)+d2(v+ε2)h2ea2τ0+l2(v+ε2)-ε1=F1(u,v),v=(1-D2)(v+ε2)h2+c2(v+ε2)+d1(u+ε1)h1ea1τ0+l1(u+ε1)-ε2=F2(u,v).
Now, we show that F(u,v) satisfies the hypotheses of Lemma 5. It is easy to see that F(u,v) is continuous, C1 in int(R+2), and F(0,0)≠0.
Since
(46)DF(u,v)=((1-D1)h1[h1+c1(u+ε1)]2d2h2ea2τ0[h2ea2τ0+l2(v+ε2)]2d1h1ea1τ0[h1ea1τ0+l1(u+ε1)]2(1-D2)h2[h2+c2(v+ε2)]2),DF(0,0)=((1-D1)h1(h1+c1ε1)2d2h2ea2τ0(h2ea2τ0+l2ε2)2d1h1ea1τ0(h1ea1τ0+l1ε1)2(1-D2)h2(h2+c2ε2)2),
so, lim(u,v)→(0+,0+)DF(u,v)=DF(0,0). Obviously, if (u,v)>0, then DF(u,v)>0; if (0,0)<(u1,v1)<(u2,v2), then DF(u1,v1)>DF(u2,v2). It satisfies all the conditions of Lemma 5; then, for every u>0,v>0, we have Fn(u,v)→(x1*-ε1,x2*-ε2) as n→∞. Corresponding to x-y coordinate, this means, for x1>ε1, x2>ε2, the system (30) tends to the unique fixed point.
It follows from the permanence of system (12) that we have x1n>ε1, x2n>ε2 for any initial value (x1(0+),x2(0+))>(0,0).
From the above analysis, we can know that, for every x1(0+)>0, x2(0+)>0, the trajectory of system (12) will tend to (x1*,x2*). The proof is completed.
6. Numerical Simulation and Discussion
In order to test the validity of our results, first, for (4) we use the parameters values (Val. 1) in Table 1. We can easily test that the assumptions in Theorems 8 and 10 hold, which means the populations x(t)=(x1(t),x2(t)) in the two patches are permanent and have a unique periodic solution x*(t)=(x1*(t),x2*(t)) which is globally stable (see Figure 1(a)). Moreover, if, in Table 1, we consider the influence of time delay, then we can see that the permanence and stability for species x unchanged. The details are given in Table 2. However, the longer the duration of the time delay, the lower the limit inferior and the limit superior of x (see Figures 1(b), 1(c), and 1(d)). This implies that the case with dispersal delay is harmful to live for species x.
Parameter values used in the simulations of Model (4).
Parameter
Interpretation
Val. 1
Val. 2
a1
Intrinsic growth rate of populations in patch 1
0.9
0.3
a2
Intrinsic growth rate of populations in patch 2
0.8
0.4
b1
Density dependence rate of populations in patch 1
0.75
0.1
b2
Density dependence rate of populations in patch 2
0.5
0.2
D1
Successfully emigrant rate of populations from patch 1
0.8
0.9
D2
Successfully emigrant rate of populations from patch 2
0.7
0.8
d1
Successfully immigrant rate from patch 1 to patch 2
0.6
0.7
d2
Successfully immigrant rate from patch 2 to patch 1
0.1
0.5
τ
The period of dispersal between two pulse events
2
2
τ0
Dispersal delay between two pulse events
0
1
Val: value.
Simulations of Model (4).
Case
τ0
x1
x2
Figure
1
0
Permanent
Permanent
Figure 1(a)
2
1
Permanent
Permanent
Figure 1(b)
3
1.5
Permanent
Permanent
Figure 1(c)
4
2
Permanent
Permanent
Figure 1(d)
(a), (b), (c), and (d) Dynamical behavior of system (4). Here, we take three sets of initial values (1, 1.2), (0.6, 0.8), and (0.2, 1).
Next, we take the parameters values (Val. 2) in Table 1. We can easily test that assumption in Theorem 9[(d1d2/e(a1+a2)τ0)-(1-D1)(1-D2)]e(a1+a2)τ+ea1τ(1-D1)+ea2τ(1-D2)=0.5865<1 holds, which implies the populations x(t)=(x1(t),x2(t)) in the two patches are extinct (see Figure 2). Comparing Figure 2(a) and Figure 2(b), we realize that system (4) with time delay accelerates the extinction comparing with no delay (see Figure 2). This is reasonable from a biological point of view. Without delay means less loss during dispersion, which implies that more members can arrive to other patches. Otherwise, populations go extinct due to much loss. The details are given in Table 3.
Simulations of Model (4).
Case
τ0
x1
x2
Figure
1
0
Extinct
Extinct
Figure 2(a)
2
1
Extinct
Extinct
Figure 2(b)
(a) and (b) Dynamical behavior of system (4). Here, we take three sets of initial values (1.7, 2.1), (1.9, 1.5), and (1.4, 2.2).
