# Analysis of Wave Solutions of an Adhenovirus-Tumor Cell System

Authors: Baba Issa Camara; Houda Mokrani
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We discuss biological background and mathematical analysis of glioma gene therapy for contributing to cancer treatment. By a reaction-diffusion system, we model interactions between gliom cells and viruses. We establish some sufficient conditions on model parameters which guarantee the permanence of the system and the existence of periodic solutions. Our study has experimental and theoretical implication in the perspective management strategy of therapy.


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## Body

## 1. Introduction

Diffuse infiltrative gliomas are the most frequent primary central nervous system (CNS) tumors in adults. Their deserved reputation as devastating diseases is due in large part to their widespread invasiveness in the neuropil, that is, the dense network of interwoven neuronal and glial cell processes. Partially because of their growth pattern, curative treatment for diffuse gliomas is generally impossible. Although patients with low-grade diffuse gliomas may survive for multiple years, these tumors lead to death of the patient sooner or later, often after progression to high-grade malignancy. Whereas surgery of most other tumors aims at complete resection, the diffuse growth of gliomas in the brain parenchyma precludes complete tumor removal. So to increase the length of survival time of patients with malignant brain tumors, novel therapeutic alternatives are currently being explored. Some of these experimental treatment strategies are based on advances in immunotherapy, stem cell therapy, local chemotherapy, and radiotherapy. However, in areas where the original tissue architecture is relatively preserved, the blood-brain barrier that may form an obstacle for optimal delivery of chemotherapeutics to diffuse infiltrative tumor cells eradicating diffuse infiltrative glioma cells by radiotherapy without significantly damaging the infiltrated brain parenchyma has been difficult to achieve [1].In addition, gene therapy is becoming a promising alternative. In essence, gene therapy consists of the delivery of a gene of interest to tumor cell populations to control and, when possible, kill the growing tumor. Viruses are prominent vehicles for gene therapy, and some adenoviral vectors exhibit oncolytic properties. To this end, a variety of viral vectors have been developed, with oncolytic viruses emerging as an innovative therapeutic tool for these tumors. To be effective, a virus used for oncolytic therapy must have several features. The desired properties of these vectors include selectivity for the tumor target, minimal brain and systemic toxicities, and the capacity to penetrate and diffuse throughout the brain to reach all neoplastic foci residing beyond the resection border of the tumor. In addition, the viral vector needs to remain active despite evoking an immune response. The goal of developing an ideal vehicle for treatment of malignant brain tumors remains to be achieved. A wide variety of viral vectors have been developed and tested in the setting of gene therapy for malignant gliomas. These are based on different kinds of viruses, such as herpes simplex virus, retrovirus, measles virus, reovirus, and adenovirus. Some have shown promising results when tested in animal models of intracranial gliomas, but, to date, clinical trials performed in humans have not shown a significant increase in survival.Various vectors have been targeted toward cancer cells by deleting the genes responsible for bypassing those cells’ antiviral proteins. Without these genes, the designed vectors will only be able to replicate within cancer cells with disrupted antiviral mechanisms. The deletion of viral genes to enhance specificity for the killing of neoplastic cells is a principle well exemplified by the actions of two oncolytic adenoviruses, ONYX 015 and Ad5-Delta24. ONYX 015 has a deletion in the viral genomic region coding E1B 55kd. This deletion effectively limits the replication of the virus to neoplastic cells that have a defective p53 pathway [2, 3]. A Phase I clinical trial to examine the effects of injection of ONYX 015 into peritumoral regions of recurrent malignant gliomas was recently completed and published [4]. In that study, ONYX 015 was injected into the walls of tumor resection cavities. This trial proved that the injection of up to 1010 plaque-forming units of ONYX 015 into brain tissue surrounding a resected malignant glioma is safe in humans.Extensive efforts have been dedicated over many years to mathematical modelling of cancer development [5–7]. These mathematical models serve as valuable tools to predict possible outcomes of virus infection and propose the optimal strategy of anti-virus therapy. Wodarz [8, 9] presented a mathematical model that describes interaction between two types of tumor cells, the cells that are infected by the virus and the cells that are not infected but are susceptible to the virus so far as they have cancer phenotype and the immune system. Our system is more general than the one considered in [8, 9] even when there is no diffusion. Because the free virus particles are very small, they disperse in the fluid tissue like Brownian particles. Therefore, we have incorporated into our model a diffusion term for the free viruses. We also assume that the tumor has a logistic growth, which can be slowed down by the inhibitor, captured in the expression 1-u. Thus, the tumor admits a maximum size and density defined by the carrying capacity K. When the virus is administered, the dynamic interactions between the virus and tumor cell populations are described by the following diffusive ratio-dependent predator-prey model of reaction-diffusion equations in the tumor region Ω:(1.1)∂V1∂t=μ1ΔV1+V1[ρ(t,x)(1-u)(1-V1K)-d(x,t)]-βr(t,x)V1V31+εV3,x∈Ω,t>0,(1.2)∂V2∂t=μ2ΔV2-V2[a(t,x)(1-u)+d(t,x)]+βr(t,x)V1V31+εV3,x∈Ω,t>0,(1.3)∂V3∂t=μ3ΔV3+k(t,x)(1-u)V2-b(t,x)V3,x∈Ω,t>0,(1.4)∂V1∂n|∂Ω=0,∂V2∂n|∂Ω=0,∂V2∂n|∂Ω=0,t>0,
where V1 is the number density of susceptible and uninfected tumor cells, V2 is the number density of infected tumor cells, and V3 is the number density of free virus, that is, virus in the extracellular tissue. When parameters of system (1.1)–(1.4) are constant, we determined in [10] the conditions for optimal therapy and, estimated by numerical simulations, the patient survival time when tumor cannot be cured. This paper is organized as follows Section 2 is devoted to some preliminaries, which are needed in next sections, including some lemmas, due to Walter and Smith. In Section 3, some conditions for the ultimate boundedness of solutions and permanence of this system are established


