# Dynamics of Mutualism-Competition-Predator System with Beddington-DeAngelis Functional Responses and  Impulsive Perturbations

Authors: Xiaoming Fan; Zhigang Wang; Fuquan Jiang
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We explore the dynamics of a class of mutualism-competition-predator interaction models with Beddington-DeAngelis functional responses and impulsive perturbations. Sufficient conditions for existence of positive periodic solution are established by using a continuation theorem in coincidence degree theory, which have been extensively applied in studying existence problems in differential equations and difference equations. In addition, sufficient criteria are given for the global stability and the globally exponential stability of system by employing comparison principle and Lyapunov method.


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## Body

## 1. Introduction

The ecological predator-prey systems and impulsive functional differential equations have been studied extensively by many authors [1–7]. A predator's per capita feeding rate on prey, or its functional response, provides a foundation for predator-prey theory. Since 1959, Holling's prey-dependent type  II functional response, a model that is a function of prey abundance only, has served as the basis for a large literature on predator-prey theory. The traditional Kolmogorov type predator-prey model with Holling's type  II functional response:
(1.1)x′=rx(1-xK)-cy(xm+x),y′=-dy+fy(xm+x),
and its various generalized forms have received great attention from both theoretical and mathematical biologists and have been well studied [1, 8, 9]. In (1.1), x and y represent the densities of prey specie and predator specie at time t, respectively. Predator-prey model with Holling's type  II functional response assumes that predators do not interfere with one another's activities; thus competition among predators for food occurs only via the depletion of prey. However, when predators have to search for food (and therefore, have to share or compete for food), the functional response in a prey-predator model should be predator-dependent. The predator-prey system with the Beddington-DeAngelis functional response:
(1.2)x′=rx(1-xK)-αxya+bx+cy,y′=-dy+βxya+bx+cy
was originally introduced by Beddington [10] and DeAngelis et al. [11], independently, where x and y represent the densities of prey specie and predator specie at time t, respectively. The main difference of this functional response from Holling's type  II functional response is that it contains an extra term presenting mutual interference by predators. In this model, individuals from a population of two or more predators not only allocate time to searching for and processing prey, but also spend some time engaging in encounters with other predators, the Beddington-DeAngelis type can provide better descriptions of predator feeding over a range of predator-prey abundances.On the other hand, differential equations with impulsive effects form a wide set of different problems. During the last three decades those problems were intensively studied. Some authors devote themselves to the study of impulsive differential equation [12–22]. The main definitions and results of the theory of systems of ordinary differential equations with impulse effects were given in [12, 13, 15]. Similarity and differentiality of such problems of applied mathematics with corresponding problems of ordinary differential equations (and without the conditions of impulsive effects) were demonstrated, and general characteristics of these systems were described. Periodic and almost-periodic solutions of differential equations with impulsive effects were studied in [18]. De La Sen investigated time-varying systems with nonnecessarily bounded everywhere continuous time-differentiable time-varying point delays [19]. The delay-free and delayed dynamics are assumed to be time-varying and impulsive, the constructed solution trajectories of both the unforced and forced systems are obtained from different (input-state space/output space and state space to output space) operators. The system stability and the compactness of the operators describing the solution trajectories are well investigated. De La Sen and Luo also obtain sufficiency-type stability results for time-delay linear systems with constant point delays under impulsive inputs of impulses of state-dependent amplitudes occurring separately through time. They proved that the amplitudes of the impulses and the time intervals between impulses may be chosen sufficiently large if the delay-free dynamics is sufficiently stable compared to the delayed one [20]. Xu and Sun investigated the problem of finite-time stability of linear time-varying singular systems with impulses at fixed times, they have proposed a sufficient condition for linear singular impulsive systems to be finite-time stable in terms of a set of coupled matrix inequalities [21]. Zhang and Sun also considered the stability of impulsive linear differential equations with time delay. By using Lyapunov functions and analysis technique, they get some results for the stability of impulsive linear differential equations with time delay [22].Some impulsive factors have also great impact on the growth of a population. For example, we notice that the births of many species are not continuous but happen at some regular time (e.g., the births of some wildlife are seasonal). Moreover, the human beings have been harvesting or stocking species at some time, then the species is affected by another type of impulse. If we incorporate these impulsive factors into the models of population interactions, the models must be governed by impulsive ordinary differential equations.Population communities are embedded in periodically varying environments. Therefore, this study should take into account the biological and environmental periodicity (e.g., seasonal effects of weather, food supplies, and  mating habits), we focus on the existence of periodic solution with strictly positive components. In real world, any biological or environmental parameters are naturally subject to fluctuation in time, so it is reasonable to study the corresponding nonautonomous system.The first author investigated the existence of positive periodic solutions of a nonautonomous competitive Lotka-Volterra system with impulse and Holling type  III functional response [23]. An N-dimensional Lotka-Volterra system with fixed moments of impulsive perturbations is given in [24] by Ahmad and Stamova. By means of piecewise continuous functions which are modifications of classical Lyapunov’ functions they give sufficient conditions for asymptotic stability of the solutions. Inspired by [23, 24], in this paper, we focus our attention on the existence of periodic solution and globally asymptotic stability of solutions for multispecies mutualism-competition-predator system with impulses in which the competition among predator species and the mutualism among prey species are simultaneously considered. In general, mutualism is relevant to two species [25, 26], thus we focus on two prey species and assume that there exists the relation of mutualism between two prey species. The primary approach is based on the coincidence degree and its related continuation theorem [27], which has been widely used in dealing with the existence of periodic solutions of differential equations. We will investigate the following nonautonomous system:
(1.3)x˙1(t)=x1(t)(∑k=1nc1k(t)yk(t)α1k(t)+β1k(t)x1(t)+γ1k(t)yk(t)b1(t)-a1(t)x1(t)-x1(t)μ1(t)+ω1(t)x2(t)-∑k=1nc1k(t)yk(t)α1k(t)+β1k(t)x1(t)+γ1k(t)yk(t)‍),x˙2(t)=x2(t)(∑k=1nc1k(t)yk(t)α1k(t)+β1k(t)x1(t)+γ1k(t)yk(t)b2(t)-a2(t)x2(t)-x2(t)μ2(t)+ω2(t)x1(t)-∑k=1n‍c2k(t)yk(t)α2k(t)+β2k(t)x2(t)+γ2k(t)yk(t)),y˙j(t)=yj(t)(∑k=1nc1k(t)yk(t)α1k(t)+β1k(t)x1(t)+γ1k(t)yk(t)-rj(t)+∑k=12‍djk(t)xk(t)αjk(t)+βjk(t)xk(t)+γjk(t)yj(t)-∑k=1n‍δjk(t)yk(t)),t≠tk,(k∈Z+),j=1,…,n,Δxi(t)=xi(t+)-xi(t-)=bikxi(t),Δyj(t)=yj(t+)-yj(t-)=b~jkyj(t),t=tk,(k∈Z+),i=1,2,j=1,…,n,
where(i)
xi(t)  (i=1,2) denote the densities of prey species at time t, respectively;(ii)
yj(t)  (j=1,…,n) denote the density of predator species at time t, respectively;(iii)
bik represent the sum of the birth rate and the harvesting (or stocking) rate of xi(t) at time t, respectively;(iv)
b~jk represent the sum of the birth rate and the harvesting (or stocking) rate of yj(t) at time t, respectively;(v)
xi(t+) and xi(t-) represent the right and left limit of xi(t) at t, yj(t+) and yj(t-) represent the right and left limit of yj(t) at t.Letg(t) be a bounded continuous function on ℝ. Define (1.4)gl=inf⁡t∈Rg(t),gu=sup⁡t∈Rg(t).
Particularly, if g(t) are T-periodic functions with respect to t, we denote
(1.5)g¯=1T∫0T‍g(t)dt.