Furthermore, we take a1=1.1+0.04cos(πt), a2=0.8+0.01sin(πt), b1=0.5+0.04sin(πt), b2=0.4+0.12cos(πt), d2=0.55 and keep other parameters unchanged from in Val. 1. Here we can see that the period of individual intrinsic growth rate and density dependence rate T=2 is equal to the period of migration τ. In this case, we have let the period of the environment match the period of migration. If we take τ0=0, from simulation (see Figure 3(a)), we can see the populations x(t)=(x1(t),x2(t)) are permanent and have a uniquely periodic solution which is globally stable. However, if we take the migration period τ=2.009,2.022, respectively, with other parameters unchanged, from numerical simulations (see Figures 3(b) and 3(c)), we can see that population dynamics change from almost periodic to chaotic. For Case 4 of Table 4, we consider the influence of time delay (see Figure 3(d)). Comparing Figures 3(c) and 3(d), we realize that system (4) with time delay is more complicated than without. The details are given in Table 4.
Simulations of Model (4).
Case
τ
τ0
x1
x2
Figure
1
2
0
PP
PP
Figure 3(a)
2
2.009
0
PAP
PAP
Figure 3(b)
3
2.022
0
PC
PC
Figure 3(c)
4
2.05
1.1
PC
PC
Figure 3(d)
PP: permanent and periodic, PAP: permanent and almost periodic, and PC: permanent and chaotic.
(a), (b), (c), and (d) Dynamical behavior of system (4). Here, we take the initial value x0=(x10,x20) = (1.735, 1.873).
Lastly, if we take a1=0.9+0.04cos(πt) and keep other parameters unchanged with Figure 3, by numerical simulations (see Figure 4), we find that all of the solutions of system (4) which through the initial points will converge to the positive periodic solution (x1*,x2*). Therefore, we can guess that under the assumptions of Theorem 10 system (4) has a unique positive periodic solution which is globally asymptotically stable. In addition, the periodic solution with time delay is larger than without delay which indicates that the duration of the time delay is beneficial to live for species x2 compared with species x1.
(a) and (b) Dynamical behavior of system (4). We take a series of initial points, such as (1.935,2.3), (1.939,2.34), and (1.943,2.38).
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (nos. 10901130, 11361059, 10961022, and 11271312), the China Scholarship Council, the Natural Science Foundation of Xinjiang Province of China (2012211B07), and the Scientific Research Programmes of Colleges in Xinjiang (XJEDU2013I03).
LevinS. A.Dispersion and population interactionsTengZ.LuZ.The effect of dispersal on single-species nonautonomous dispersal models with delaysCuiJ.TakeuchiY.LinZ.Permanence and extinction for dispersal population systemsCuiJ.ChenL.Permanence and extinction in logistic and Lotka-Volterra systems with diffusionXuR.MaZ.The effect of dispersal on the permanence of a predator-prey system with time delayTakeuchiY.CuiJ.MiyazakiR.SaitoY.Permanence of delayed population model with dispersal lossBerettaE.TakeuchiY.Global stability of single-species diffusion Volterra models with continuous time delaysBerettaE.TakeuchiY.Global asymptotic stability of Lotka-Volterra diffusion models with continuous time delayBerettaE.FergolaP.TennerielloC.Ultimate boundedness for nonautonomous diffusive Lotka-Volterra patchesFreedmanH. I.ShuklaJ. B.TakeuchiY.Population diffusion in a two-patch environmentHastingsA.Dynamics of a single species in a spatially varying environment: the stabilizing role of high dispersal ratesWangW.ChenL.LuZ.Global stability of a population dispersal in a two-patch environmentDongL.ChenL.ShiP.Periodic solutions for a two-species nonautonomous competition system with diffusion and impulsesHuiJ.ChenL.-S.A single species model with impulsive diffusionWangL.LiuZ.JinghuiChenL.Impulsive diffusion in single species modelBallingerG.LiuX.Permanence of population growth models with impulsive effectsZhangL.TengZ.N-species non-autonomous Lotka-Volterra competitive systems with delays and impulsive perturbationsVandermeerJ.StoneL.BlasiusB.Categories of chaos and fractal basin boundaries in forced predator-prey modelsZhangL.TengZ.DeAngelisD. L.RuanS.Single species models with logistic growth and dissymmetric impulse dispersalZhaoZ.ZhangX.ChenL.The effect of pulsed harvesting policy on the inshore-offshore fishery model with the impulsive diffusionYanX.-P.LiW.-T.Hopf bifurcation and global periodic solutions in a delayed predator-prey systemZhangL.TengZ.Permanence for a delayed periodic predator-prey model with prey dispersal in multi-patches and predator density-independentSmithH. L.ZhaoX.-Q.Global attractivity in a class of nonmonotone reaction-diffusion equations with time delayTakeuchiY.WangW.SaitoY.Global stability of population models with patch structureMengX.ChenL.Permanence and global stability in an impulsive Lotka-Volterra n-species competitive system with both discrete delays and continuous delaysFreedmanH. I.RaiB.WaltmanP.Mathematical models of population interactions with dispersal. II. Differential survival in a change of habitatMacArthurR. H.WilsonE. O.SmithH. L.Cooperative systems of differential equations with concave nonlinearitiesLakshmikanthamV.BaĭnovD. D.SimeonovP. S.BainovD.SimeonovP.ZhangF.GaoS.ZhangY.Effects of pulse culling on population growth of migratory birds and economical birds