## 2. Preliminaries

We need the following lemmas due to Walter [11] and Smith [12], respectively.Lemma 2.1.
Suppose that vector functionsv(t,x)=(v1(t,x),…,vm(t,x)) and w(t,x)=(w1(t,x),…,wm(t,x)),  m≥1, satisfy the following conditions: (i)
they are of classC2 in x, x∈X, and of class C1 in (t,x)∈[a,b]×Ω̅, where Ω⊂ℝn is a bounded domain with smooth boundary;(ii)
vt-μΔv-g(t,x,v)≤wt-μΔw-g(t,x,w), where (t,x)∈[a,b]×Ω, μ=(μ1,…,μm)>0, vector function g(t,x,v)=(g1(t,x,v),…,gm(t,x,v)) is continuously differential and quasi-monotonically increasing with respect to u=(u1,…,um) and ∂gi(t,x,u1,…,um)/∂uj≥0, i,j=1,…,m, i≠j;(iii)
∂v/∂n=∂w/∂n=0, (t,x)∈[a,b]×∂Ω. Then, v(t,x)≤w(t,x), for all (t,x)∈[a,b]×Ω̅.Lemma 2.2.
Assume thatT and μ are positive real numbers, a function u(t,x) is continuous on [0,T]×Ω̅, continuously differential in x∈Ω̅, with continuous derivatives ∂2u/∂xi∂xj and ∂u/∂t on (0,T]×Ω, and u(t,x) satisfies the following inequalities:
(2.1)ut-μΔu-c(t,x,u)≥0,(t,x)∈(0,T]×Ω,∂u∂n≥0,(t,x)∈(0,T]×∂Ω,u(0,x)≥0,x∈Ω,
where c(t,x,u) is bounded on (0,T]×Ω. Then u(t,x)≥0 on x∈(0,T]×Ω̅. Moreover, u(t,x) is strictly positive on x∈(0,T]×Ω̅ if u(t,x) is not identically zero.Consider the following logistic differential equation:(2.2)dzdt=z(a-bz),
where z∈ℝ+; a and b are positive constants.Lemma 2.3.
Every solutionz(t,0,z0), z0>0 of (2.2) satisfies
(2.3)lim⁡t→+∞z(t)=ab.