The range of the indices i∈{1,2} and j,k∈{1,…,n} are used in this paper unless otherwise stated.Throughout the paper, we give the hypotheses as follows(A1)
for anyt∈ℝ, bi(t), ai(t), rj(t), μi(t), ωi(t), dji(t), δjk(t), cik(t), αik, βik, γik are nonnegative continuous T-periodic functions and αij=αji, βij=βji, γij=γji;(A2)
bik, b~jk>0, 1+bik>0, 1+b~jk>0, bik, b~jk(k∈ℤ+) are constants. There exists a positive integer q, such that tk+q=tk+T,bi(k+q)=bik,b~j(k+q)=b~jk,(k∈ℤ+). Without loss of generality, we also suppose that tk≠0 and [0,T]∩{tk:k∈ℤ+}={t1,t2,…,ts}, then it follows that q=s;(A3)
xi(t),yj(t) is left-continuous at tk, that is, the following relations are satisfied:
(1.6)xi(tk-)=xi(tk),xi(tk+)=(1+bik)xi(tk),k∈Z+,yj(tk-)=yj(tk),yj(tk+)=(1+b~jk)yj(tk),k∈Z+;(A4)
t1<t2<⋯ and limk→∞tk=∞;(A5)
xi(t0+)>0,yj(t0+)>0.A brief description of the organization of this paper is as follows. The basic concepts and lemmas are given in Section2. The main results in this paper are stated in Theorems 3.1, 4.5, and 4.6.


## 2. Basic Concepts and Lemma

Letφ:[0,T]⊂J→ℝ be a piecewise continuous function with points of discontinuity t1,…,ts, we denote
(2.1)PCT={φ:φ(0)=φ(T)},PCT1={φ∈PCT:(dφdt)∈PCT}.
Suppose
(2.2)‖φ‖PCT=sup⁡t∈[0,T]|φ(t)|,‖φ‖PCT1=max⁡{‖φ‖PCT,‖φ˙‖PCT}.
It can be easily proved that PCT and PCT1 are Banach spaces under the condition endowed the above norms.Make the change of variablesxi(t)=exp{ui(t)},yj(t)=exp{vj(t)}, then (1.1) can be reformulated as
(2.3)u˙1(t)=b1(t)-a1(t)exp⁡{u1(t)}-exp⁡{u1(t)}μ1(t)+ω1(t)exp⁡{u2(t)}-∑k=1n‍c1k(t)exp⁡{vk(t)}α1k(t)+β1k(t)exp⁡{u1(t)}+γ1k(t)exp⁡{vk(t)},u˙2(t)=b2(t)-a2(t)exp⁡{u2(t)}-exp⁡{u2(t)}μ1(t)+ω1(t)exp⁡{u1(t)}-∑k=1n‍c2k(t)exp⁡{vk(t)}α2k(t)+β2k(t)exp⁡⁡{u2(t)}+γ2k(t)exp⁡{vk(t)},y˙j(t)=-rj(t)+∑k=12‍djk(t)exp⁡⁡{uk(t)}αjk(t)+βjk(t)exp⁡⁡{uk(t)}+γjk(t)exp⁡⁡{vj(t)}-∑k=1n‍δjk(t)exp⁡{vk(t)},t≠tk,(k∈Z+),j=1,…,n,Δui(t)=ui(t+)-ui(t-)=ln⁡(1+bik),Δvj(t)=vj(t+)-vj(t-)=ln⁡(1+b~jk),t=tk(k∈Z+),i=1,2,j=1,…,n.
For (1.3) and (2.3), we have similar lemma and definitions. So we only relate such results for (1.3).Definition 2.1.
The mappingx:[0,T]→ℝn+2,x(t)=(x1(t),x2(t),y1(t),…,yn(t)) is called a solution of system (1.1) in [0,T], if (1)
x(t) is piecewise continuous in [0,T] and [0,T]∩{tk:k∈ℤ+} are discontinuous points of the first kind of x(t), and x(t) are left continuous at [0,T]∩{tk:k∈ℤ+};(2)
x(t) satisfies system in [0,T].Definition 2.2.
The mappingx:J→ℝn+2 is called a T-periodic solution of system (1.1) if (1)
x(t)=(x1(t),x2(t),y1(t),…,yn(t)) is a solution of system (1.1);(2)
x(t) satisfies x(t+T-0)=x(t-0),t∈ℝ.Obviously, ifx(t) is a solution of (1.1) satisfying x(0)=x(T) in [0,T], then from the periodicity of the vector field of (1.1), we know that
(2.4)x*(t)={x(t-lT),t∈[lT,(l+1)T]∖{tk:k∈Z+},l∈N,x*,(t)isleftcountiousattk
is an T-periodic solution for (1.1). Thus, the problem which discusses the existence of solution of (1.1) will be transformed to discuss the existence of periodic solution for (1.1) in [0,T] satisfying x(0)=x(T). In order to explore the existence of positive periodic solutions of (1.1), we recall some concepts and results on coincidence degree from [27, pages 39-40], borrowing notations there.Let𝕏,ℤ be normed vector spaces, ℒ:Domℒ⊆𝕏→ℤ a linear mapping, and 𝒩:𝕏→ℤ a continuous mapping. If dimkerℒ=comdimImℒ<+∞ and Imℒ is a closed in ℤ, then the mapping ℒ will be called a Fredholm mapping of index zero. If ℒ is a Fredholm mapping of index zero, there exists continuous projects 𝒫:𝕏→𝕏 and 𝒬:ℤ→ℤ such that Im𝒫=Kerℒ,Imℒ=Ker𝒬=Im(I-𝒬). It follows that ℒ|Domℒ∩Ker𝒫:(I-𝒫)𝕏→Imℒ has an inverse which is denoted by Kp. If Ω be an open bounded subset of 𝕏, the mapping 𝒩 will be called ℒ-compact on Ω¯ provided that 𝒬𝒩(Ω¯) is bounded, and Kp(I-𝒬)𝒩:Ω¯→𝕏 is compact. Since Im𝒬 is isomorphic to Kerℒ, there exists an isomorphism 𝒥:Im𝒬→Kerℒ.The following continuation theorem is due to Gaines and Mawhin [27].Lemma 2.32.3 (Continuation Theorem in [27]).
Letℒ a Fredholm mapping of index zero and 𝒩 be ℒ-compact on Ω¯. Suppose that (a)
for eachλ∈(0,1), every solution x of ℒx=λ𝒩x such that x∉∂Ω;(b)
𝒬𝒩x≠0 for each x∈Kerℒ∩∂Ω and deg{𝒥𝒬𝒩,Ω∩Kerℒ,0}≠0.Then the operator equation ℒx=𝒩x has at least one solution lying in Domℒ∩Ω¯.We can show that the solution of (1.1) with positive initial value remains positive too, that is, the following Lemma 2.4 holds.Lemma 2.4.
Suppose the hypotheses (A1)–(A5) hold.xi(t),yj(t) is a solution of (1.1), then
(2.5)xi(t),yj(t)>0,t∈J.Proof.
By integrating of (1.1) in the interval (t0,t1], we have
(2.6)x1(t)=xi(t0+)exp⁡{∑k=1n‍∫t0t‍(∑k=1n‍b1(t)-a1(t)x1(t)-x1(t)μ1(t)+ω1(t)x2(t)-∑k=1n‍c1k(t)yk(t)α1k(t)+β1k(t)x1(t)+γ1k(t)yk(t))dt},x2(t)=xi(t0+)exp⁡{∑k=1n‍∫t0t‍(∑k=1n‍b2(t)-a2(t)x2(t)-x2(t)μ2(t)+ω2(t)x1(t)-∑k=1n‍c2k(t)yk(t)α2k(t)+β2k(t)x2(t)+γ2k(t)yk(t))dt},yj(t)=yj(t0+)exp⁡{∑k=1n‍∫t0t‍(∑k=1n‍-rj(t)+∑k=12‍djk(t)xk(t)αjk(t)+βjk(t)xk(t)+γjk(t)yj(t)-∑k=1n‍δjk(t)yk(t))dt},
for t∈(t0,t1].
There does not exist point of discontinuity ofxi(t), yj(t) in the interval (t0,t1], it is obvious that xi(t)>0, yj(t)>0 for t∈(t0,t1], hence x(t1)>0. We have from (1.1) that
(2.7)xi(t1+)=(1+bik)xi(t1),yj(t1+)=(1+b~jk)yj(t1).
It follows from (A2) that
(2.8)xi(t1+)=(1+bik)xi(t1)>0,yj(t1+)=(1+b~jk)yj(t1)>0.
We now integrate (1.1) in the interval (t1,t2], and we have
(2.9)x1(t)=xi(t1+)exp⁡{∑k=1n‍∫t0t‍(∑k=1n‍b1(t)-a1(t)x1(t)-x1(t)μ1(t)+ω1(t)x2(t)-∑k=1n‍c1k(t)yk(t)α1k(t)+β1k(t)x1(t)+γ1k(t)yk(t))dt},x2(t)=xi(t1+)exp⁡{∑k=1n‍∫t0t‍(∑k=1n‍b2(t)-a2(t)x2(t)-x2(t)μ2(t)+ω2(t)x1(t)-∑k=1n‍c2k(t)yk(t)α2k(t)+β2k(t)x2(t)+γ2k(t)yk(t))dt},yj(t)=yj(t1+)exp⁡{∑k=1n‍∫t0t‍(∑k=1n‍-rj(t)+∑k=12‍djk(t)xk(t)αjk(t)+βjk(t)xk(t)+γjk(t)yj(t)-∑k=1n‍δjk(t)yk(t))dt},
for t∈(t1,t2]. From the above relation it follows that xi(t)>0 for t∈(t1,t2].
By similar arguments, we can obtain that(2.10)x1(t)=xi(tk+)exp⁡{∑k=1n‍∫t0t‍(∑k=1n‍b1(t)-a1(t)x1(t)-x1(t)μ1(t)+ω1(t)x2(t)-∑k=1n‍c1k(t)yk(t)α1k(t)+β1k(t)x1(t)+γ1k(t)yk(t))dt},x2(t)=xi(tk+)exp⁡{∑k=1n‍∫t0t‍(∑k=1n‍b2(t)-a2(t)x2(t)-x2(t)μ2(t)+ω2(t)x1(t)-∑k=1n‍c2k(t)yk(t)α2k(t)+β2k(t)x2(t)+γ2k(t)yk(t))dt},yj(t)=yj(tk+)exp⁡{∑k=1n‍∫t0t‍(-rj(t)+∑k=12‍djk(t)xk(t)αjk(t)+βjk(t)xk(t)+γjk(t)yj(t)-∑k=1n‍δjk(t)yk(t))dt},
for t∈(tk,tk+1]. So xi(t)>0,yj(t)>0 for t∈J.


## 3. Existence of Positive Periodic Solution

We denote(3.1)Δi=b¯i+1T∑k=1q‍ln⁡(1+bik),Δ~j=1T∑k=1q‍ln⁡(1+b~jk)-r¯j,B=(a¯i*j*00δ¯i′j′),Δ=(ΔiΔ~j),
where i*,j*∈{1,…,n},i′,j′∈{1,…,n}. We also denote Bk(k∈{1,…,n+2}) the matrix obtained by replacing the kth column of B with Δ.Theorem 3.1.
If system (2.3) satisfies (A1)–(A5) and following conditions(A6)
Δ1>((exp{H1}/K2)(1/ω1-)-∑k=1n(c1k/γ1k-))/a¯1,Δ2>((exp{H2}/K1)(1/ω2-)-∑k=1n(c2k/γ2k-))/a¯2,Δ~j>∑k=1,k≠jnδ¯jkexp{H~k}/δ¯jj,  where
(3.2)Hi=max⁡{ln⁡(Δia¯ii)}+2∑k=1q‍ln⁡(1+bik),H~j=max⁡{ln⁡Δ~j+∑k=1n(djk/βjk-)δ¯jj}+2∑k=1q‍ln⁡(1+bjk),(A7)
B>0,Bk>0,k∈{1,…,n+2}.Then system (1.1) has at least one positive T-periodic solution.Proof.
Let𝕏=PCT1×PCT1×⋯×PCT1 be the Cartesian production of (n+2)-tuples, ℤ=𝕏×ℝ(n+2)q, where ℝ(n+2)q is (n+2)q-dimensional Euclidean space. It is clear that x(0)=x(T) for any x∈𝕏.
We define(3.3)‖x‖=∑i=12‍‖xi‖PCT1+∑j=1n‍‖yj‖PCT1=∑i=12‍sup⁡[0,T]|xi(t)|+∑j=1n‍sup⁡[0,T]|yj(t)|,‖z‖Z=‖x‖+‖c‖*,
for any x=(x1,x2,y1,…,yn)∈𝕏,c=(c1,…,c(n+2)q)∈ℝ(n+2)q, and z=(x,c)∈ℤ, where ∥·∥ is the norm of 𝕏, and ∥·∥* is any norm of ℝ(n+2)q. Then it is trivial to check that 𝕏,ℤ are both Banach spaces when they are endowed with the above norms ∥·∥ and ∥·∥ℤ, respectively.
Let(3.4)L:Dom⁡L⊂X→Z,(ui(t)vj(t))⟶((u˙i(t)v˙j(t)),{(Δui(tk)Δvj(tk))}k=1q),N:X→Z,N(ui(t)vj(t))=((u˙i(t)v˙j(t)),{(Δui(tk)Δvj(tk))}k=1q).
Then
(3.5)Ker⁡L={(ui(t)vj(t)):(ui(t)vj(t))=(c1⋮cn+2)∈Rn+2},Im⁡L={z=((fif~j),{(likl~jk)}k=1q)∈Z:∫0T‍fidt+∑k=1q‍lik=0,∫0T‍f~jdt+∑k=1q‍l~jk=0},dim⁡Ker⁡L=n+2=codimIm⁡L.
Since Imℒ is closed in ℤ,ℒ is a Fredholm mapping of index zero. We respectively define 𝒫 and 𝒬 in the following:
(3.6)P(ui(t)vj(t))=1T(∫0T‍ui(t)dt∫0Tvj(t)dt‍),Qz=Q((fif~j),{(likl~jk)}k=1q)=(1T(∫0Tfidt+∑k=1q‍lik‍∫0T‍f~jdt+∑k=1q‍l~j,k),{(0⋮0)}k=1q).
It is easy to show that 𝒫, 𝒬 are continuous projectors such that Imℒ=Ker𝒬=Im(I-𝒬). Furthermore, the generalized inverse (ℒ)Kp:Imℒ→Domℒ⋂‍Ker𝒫 exists. That follows one will calculate Kp.
Let(3.7)z=((fif~j),{(likl~jk)}k=1q)∈Im⁡L,
then
(3.8)(u˙i(t)v˙j(t))=(fi(t)f~j(t))t≠tk,(Δui(t)Δvj(t))t=tk=(likl~jk),
that is,
(3.9)(ui(t)vj(t))=(∫0t‍fi(s)ds+∑t>tk‍lik+ui(0)∫0t‍f~j(s)ds+∑t>tk‍l~jk+vj(0)).
Note that
(3.10)(ui(t)vj(t))∈Ker⁡P,thatis,(1T∫0T‍ui(s)ds1T∫0T‍vj(s)ds)=(00).
From(3.9), we obtain
(3.11)(1T∫0T‍∫0t‍ui(s)dsdt+∫0T‍∑t>tk‍likdt+Tui(0)1T∫0T‍∫0t‍vj(s)dsdt+∫0T‍∑t>tk‍l~jkdt+Tvj(0))=(00),
and hence,
(3.12)(ui(t)vj(t))=(∫0t‍ui(s)ds+∑t>tk‍lik-1T∫0T‍∫0t‍ui(s)dsdt-∑k=1q‍lik+1T∑k=1q‍liktk∫0t‍vj(s)ds+∑t>tk‍l~jk-1T∫0T‍∫0t‍vj(s)dsdt-∑k=1q‍l~jk+1T∑k=1q‍l~jktk),
that is,
(3.13)Kpz=(∫0t‍fi(s)ds+∑t>tk‍lik-1T∫0T‍∫0t‍fi(s)dsdt-∑k=1q‍lik∫0t‍f~j(s)ds+∑t>tk‍l~jk-1T∫0T‍∫0t‍f~j(s)dsdt-∑k=1q‍l~jk).
Thus
(3.14)QN=(ui(t)vj(t))=((hih~j),{(00)}k=1q),
where
(3.15)h1=Δ1-1T∫0T‍(∑k=1n‍a1(t)exp⁡{u1(t)}+exp⁡⁡{u1(t)}μ1(t)+ω1(t)exp⁡⁡{u2(t)}+∑k=1n‍c1k(t)exp⁡⁡{vk(t)}α1k(t)+β1k(t)exp⁡⁡{u1(t)}+γ1k(t)exp⁡⁡{vk(t)})dt,h2=Δ2-1T∫0T‍(∑k=12‍a2(t)exp⁡{u2(t)}+exp⁡{u2(t)}μ1(t)+ω1(t)exp⁡{u1(t)}+∑k=1n‍c2k(t)exp⁡{vk(t)}α2k(t)+β2k(t)exp⁡{u2(t)}+γ2k(t)exp⁡{vk(t)})dt,h~j=Δ~j+1T∫0T‍(∑k=12‍djk(t)exp⁡{uk(t)}αjk(t)+βjk(t)exp⁡{uk(t)}+γjk(t)exp⁡{vj(t)}-∑k=1n‍δjk(t)exp⁡⁡{vk(t)})dt.