## 3. Permanence

Throughout the paper we always assume that(H): a(t,x), b(t,x), d(t,x), k(t,x), r(t,x) and ρ(t,x) are bounded positive-valued functions on ℝ×Ω̅ continuously differential in t and x and are periodic in t with a period ω>0.Definition 3.1.
Solutions of system (1.1)–(1.4) are said to be ultimately bounded if there exist positive constants N1,N2,N3 such that for every solution (V1(t,x,V01,V03,V03);V2(t,x,V01,V03,V03);V3(t,x,V01,V03,V03)) there exists a moment of time t̅=t̅(V01,V03,V03)>0 such that V1(t,x,V01,V03,V03)≤N1, V2(t,x,V01,V03,V03)≤N2, V3(t,x,V01,V03,V03)≤N3, for all x∈Ω̅ and t>t̅.Definition 3.2.
System (1.1)–(1.4) is said to be permanent if there exist positive constants ζ and η such that for every solution with nonnegative initial functions V01(x),V02(x), and V03(x),V01(x)≠0,V02(x)≠0,V03(x)≠0,there exists a moment of time t̂=t̂(V01,V03,V03) such that ζ≤V1(t,x,V01,V03,V03)≤η,ζ≤V2(t,x,V01,V03,V03)≤η,ζ≤V3(t,x,V01,V03,V03)≤η,for all x∈Ω̅ and t>t̂.
For simplicity, for a bounded functionφ(t,x), we denote φm=inf(t,x)φ(t,x) and φM=sup(t,x)φ(t,x). Now we have the following positively invariant principle for system (1.1)–(1.4).Theorem 3.3.
Assume that conditions(H) hold, then nonnegative and positive quadrants of ℝ3 are positively invariant for system (1.1)–(1.4).Proof.
Suppose(V1(t,x,V01,V03,V03);V2(t,x,V01,V03,V03);V3(t,x,V01,V03,V03)) is a solution of system (1.1)–(1.4) with initial condition V01(x)≥0 (V01(x)≠0), V02(x)≥0 (V02(x)≠0), V03(x)≥0 (V03(x)≠0). Let V̂1(t,x) be a solution of
(3.1)∂V̂1∂t-μ1ΔV̂1+V̂1[dM-ρm(1-u)(1-V̂1K)+ε-1βMrM]=0,V̂1(0,x)=V01(x),∂V̂1∂n|∂Ω≥0,t>0.
It holds that
(3.2)∂V̂1∂t-μ1ΔV̂1+V̂1[d(x,t)-ρ(t,x)(1-u)(1-V̂1K)]+βr(t,x)V̂1V31+εV3≤∂V̂1∂t-μ1ΔV̂1+V̂1[dM-ρm(1-u)(1-V̂1K)+ε-1βMrM],
which implies that V̂1(t,x) is a lower solution of (1.1). By Lemma 2.2, we have V̂1(t,x)≥0, for all t>0 and x∈Ω̅. In addition, since V01(x)≥0 (V01(x)≠0), then V̂1(t,x)>0, for all t>0 and x∈Ω̅. Thus by Lemma 2.1, V1(t,x) is bounded from below by positive function V̂1(t,x), and so V1(t,x)>0.
LetV̂2(t,x) be a solution of
(3.3)∂V̂2∂t-μ2ΔV̂2+V̂2[aM(1-u)+dM]=0,V̂2(0,x)=V02(x),∂V̂2∂n|∂Ω≥0,t>0.
It holds that(3.4)∂V̂2∂t-μ2ΔV̂2+V̂2[a(t,x)(1-u)+d(t,x)]-βr(t,x)V1V31+εV3≤∂V̂2∂t-μ2ΔV̂2+V̂2[aM(1-u)+dM]=0,
which implies that V̂2(t,x) is a lower solution of (1.2).
LetV̂3(t,x) be a solution of
(3.5)∂V̂3∂t-μ3ΔV̂3+bMV̂3=0,V̂3(0,x)=V03(x),∂V̂3∂n|∂Ω≥0,t>0.
It holds that(3.6)∂V̂3∂t-μ3ΔV̂3+b(t,x)V̂3-k(t,x)(1-u)V2≤∂V̂3∂t-μ3ΔV̂3+bMV̂3=0,
which implies that V̂3(t,x) is a lower solution of (1.3).
A similar argument toV1(t,x) leads to that V2(t,x) and V3(t,x) are bounded from below, respectively, by positive functions V̂2(t,x) and V̂3(t,x).Theorem 3.4.
Assume that conditions(H) hold then all solutions of system (1.1)–(1.4) with nonnegative initial functions are ultimately bounded.Proof.
LetV¯1(t,x) be a solution of
(3.7)∂V¯1∂t-μ1ΔV¯1-ρM(1-u)(1-V¯1K)V¯1=0,V¯1(0,MV1)=MV1,
where MV1 is such that maxx∈Ω̅|V01(x)|≤MV1.
It holds that(3.8)0=∂V1∂t-μ1ΔV1+V1[d(x,t)-ρ(t,x)(1-u)(1-V1K)]+βr(t,x)V1V31+εV3≥∂V1∂t-μ1ΔV1-ρM(1-u)(1-V1K)V1.
So, we get
(3.9)0=∂V¯1∂t-μ1ΔV¯1-ρM(1-u)(1-V¯1K)V¯1≥∂V1∂t-μ1ΔV1-ρM(1-u)(1-V1K)V1.
Therefore, Lemma2.1 gives V1(t,x,V01,V02,V03)≤V¯1(t,MV1).
Note that, according to the uniqueness theorem, the solutionV¯1(t,MV1) of (3.7) does not depend on x for t>0, and so V¯1(t,MV1) satisfies the ordinary differential equation:
(3.10)∂V¯1∂t-ρM(1-u)(1-V¯1K)V¯1=0.
By Lemma 2.3, we have
(3.11)V¯1(t,MV1)⟶K,ast⟶+∞.
So, there exists N1 and t1>0 such that V1(t,x)≤N1.
For the infected tumor cells, we have the following inequality:(3.12)0=∂V2∂t-μ2ΔV2+V2[a(t,x)(1-u)V2+d(t,x)]-βr(t,x)V1V31+εV3≥∂V2∂t-μ2ΔV2+V2[am(1-u)+dm]-ε-1βMrMN1.
So, we get
(3.13)0=∂V¯2∂t-μ2ΔV¯2+V¯2[am(1-u)+dm]-ε-1βMrMN1≥∂V2∂t-μ2ΔV2+V2[am(1-u)+dm]-ε-1βMrMN1.
Therefore, Lemma 2.1 gives V2(t,x,V01,V02,V03)≤V¯2(t,MV2), where V¯2(t,MV2) satisfies the ordinary differential equation:
(3.14)∂V¯2∂t=-V¯2[am(1-u)+dm]+ε-1βMrMN1,V¯2(0,MV2)=MV2.
Since
(3.15)V¯2(t,MV2)⟶ε-1βrMN1am(1-u)+dmast⟶+∞,
thus, there exists N2 and t2>0 such that V2(t,x)≤N2.
For the virus in the extracellular tissue, we have the following inequality:(3.16)0=∂V3∂t-μ3ΔV3+b(t,x)V3-k(t,x)(1-u)V2≥∂V3∂t-μ3ΔV3+bmV3-kM(1-u)N2.
So, we get
(3.17)0=∂V¯3∂t-μ3ΔV¯3+bmV¯3-kM(1-u)N2≥∂V3∂t-μ3ΔV3+bmV3-kM(1-u)N2.
Therefore, Lemma 2.1 gives V3(t,x,V01,V02,V03)≤V¯3(t,MV3), where V¯3(t,MV3) satisfies the ordinary differential equation:
(3.18)∂V¯3∂t=-bmV¯3+kM(1-u)N2,V¯3(0,MV3)=MV3.
Since
(3.19)V¯3(t,MV3)⟶kM(1-u)N2bm,ast⟶+∞,
thus, there exists N3 and t3>0 such that V3(t,x)≤N3.Theorem 3.5.
Assume that conditions(H) hold; in addition, if dM+βε-1rM<ρm(1-u), then the system (1.1)–(1.4) is permanent.Proof.
Theorem3.4 implies that there exists η>0 such that V1(t,x)≤η, V2(t,x)≤η, and V3(t,x)≤η starting with a certain moment of time. Note that, by comparison principle, if V01(x)≥0 (V01(x)≠0), V02(x)≥0 (V02(x)≠0), and V03(x)≥0 (V03(x)≠0), then V1(t,x,V01,V03,V03)>0, V2(t,x,V01,V03,V03)>0, and V3(t,x,V01,V03,V03)>0 for all x∈Ω̅ and t>0. Considering the solution on the interval t≥ϵ with some ϵ>0, we get (V1(ϵ,x,V01,V03,V03);V2(ϵ,x,V01,V03,V03);V3(ϵ,x,V01,V03,V03)) separated from zero. Therefore, we can assume that minx∈Ω̅V01(x)=mV1>0, minx∈Ω̅V02(x)=mV2>0 and, minx∈Ω̅V03(x)=mV3>0. Let V̂1(t,x) be a solution of
(3.20)∂V̂1∂t+V̂1[dM-ρm(1-u)(1-V̂1K)+ε-1βMrM]=0,V̂1(0,MV1)=mV1.
So using the inequality
(3.21)0=∂V1∂t-μ1ΔV1+V1[d(x,t)-ρ(t,x)(1-u)(1-V1K)]+βr(t,x)V1V31+εV3≤∂V1∂t-μ1ΔV1+V1[dM-ρm(1-u)(1-V1K)+ε-1βMrM],
one has, by (3.20),
(3.22)0=∂V̂1∂t-μ1ΔV̂1+V̂1[dM-ρm(1-u)(1-V̂1K)+ε-1βMrM]≤∂V1∂t-μ1ΔV1+V1[dM-ρm(1-u)(1-V1K)+ε-1βMrM].
Therefore, Lemma 2.1 gives V1(t,x,V01,V02,V03)≥V̂1(t,mV1). By the condition dM+ε-1βMrM<ρm(1-u), we have
(3.23)V̂1(t,mV1)⟶K[ρm(1-u)-dM-ε-1βMrM]ρm(1-u),t⟶+∞.
Therefore, there exists ζ1>0 such that V1(t,x,V01,V02,V03)≥ζ1 for t large enough.
LetV̂2(t,x) be a solution of
(3.24)∂V̂2∂t+V̂2[aM(1-u)+dM]-ε-1βMrMζ1=0,V̂2(0,x)=mV2.
Using the inequality
(3.25)0=∂V2∂t-μ2ΔV2+V2[a(t,x)(1-u)+d(t,x)]-βr(t,x)V1V31+εV3≤∂V2∂t-μ2ΔV2+V2[aM(1-u)+dM]-ε-1βMrMζ1,
we have, by (3.24),
(3.26)0=∂V̂2∂t-μ2ΔV̂2+V̂2[aM(1-u)+dM]-ε-1βMrMζ1≤∂V2∂t-μ2ΔV2+V2[aM(1-u)+dM]-ε-1βMrMζ1.
Therefore, V2(t,x,V01,V02,V03)≥V̂2(t,mV2)→(ε-1βMrMζ1)/(aM(1-u)+dM), as t→+∞. Therefore, there exists ζ2>0 such that V2(t,x,V01,V02,V03)≥ζ2 for t large enough.
Now, letV̂3(t,x) be a solution of
(3.27)∂V̂3∂t+bMV̂3-kM(1-u)ζ2=0,V̂3(0,x)=mV3.
By the following inequality,
(3.28)0=∂V3∂t-μ3ΔV3+b(t,x)V3-k(t,x)(1-u)V2≤∂V3∂t-μ3ΔV3+bMV3-kM(1-u)ζ2,
we have
(3.29)∂V̂3∂t-μ3ΔV̂3+bMV̂3-kM(1-u)ζ2≤∂V3∂t-μ3ΔV3+bMV3-kM(1-u)ζ2.
Therefore, V3(t,x,V01,V02,V03)≥V̂3(t,mV3)→(kM(1-u)ζ2)/(bM), as t→+∞. Therefore, there exists ζ3>0 such that V3(t,x,V01,V02,V03)≥ζ3 for t large enough.