Clearly, 𝒬𝒩 and Kp(I-𝒬)𝒩 are continuous. Using the Arzela-Ascoli theorem, it is not diffcult to show that Kp(I-𝒬)𝒩(Ω¯)¯ is compact for any open bounded set Ω⊂𝕏. Moreover, 𝒬𝒩(Ω) is bounded. Thus, 𝒩 is ℒ-compact on Ω¯ with any open bounded set Ω⊂𝕏. A isomorphism 𝒥 of Im𝒬 onto Kerℒ can be chosen to be the identity mapping; since Im𝒬=Kerℒ, there exists an isomorphism 𝒥:Im𝒬→𝕏 given by
(3.16)J:Im⁡Q⟶X,((g1⋮gn+2),{(0⋮0)}k=1q)⟶(g1⋮gn+2).
Corresponding to operator equation ℒu=λ𝒩u,u(t)=(u1(t),u2(t),v1(t),…,vn(t))T,λ∈(0,1), we have
(3.17)u˙1(t)=λ(∑k=1n‍b1(t)-a1(t)exp⁡{u1(t)}-exp⁡⁡{u1(t)}μ1(t)+ω1(t)exp⁡⁡{u2(t)}-∑k=1n‍c1k(t)exp⁡⁡{vk(t)}α1k(t)+β1k(t)exp⁡⁡{u1(t)}+γ1k(t)exp⁡⁡{vk(t)}),u˙2(t)=λ(∑k=1n‍b2(t)-a2(t)exp⁡{u2(t)}-exp⁡⁡{u2(t)}μ1(t)+ω1(t)exp⁡⁡{u1(t)}-∑k=1n‍c2k(t)exp⁡⁡{vk(t)}α2k(t)+β2k(t)exp⁡⁡{u2(t)}+γ2k(t)exp⁡⁡{vk(t)}),v˙j(t)=λ(-rj(t)+∑k=12‍djk(t)exp⁡⁡{uk(t)}αjk(t)+βjk(t)exp⁡⁡{uk(t)}+γjk(t)exp⁡⁡{vj(t)}-∑k=1n‍δjk(t)exp⁡{vk(t)}),t≠tk,(k∈N+),Δui(t)=ui(t+)-ui(t-)=λln⁡(1+bik),Δvj(t)=vj(t+)-vj(t-)=λln⁡(1+b~jk),t=tk,(k∈N+).
Suppose thatu(t)=(u1(t),u2(t),v1(t),…,vn(t))T is a solution of system (3.17) for a fixed λ∈(0,1). Integrating on both sides of (3.17) from 0 to T, we obtain
(3.18)∫0T‍(∑k=1n‍-b1(t)+a1(t)exp⁡{u1(t)}+exp⁡⁡{u1(t)}μ1(t)+ω1(t)exp⁡⁡{u2(t)}+∑k=1n‍c1k(t)exp⁡⁡{vk(t)}α1k(t)+β1k(t)exp⁡⁡{u1(t)}+γ1k(t)exp⁡⁡{vk(t)})dt=∑k=1q‍ln⁡(1+b1k),∫0T‍(∑k=1n‍-b2(t)+a2(t)exp⁡{u2(t)}+exp⁡⁡{u2(t)}μ2(t)+ω2(t)exp⁡⁡{u1(t)}+∑k=1n‍c2k(t)exp⁡⁡{vk(t)}α2k(t)+β2k(t)exp⁡⁡{u2(t)}+γ2k(t)exp⁡⁡{vk(t)})dt=∑k=1q‍ln⁡(1+b2k),∫0T‍(rj(t)-∑k=12‍djk(t)exp⁡⁡{uk(t)}αjk(t)+βjk(t)exp⁡⁡{uk(t)}+γjk(t)exp⁡⁡{vj(t)}+∑k=1n‍δjk(t)exp⁡{vk(t)})dt=∑k=1q‍ln⁡(1+b~jk).
Since u∈𝕏, there exist ξi,ξj∈[0,T] such that
(3.19)ui(ξi)=min⁡t∈[0,T]ui(t),vj(ξj)=min⁡t∈[0,T]vj(t).
On the other hand, note that there exist ζi+,ζj+∈[0,T] such that
(3.20)ui(ζi+)=sup⁡t∈[0,T]ui(t),vj(ζj+)=sup⁡t∈[0,T]vj(t).
If ζi≠tk, then ui(ζi+)=ui(ζi),vj(ζj+)=vj(ζj).  While if ζj=tk, we have ui(ζi+)=ui(tk+),vj(ζj+)=vj(tk+). Thus we can obtain from (3.18) and (3.19) that
(3.21)∫0T‍ai(t)exp⁡{ui(ξi)}dt⩽∫0T‍ai(t)eui(t)dt⩽∑k=1q‍ln⁡(1+bik)+∫0T‍bi(t)dt,∫0T‍δjj(t)exp⁡{vj(ξj)}dt⩽∫0T‍δjj(t)exp⁡{vj(t)}dt⩽∑k=1q‍ln⁡(1+b~jk)-∫0T‍rj(t)dt+∑k=1n‍∫0T‍djk(t)βjk(t)dt.
It follows that
(3.22)ui(ξi)⩽max⁡{ln⁡Δia¯i}≔p,vj(ξj)⩽max⁡{ln⁡Δ~j+∑k=1n(djk/βjk-)δ¯jj}:=g.
From (3.17) and (3.18), we obtain
(3.23)∫0T‍|u˙i(t)|dt⩽2∑k=1q‍ln⁡(1+bik),∫0T‍|v˙j(t)|dt⩽2∑k=1q‍ln⁡(1+b~jk).
By (3.22) and (3.23), we have that
(3.24)ui(t)⩽ui(ξi)+∫0T‍|u˙i(t)|dt⩽p+2∑k=1q‍ln⁡(1+bik)∶=Hi,vj(t)⩽vj(ξj)+∫0T‍|v˙j(t)|dt⩽g+2∑k=1q‍ln⁡(1+b~jk)∶=H~j.
On the other hand, by (3.18) and (3.20), we also have
(3.25)∫0T‍δjj(t)exp⁡{vj(ζj+)}dt⩾∫0T‍δjj(t)exp⁡{vj(t)}dt⩾∑k=1q‍ln⁡(1+b~jk)-∫0T‍rj(t)dt-∑k=1,k≠jn‍∫0T‍δjk(t)exp⁡{H~k}dt,vj(ζj+)⩾ln⁡(Δ~j-∑k=1,k≠jnδ¯jkexp⁡{H~k}δ¯jj),
and hence,
(3.26)vj(t)⩾vj(ζj+)-∫0T‍|v˙j(t)|dt⩾ln⁡(Δ~j-∑k=1,k≠jnδ¯jkexp⁡{H~k}δ¯jj)-2∑k=1q‍ln⁡(1+b~jk)∶=M~j.By (3.18), we have(3.27)∫0T‍a1(t)exp⁡{u1(ζ1+)}dt⩾∫0T‍a1(t)exp⁡{u1(t)}dt⩾∑k=1q‍ln⁡(1+b1k)+∫0T‍b1(t)dt-∫0T‍exp⁡{u1(t)}μ1(t)+ω1(t)exp⁡{u2(t)}dt-∑k=1n‍∫0T‍c1k(t)exp⁡{vk(t)}α1k(t)+β1k(t)exp⁡{u1(t)}+γ1k(t)exp⁡{vk(t)}dt⩾∑k=1q‍ln⁡(1+b1k)+∫0T‍b1(t)dt-exp⁡{H1}K2∫0T‍1ω1(t)dt-∑k=1n‍∫0T‍c1k(t)γ1k(t)dt,∫0T‍a2(t)exp⁡{u2(ζ2+)}dt⩾∫0T‍a2(t)exp⁡{u2(t)}dt⩾∑k=1q‍ln⁡(1+b2k)+∫0T‍b2(t)dt-∫0T‍exp⁡{u2(t)}μ2(t)+ω2(t)exp⁡{u1(t)}dt-∑k=1n‍∫0T‍c2k(t)exp⁡{vk(t)}α2k(t)+β1k(t)exp⁡{u2(t)}+γ2k(t)exp⁡{vk(t)}dt⩾∑k=1q‍ln⁡(1+b2k)+∫0T‍b2(t)dt-exp⁡{H2}K1∫0T‍1ω2(t)dt-∑k=1n‍∫0T‍c2k(t)γ2k(t)dt,
where Ki is a positive number satisfied exp{ui(t)}⩾Ki for t∈[0,T]; therefore, (3.28)u1(ζ1+)⩾ln⁡(Δ1-(exp⁡⁡{H1}/K2)(1/ω1¯)-∑k=1n(c1k/γ1k¯)a¯1),u2(ζ2+)⩾ln⁡(Δ2-(exp⁡⁡{H2}/K1)(1/ω2¯)-∑k=1n(c2k/γ2k¯)a¯2).
So
(3.29)u1(t)⩾u1(ζ1+)-∫0T‍|u˙1(t)|dt⩾ln⁡(Δ1-(exp⁡⁡{H1}/K2)(1/ω1¯)-∑k=1n(c1k/γ1k¯)a¯1)-2∑k=1q‍ln⁡(1+b1k)∶=N1,u2(t)⩾u2(ζ2+)-∫0T‍|u˙2(t)|dt⩾ln⁡(Δ2-(exp⁡⁡{H2}/K1)(1/ω2¯)-∑k=1n(c2k/γ2k¯)a¯2)-2∑k=1q‍ln⁡(1+b2k)∶=N2.
Take Mi=min{Ki,Ni}. By (3.24)–(3.29),  we have(3.30)|ui(t)|<max⁡{|Mi|,|Hi|}∶=Gi,|vj(t)|<max⁡{|M~j|,|H~j|}∶=G~j.
Here, Gi,G~j are independent of the choice of λ.
Letu=(u1(t),u2(t),v1(t),…,vn(t))T∈ℝn+2, set G=∥u∥+G0, where G0 is taken sufficiently large, such that the solution u*=(u1*,u2*,v1*,…,vn*)T of the equation
(3.31)b¯1-a¯1exp⁡{u1(t)}-1T∫0T‍exp⁡⁡{u1(t)}dtμ1(t)+ω1(t)exp⁡⁡{u2(t)}-1T∑k=1n‍∫0T‍c1k(t)exp⁡⁡{vk(t)}dtα1k(t)+β1k(t)exp⁡⁡{u1(t)}+γ1k(t)exp⁡{vk(t)}+1T∑k=1q‍ln⁡(1+b1k)=0,b¯2-a¯2exp⁡⁡{u2(t)}-1T∫0T‍exp⁡⁡{u2(t)}dtμ2(t)+ω2(t)exp⁡⁡{u1(t)}-1T∑k=1n‍∫0T‍c2k(t)exp⁡⁡{vk(t)}dtα2k(t)+β2k(t)exp⁡⁡{u2(t)}+γ2k(t)exp⁡⁡{vk(t)}+1T∑k=1q‍ln⁡⁡(1+b2k)=0,-r¯j+1T∑k=12‍∫0T‍djk(t)exp⁡⁡{uk(t)}dtαjk(t)+βjk(t)exp⁡⁡{uk(t)}+γjk(t)exp⁡⁡{vj(t)}-∑k=1n‍δ¯jkexp⁡⁡{vk(t)}+1T∑k=1q‍ln⁡⁡(1+b~jk)=0
satisfies
(3.32)‖(u1*,u2*,v1*,…,vn*)T‖<G0,‖u*(tk+0)‖<G,(k=1,2,…,q).
LetΩ={u=(u1,u2,v1,…,vn)T∈𝕏∣∥(u1,u2,v1,…,vn)T∥<G}. Then it is clear that Ω satisfies the requirement (a) of Lemma 2.3.
Whenu∈∂Ω∩Kerℒ=∂Ω∩ℝn+2, u is a constant vector in ℝn+2 with ∥u∥=G, then 𝒬𝒩u≠0, we check 𝒬𝒩u and according to mean value theorem, there exist θ1,θ2,θ~j,η1,η2∈[0,T] such that
(3.33)QNu=((Δ1-a¯1exp⁡{u1(t)}-exp⁡{u1(t)}μ1(θ1)+ω1(θ1)exp⁡{u2(t)}-∑k=1n‍c¯1kexp⁡{vk(t)}α1k(η1)+β1k(η1)exp⁡{u1(t)}+γ1k(η1)exp⁡{vk(t)}Δ2-a¯2exp⁡{u2(t)}-exp⁡{u2(t)}μ2(θ2)+ω2(θ2)exp⁡{u1(t)}-∑k=1n‍c¯2kexp⁡{vk(t)}α2k(η2)+β2k(η2)exp⁡{u2(t)}+γ2k(η2)exp⁡{vk(t)}Δ~j+∑k=12‍d¯jkexp⁡{uk(t)}αjk(θ~j)+βjk(θ~j)exp⁡{uk(t)}+γjk(θ~j)exp⁡{vj(t)}-∑k=1n‍δ¯jkexp⁡{vk(t)}){(0⋮0)}k=1q).
For any u∈Ω∩Kerℒ, we have
(3.34)JQNu=(Δ1-a¯1exp⁡{u1(t)}-exp⁡{u1(t)}μ1(θ1)+ω1(θ1)exp⁡{u2(t)}-∑k=1n‍c¯1kexp⁡{vk(t)}α1k(η1)+β1k(η1)exp⁡{u1(t)}+γ1k(η1)exp⁡{vk(t)}Δ2-a¯2exp⁡{u2(t)}-exp⁡{u2(t)}μ2(θ2)+ω2(θ2)exp⁡{u1(t)}-∑k=1n‍c¯2kexp⁡{vk(t)}α2k(η2)+β2k(η2)exp⁡{u2(t)}+γ2k(η2)exp⁡{vk(t)}Δ~j+∑k=12‍d¯jkexp⁡{uk(t)}αjk(θ~j)+βjk(θ~j)exp⁡{uk(t)}+γjk(θ~j)exp⁡{vj(t)}-∑k=1n‍δ¯jkexp⁡{vk(t)}).
Define the map ψ:(Ω¯∩ℝn+2)×[0,1]→ℝn+2 by
(3.35)ψ(u1u2vj)=(Δ1-a¯1exp⁡{u1(t)}Δ2-a¯2exp⁡{u2(t)}Δ~j-∑k=1n‍δ¯jkexp⁡{vk(t)})-μ(exp⁡⁡{u1(t)}μ1(θ1)+ω1(θ1)exp⁡⁡{u2(t)}+∑k=1n‍c¯1kexp⁡⁡{vk(t)}α1k(η1)+β1k(η1)exp⁡⁡{u1(t)}+γ1k(η1)exp⁡⁡{vk(t)}exp⁡⁡{u2(t)}μ2(θ2)+ω2(θ2)exp⁡⁡{u1(t)}+∑k=1n‍c¯2kexp⁡⁡{vk(t)}α2k(η2)+β2k(η2)exp⁡⁡{u2(t)}+γ2k(η2)exp⁡⁡{vk(t)}-∑k=12‍d¯jkexp⁡⁡{uk(t)}αjk(θ~j)+βjk(θ~j)exp⁡⁡{uk(t)}+γjk(θ~j)exp⁡⁡{vj(t)}),
where (u1,u2,v1,…,vn)∈ℝn+2,μ∈[0,1]. We can show that
(3.36)ψ(u1,u2,v1,…,vn,μ)≠0,
for any u=(u1,u2,v1,…,vn)T∈∂Ω∩ℝn+2, μ∈[0,1]. Otherwise, it is similar to above  mentioned discussion, we obtain that the solution u=(u1,u2,v1,…,vn)T of ψ(u1,u2,v1,…,vn,μ)=0 satisfies ∥u∥<G; therefore, this contradicts the fact that u∈∂Ω∩ℝn+2.
Sinceψ(u1,u2,v1,…,vn,μ) is a homotopic mapping and topological degree is invariant under homotopic mapping, thus we can show the topological degree as folows:
(3.37)deg⁡((JQN,Ω∩Ker⁡L,(0,…,0)T))=deg⁡((ψμ=1,Ω∩Ker⁡L,(0,…,0)T))=deg⁡((ψμ=0,Ω∩Ker⁡L,(0,…,0)T)).
On the other hand, the differential equations
(3.38)b¯i-a¯ixk*+1T∑k=1q‍ln⁡(1+bik)=0,-r¯j-∑k=1n‍δ¯jkyk*+1T∑k=1q‍ln⁡(1+b~jk)=0
satisfies condition (A7), by the law of Crammer, one can easily obtain that (3.38) has unique solution
(3.39)(x1*~,x2*~,y1*~,…,yn*~),
where xi*~=det(Bi)/det(B),yj*~=det(Bi)/det(B). Therefore, one can check
(3.40)deg⁡(JQN,Ω∩Ker⁡L,(0,…,0)T)=deg⁡(ψμ=0,Ω∩Ker⁡L,(0,…,0)T)=sgn⁡((-1)n+2x1*x2*∏j=1n‍yj*det⁡(B))≠0.
By now we have proved thatΩ satisfies all requirements of Lemma 2.3, then ℒx=𝒩x has at least one solution in Domℒ∩Ω¯, that is, (2.3) has at least one T-periodic solution in Domℒ∩Ω¯ and denote u^=(u^1,u^2,v^1,…,v^n)T by the solution of system (2.3). Set x^i=exp{u^i},y^j=exp{v^j}, then x^=(x^1,x^2,y^1,…,y^n)T is one positive T-periodic solution of system (1.3). The proof is complete.


## 4. Global Stability and Globally Exponential Stability of Solutions

Letx0=(x10,x20,y1,0,…,yn,0) and xi0,yj0∈R+. We denote by
(4.1)x(t)=x(t;t0,x0)=(x1(t),x2(t),y1(t),…,yn(t))
the solution of system (1.3) satisfying the initial conditions:
(4.2)x(t0+0;t0,x0)=x0
and by J=J(t0,x0) the maximal interval of type [t0,η) in which the solution x(t;t0,x0) is defined.Letx(t;t0,x0),x(t)=(x1(t),x2(t),y1(t),…,yn(t)) and x*(t;t0,x0*), x*(t)=(x1*(t),x2*(t),y1*(t),…,yn*(t)) be any two solutions of (1.1) with initial conditions:
(4.3)x(t0+0;t0,x0)=x0,x*(t0+0;t0,x0*)=x0*,t0∈R+.
We introduce the following notations:
(4.4)Gk=(tk-1,tk)×R+n+2,k∈Z+,G=⋃k=1∞‍Gk.
We put forward two definitions in [24, 28].Definition 4.14.1 (Ahmad and Stamova [24]).
The system (1.3) is said to be(a)
globally stable if for all ɛ>0, there exists δ=δ(ɛ,t0)>0 such that if x(t),x*(t)∈ℝ+n+2, with ∥x0-x0*∥⩽δ then for all t⩾t0∥x(t;t0,x0)-x*(t;t0,x0*)∥<ɛ;(b)
globally asymptotically stableif it is globally stable and
(4.5)lim⁡t→∞‖x(t;t0,x0)-x*(t;t0,x0*)‖=0;(c)
globally exponentially stableif for all α>0, there exists γ=γ(α)>0 such that x(t),x*(t)∈ℝ+n+2, with ∥x0-x0*∥⩽α, then for all t⩾t0(4.6)‖x(t;t0,x0)-x*(t;t0,x0*)‖<γ‖x0-x0*‖exp⁡{-φ(t-t0)}.Definition 4.24.2 (Ahmad and Stamova [24, 28]).
We say that the functionV(t,x),V:[t0,∞)×ℝ+n+2 belongs to the class V0 if the following conditions are satisfied: (1)
the functionV is continuous in ⋃i=1∞Gi, and V(t,0)=0 for t∈[t0,∞);(2)
the function V satisfies locally the Lipschitz condition with respect tox on each of the sets Gi;(3)
for eachk∈ℤ+, there exist the finite limits:
(4.7)V(tk-0,x)=lim⁡t→tk,t<tkV(t,x),V(tk+0,x)=lim⁡t→tk,t>tkV(t,x);(4)
for eachk∈ℤ+, the following equalities are valid:
(4.8)V(tk-1,x)=V(tk,x).
In the proofs of the main theorems, we will use the following comparison results.Lemma 4.3.
Suppose the hypotheses (A1)–(A5) hold. There exist functionsPi,Qi,P~j,Q~j∈ℝn+2 such that Pi(t)⩽xi(t)⩽Qi(t),P~j(t)⩽yj(t)⩽Q~j(t) for all t⩾t0.Proof.
First we will prove that(4.9)xi(t)⩽Qi(t),yj(t)⩽Q~j(t),
for all t⩾t0, where (Qi(t),Q~j(t))=(q1(t),q2(t),q~1(t),…,q~n(t)) is the maximal solution of the system:
(4.10)q˙i(t)=qi(t)(bi(t)-ai(t)qi(t)),q~˙j(t)=q~j(t)(-rj(t)+∑k=12‍djk(t)q~k(t)αjk(t)+βjk(t)q~k(t)+γjk(t)q~j(t)),t≠tk,(k∈Z+),qi(t0+)=qi0>0,q~j(t0+)=q~j0>0,qi(tk+)=(1+bik)qi(tk),q~j(tk+)=(1+b~jk)q~j(tk),t=tk,(k∈Z+).
The maximal solution
(4.11)Qi(t)=Qi(t;t0,q0),Q~j(t)=Qi(t;t0,q0),q0=(q10,q20,q~30,…,q~(n+2)0)
of (4.10) is defined by the equality:
(4.12)Qi(t;t0,q0)={Qi0(t;t0,Qi0+0),t0<t⩽t1,Qi1(t;t1,Qi1+0),t1<t⩽t2,…,Qik(t;tk,Qik+0),tk<t⩽tk+1,…,Q~j(t;t0,q0)={Q~j0(t;t0,Q~j0+0),t0<t⩽t1,Q~j1(t;t1,Q~j1+0),t1<t⩽t2,…,Q~jk(t;tk,Q~jk+0),tk<t⩽tk+1,…,
where Qki(t;tk,Qik+0), Q~kj(t;tk,Q~jk+0) is the solution of the equation without impulses
(4.13)q˙i(t)=qi(t)(bi(t)-ai(t)qi(t)),q~˙j(t)=q~j(t)(-rj(t)+∑k=12‍djk(t)qk(t)αjk(t)+βjk(t)qk(t)+γjk(t)q~j(t))
in the interval (tk,tk+1], k∈ℤ+, for which Qik+0=(1+bik)Qik(tk;tk-1,Qik-1+0), Q~jk+0=(1+b~jk), Q~jk(tk;tk-1,Q~jk-1+0), k∈ℤ+, and Qi0+0=qi0, Q~j0+0=q~j0.
We note that the solutionsxi(t),yj(t) of (1.1) are functions which for t≠tk,k∈ℤ+ satisfy
(4.14)x˙1(t)=x1(t)(∑k=1n‍b1(t)-a1(t)x1(t)-x1(t)μ1(t)+ω1(t)x2(t)-∑k=1n‍c1k(t)yk(t)α1k(t)+β1k(t)x1(t)+γ1k(t)yk(t)),x˙2(t)=x2(t)(∑k=1n‍b2(t)-a2(t)x2(t)-x2(t)μ2(t)+ω2(t)x1(t)-∑k=1n‍c2k(t)yk(t)α2k(t)+β2k(t)x2(t)+γ2k(t)yk(t)),y˙j(t)=yj(t)(-rj(t)+∑k=12‍djk(t)xk(t)αjk(t)+βjk(t)xk(t)+γjk(t)yj(t)-∑k=1n‍δjk(t)yk(t)),
and for t=tk, k∈ℤ+, satisfying the conditions xi(tk+)=(1+bik)xi(tk), yj(tk+)=(1+b~jk)yj(tk), By hypothesis (A1), it follows from (1.1) that
(4.15)x˙i(t)⩽xi(t)(bi(t)-ai(t)xi(t)),y˙j(t)⩽yj(t)(-rj(t)+∑k=12‍djk(t)qk(t)αjk(t)+βjk(t)qk(t)+γjk(t)q~j(t)),
for t≠tk. The elementary differential inequality (4.15) yields that
(4.16)xi(t)⩽Qi(t),yj(t)⩽Q~j(t),
for all t∈(t0,t1], that is, the inequality (4.9) is valid for t∈(t0,t1]. Suppose that (4.9) is satisfied for t∈(tk-1,tk],k>1. Then using hypotheses (A2)–(A4) and the fact that (4.9) is satisfied for t=tk, we obtain
(4.17)xi(tk+)=(1+bik)xi(tk)⩽(1+bik)Qi(tk)=Qi(tk+),yj(tk+)=(1+b~jk)yj(tk)⩽(1+b~jk)Q~j(tk)=Q~j(tk+).
We apply again the comparison result (4.15) in the interval (tk,tk+1] and obtain
(4.18)xi(t;t0,xi0)⩽Qik(t;tk,Qik+0)=Qi(t;t0,q0),yj(t;t0,yj0)⩽Q~jk(t;tk,Q~jk+0)=Q~j(t;t0,q0),
that is, the inequality (4.9) is valid for (tk,tk+1]. The proof of (4.9) is completed by induction. Further, by analogous arguments, using (A1)–(A4), we obtain from (1.1) and (4.15) the following:
(4.19)x˙1(t)⩾x1(t)(∑k=1n‍b1(t)-a1(t)x1(t)-Q1(t)μ1(t)+ω1(t)x2(t)-∑k=1n‍c1k(t)Q~k(t)α1k(t)+β1k(t)x1(t)+γ1k(t)yk(t)),x˙2(t)⩾x2(t)(∑k=1n‍b2(t)-a2(t)x2(t)-Q2(t)μ2(t)+ω2(t)x1(t)-∑k=1n‍c2k(t)Q~k(t)α2k(t)+β2k(t)x1(t)+γ2k(t)yk(t)),y˙j(t)⩾yj(t)(-rj(t)+∑k=12‍djk(t)xk(t)αjk(t)+βjk(t)xk(t)+γjk(t)yj(t)-∑k=1n‍δjk(t)Q~k(t)),t≠tk,(k∈Z+),xi(tk+)=(1+bik)xi(tk),yj(tk+)=(1+b~jk)yj(tk),t=tk,(k∈Z+),
and hence
(4.20)xi(t)⩾Pi(t),yj(t)⩾P~j(t),
for all t⩾t0, where (Pi(t),P~j(t))=(p1(t),p2(t),p~1(t),…,p~n(t)) is the minimal solution of the system:
(4.21)p˙1(t)=p1(t)(b1(t)-a1(t)x1(t)-Q1(t)μ1(t)+ω1(t)x2(t)-∑k=1n‍c1k(t)Q~k(t)α1k(t)+β1k(t)x1(t)+γ1k(t)yk(t)),p˙2(t)=p2(t)(b2(t)-a2(t)x2(t)-Q2(t)μ2(t)+ω2(t)x1(t)-∑k=1n‍c2k(t)Q~k(t)α2k(t)+β2k(t)x1(t)+γ2k(t)yk(t)),p~˙j(t)=p~j(t)(-rj(t)+∑k=12‍djk(t)pk(t)αjk(t)+βjk(t)pk(t)+γjk(t)p~j(t)-∑k=1n‍δjk(t)Q~k(t)),t≠tk,(k∈Z+),pi(t0+)=pi0>0,p~j(t0+)=p~j0>0,pi(tk+)=(1+bik)pi(tk),p~j(tk+)=(1+b~jk)p~j(tk),t=tk,(k∈Z+).
Thus, the proof follows from (4.9) and (4.20).Lemma 4.4.
Suppose the hypotheses (A1)–(A5) hold.x(t)=x(t;t0,x0)=(x1(t),x2(t),y1(t),…,yn(t)) is a solution of (1.1), then there exist positive constants: αi,βi,α~j, and β~j such that
(4.22)αi⩽xi(t)⩽βi,α~j⩽yj(t)⩽β~j,
for all t∈(tk-1,tk],k∈ℤ+ and if in addition
(4.23)0<1+bik⩽1,0<1+b~jk⩽1,
then
(4.24)αi⩽xi(t)⩽βi,α~j⩽yj(t)⩽β~j,
for all t∈J.Proof.
From Lemma4.3, we have Pi(t)⩽xi(t)⩽Qi(t),P~j(t)⩽yj(t)⩽Q~j(t) for all t⩾t0, where Pi(t),P~j(t) is the minimal solution of the system (4.21), and Qi(t),Q~j(t) is the maximal solution of the system (4.10). Under the conditions of Lemma 4.4 for the solutions of (4.10) and (4.21) with initial functions, it is valid that
(4.25)αi⩽Pi(t),Qi(t)⩽βi,α~j⩽P~j(t),Q~j(t)⩽β~j,αi,α~j>0,  0<βi,β~j<∞, for all t⩾t0, t∈(tk-1,tk], k∈ℤ+, then αi⩽xi(t)⩽βi, α~j⩽yj(t)⩽β~j, for all t⩾t0, t∈(tk-1,tk], k∈ℤ+. If in addition 0<1+bik⩽1, 0<1+b~jk⩽1, then from the left continuity of xi(t) at the points tk, we have
(4.26)αi=(1+bik)αi-(bik)αi⩽(1+bik)xi(t)-(bik)xi(t)⩽(1+bik)βi-(bik)βi=βi,α~j=(1+b~jk)α~j-(b~jk)α~j⩽(1+b~jk)yj(t)-(b~jk)yj(t)⩽(1+b~jk)β~j-(b~jk)β~j=β~j,
that is,
(4.27)αi⩽xj(tk+)⩽βi,α~j⩽yj(tk+)⩽β~j,
hence
(4.28)αi⩽xi(t)⩽βi,α~j⩽yj(t)⩽β~j,
for all t⩾t0.LetV∈V0, for any (t,x)∈[tk-1,tk)×ℝ+n+2, the right-hand derivative D+V(t,x(t)) along the solution x(t,t0,x0) of (1.1) is defined by
(4.29)D+V(t,x(t))=lim⁡h→0+inf⁡1h[V(t+h,x(t+h))-V(t,x(t))].
Define
(4.30)m(t)=∑i=12‍|xi(t)-xi*(t)|+∑j=1n‍|yj(t)-yj*(t)|
and consider a Lyapunov function:
(4.31)V(x(t),x*(t))=∑i=1n+2‍Vi(t)=∑i=12‍|ln⁡xi(t)xi*(t)|+∑j=1n‍|ln⁡yj(t)yj*(t)|.Theorem 4.5.
Let the following conditions hold:(1)
the hypotheses (A1)–(A5) hold;(2)
there exist nonnegative continuous functionsδi(t), and δj(t) such that
(A8)
(4.32)a1l+μ1l+ω1lα2*(μ1u+ω1uβ2)(μ1u+ω1uβ2*)>∑k=1n‍c1kuβ1kuβ~k(α1kl+β1klα1+γ1klα~k)(α1kl+β1klα1*+γ1klα~k*)+ω2uβ1*(μ2l+ω2lα1)(μ2l+ω2lα1*)+∑j=1n‍dj1u(αj1u+γj1uβ~j)(αj1l+βj1lα1+γj1lα~j)(αj1l+βj1lα1*+γj1lα~j*),a2l+μ2l+ω2lα1*(μ2u+ω1uβ1)(μ2u+ω2uβ1*)>∑k=1n‍c2kuβ2kuβ~k(α2kl+β2klα2+γ2klα~k)(α2kl+β2klα2*+γ2klα~k*)+ω1uβ2*(μ1l+ω1lα2)(μ1l+ω1lα2*)+∑j=1n‍dj2u(αj2u+γj2uβ~j)(αj2l+βj2lα2+γj2lα~j)(αj2l+βj2lα2*+γj2lα~j*),δjjl+∑k=12‍djklγjklαk(αjku+βjkuβk+γjkuβ~j)(αjku+βjkuβk*+γjkuβ~j*)>∑k=1,k≠jn‍δjku+∑i=12‍ciju(αiju+βijuβi)(αijl+βijlαi+γijlα~j)(αijl+βijlαi*+γijlα~j*);(3)
0<1+bik⩽1, 0<1+b~jk⩽1.
Then, the solutionx(t) of (1.1) is globally stable.Proof.
Consider the upper right derivativeD+V(x(t),x*(t)) along the solution of system (1.1). For t⩾t0 and t≠tk,k∈ℤ+, we derive the estimate as follows:
(4.33)D+V(x(t),x*(t))=(-a1(t)(xk(t)-xk*(t))-∑k=1n(c1k(t)yk(t)α1k(t)+β1k(t)x1(t)+γ1k(t)yk(t)-c1k(t)yk*(t)α1k(t)+β1k(t)x1*(t)+γ1k(t)yk*(t))-(x1(t)μ1(t)+ω1(t)x2(t)-x1*(t)μ1(t)+ω1(t)x2*(t))∑k=1n)×sgn⁡(x1-x1*)+(∑k=1n-a2(t)(x2(t)-x2*(t))-∑k=1n(c2k(t)yk(t)α2k(t)+β2k(t)x2(t)+γ2k(t)yk(t)∑k=1n-c2k(t)yk*(t)α2k(t)+β2k(t)x2*(t)+γ2k(t)yk*(t))-(x2(t)μ2(t)+ω2(t)x1(t)-x2*(t)μ2(t)+ω2(t)x1*(t))∑k=1n)×sgn⁡(x2-x2*)+∑j=1n(∑k=12(djk(t)xk(t)αjk(t)+βjk(t)xk(t)+γjk(t)yj(t)-djk(t)xk*(t)αjk(t)+βjk(t)xk*(t)+γjk(t)yj*(t))-∑k=1nδjk(t)(yk(t)-yk*(t)))sgn⁡(yj-yj*),⩽∑i=12‍(-ail|xi(t)-xi*(t)|+∑k=1n‍cikuαiku(αikl+βiklxi(t)+γiklyk(t))(αikl+βiklxi*(t)+γiklyk*(t))×|yk(t)-yk*(t)|+∑k=1n‍cikuβikuyk(t)(αikl+βiklxi(t)+γiklyk(t))(αikl+βiklxi*(t)+γiklyk*(t))|xi(t)-xi*(t)|+∑k=1n‍cikuβikuxi(t)(αikl+βiklxi(t)+γiklyk(t))(αikl+βiklxi*(t)+γiklyk*(t))|yk(t)-yk*(t)|)-μ1l+ω1lx2*(t)(μ1u+ω1ux2(t))(μ1u+ω1ux2*(t))|x1(t)-x1*(t)|+ω1ux2*(t)(μ1l+ω1lx2(t))(μ1l+ω1lx2*(t))|x2(t)-x2*(t)|-μ2l+ω2lx1*(t)(μ2u+ω2ux1(t))(μ2u+ω2ux1*(t))|x2(t)-x2*(t)|+ω2ux1*(t)(μ2l+ω2lx1(t))(μ2l+ω2lx1*(t))|x1(t)-x1*(t)|+∑j=1n‍(∑k=12‍djkuαjku(αjkl+βjklxk(t)+γjklyj(t))(αjkl+βjklxk*(t)+γjklyj*(t))|xk(t)-xk*(t)|+∑k=12‍djkuγjkuyj(t)(αjkl+βjklxk(t)+γjklyj(t))(αjkl+βjklxk*(t)+γjklyj*(t))|xk(t)-xk*(t)|-∑k=12‍djklγjklxk(t)(αjku+βjkuxk(t)+γjkuyj(t))(αjku+βjkuxk*(t)+γjkuyj*(t))|yj(t)-yj*(t)|-δjjl|yj-yj*|+∑k=1,k≠jn‍δjku|yk(t)-yk*|)=(-a1l+μ1l+ω1lα2*(μ1u+ω1uβ2)(μ1u+ω1uβ2*)-∑k=1n‍c1kuβ1kuβ~k(α1kl+β1klα1+γ1klα~k)(α1kl+β1klα1*+γ1klα~k*)-ω2uβ1*(μ2l+ω2lα1)(μ2l+ω2lα1*)-∑j=1n‍dj1u(αj1u+γj1uβ~j)(αj1l+βj1lα1+γj1lα~j)(αj1l+βj1lα1*+γj1lα~j*))×|x1(t)-x1*(t)|-(∑k=1n‍a2l+μ2l+ω2lα1*(μ2u+ω1uβ1)(μ2u+ω2uβ1*)-∑k=1n‍c2kuβ2kuβ~k(α2kl+β2klα2+γ2klα~k)(α2kl+β2klα2*+γ2klα~k*)-ω1uβ2*(μ1l+ω1lα2)(μ1l+ω1lα2*)-∑j=1n‍dj2u(αj2u+γj2uβ~j)(αj2l+βj2lα2+γj2lα~j)(αj2l+βj2lα2*+γj2lα~j*))×|x2(t)-x2*(t)|-∑j=1n‍(δjjl+∑k=12‍djklγjklαk(αjku+βjkuβk+γjkuβ~j)(αjku+βjkuβk*+γjkuβ~j*)-∑k=1,k≠jn‍δjku-∑i=12‍ciju(αiju+βijuβi)(αijl+βijlαi+γijlα~j)(αijl+βijlαi*+γijlα~j*))×|yj(t)-yj*|.
Thus in view of hypothesis (A8), we obtain
(4.34)D+V(x(t),x*(t))⩽-δ(t)m(t),t⩾t0,t≠tk,(k∈Z+),
where δ(t)=min{δi,δj}.
Fort⩾t0,t=tk,(k∈ℤ+), we have
(4.35)V(x(tk+),x*(tk+))=∑i=12‍|ln⁡xi(tk+)xi*(tk+)|+∑j=1n‍|ln⁡yj(tk+)yj*(tk+)|=∑i=12‍|ln⁡(1+bik)xi(tk)(1+bik)xi*(tk)|+∑j=1n‍|ln⁡(1+b~jk)yj(tk)(1+b~jk)yj*(tk)|=V(x(tk),x*(tk)).
Then the inequality,
(4.36)V(x(t),x*(t))⩽V(x(t0+),x*(t0+))-∫t0t‍δ(s)m(s)ds,t⩾t0
holds.
By Mean Value Theorem and by Lemma4.4 it follows that for any closed interval contained in t∈(tk-1,tk],k∈ℤ+, there exist positive numbers r and R such that for every i,j, r⩽xi(t),yj(t),xi*(t),yj*(t)⩽R and
(4.37)1R|xi(t)-xi*(t)|⩽|ln⁡xi(t)-ln⁡xi*(t)|⩽1r|xi(t)-xi*(t)|,1R|yj(t)-yj*(t)|⩽|ln⁡yj(t)-ln⁡yj*(t)|⩽1r|yj(t)-yj*(t)|.
Hence, we obtain
(4.38)V(x0,x0*)=∑i=12‍|ln⁡xi(t0+)-ln⁡xi*(t0+)|+∑j=1n‍|ln⁡yj(t0+)-ln⁡yj*(t0+)|⩽1r‖x0-x0*‖.
Further, from (4.34) and (4.35) we have
(4.39)D+V(x(t),x*(t))⩽0,t⩾t0,t≠tk,ΔV(x(tk),x*(tk))=0,
and hence
(4.40)V(x(t),x*(t))⩽V(x0,x0*),
for all t⩾t0. Given 0<ɛ<R, choose δ=ɛr/2R. Then from (4.37)–(4.40) it follows that
(4.41)∑i=12‍|xi(t)-xi*(t)|+∑j=1n‍|yj(t)-yj*(t)|<ɛ,
for all t⩾t0, whenever ∥x0-x0*∥⩽δ and t0∈ℝ+. Since t0∈ℝ+ is arbitrary, by Definition 4.1(a), the system (1.1) is globally stable. This proves the theorem.Theorem 4.6.
In addition to the assumptions of Theorem4.5, suppose that there exists a constant c such that
(4.42)∫t0t‍δ(s)ds=c(t-t0).
Then the system (1.1) is globally exponentially stable.Proof.
We consider again the Lyapunov function (4.31). From (4.34) and (4.37), we obtain
(4.43)D+V(x(t),x*(t))⩽-δ(t)m(t)⩽-rδ(t)V(x(t),x*(t)).
From the above mentioned estimate and (4.38), we have
(4.44)V(x(t),x*(t))⩽V(x0,x0*)exp⁡{-r∫t0t‍δ(s)ds},
for all t⩾t0. Then from (4.37),(4.40), (4.44), and (4.38), we deduce the inequality
(4.45)∑i=12‍|xi(t)-xi*(t)|+∑j=1n‍|yj(t)-yj*(t)|⩽Rr‖x0-x0*‖e-rc(t-t0)
for t⩾t0. This shows that the system (1.1) is globally exponentially stable. This proves the theorem.



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*Source: 963486-2012-07-24.xml*