## 4. Periodic Solutions

Theorem 4.1.
Assume that conditions(H) hold and system (1.1)–(1.4) is permanent. Moreover, if one assumes the following conditions
(4.1)2(ρM(1-u)(1-2ζK)-dm)-ε-1βMrM<0,βMεrM+kM-4ζaM(1-u)+2dM<0,kM-2bm<0,
then the system has a unique globally asymptotic stable strictly positive ω-periodic solution.Proof.
For convenience, we denotea=a(t,x) and similar meaning to b, d, k, r, and ρ. Let (V11(t,x),V21(t,x),V31(t,x)) and (V21(t,x),V22(t,x),V32(t,x)) be two solutions of system bounded by constants ζ and η for below and above. Consider the function
(4.2)L(t)=∫Ω[(V11(t,x)-V12(t,x))2+(V21(t,x)-V22(t,x))2+(V31(t,x)-V32(t,x))2]dx.
So,
(4.3)dL(t)dt=2∫Ω(V11-V12)(∂V11∂t-∂V12∂t)dx+2∫Ω(V21-V22)(∂V21∂t-∂V22∂t)dx+2∫Ω(V31-V32)(∂V31∂t-∂V32∂t)dx=2μ1∫Ω(V11-V12)Δ(V11-V12)dx+2μ2∫Ω(V21-V22)Δ(V21-V22)dx+2μ3∫Ω(V31-V32)Δ(V31-V32)dx+2∫Ω(V11-V12)2[ρ(1-u)(1-(V11+V12)K)-d]dx-2∫Ω(V11-V12)βrV11V311+εV31dx+2∫Ω(V11-V12)βrV12V321+εV32dx-2∫Ω(V21-V22)2[a(1-u)(V11+V12)+d]dx+2∫Ω(V21-V22)βrV11V311+εV31dx-2∫Ω(V21-V22)βrV12V321+εV32dx+2∫Ωk(1-u)(V21-V22)(V31-V32)dx-2∫Ωb(V31-V32)2dx:=I1+⋯+I11.
It follows from the boundary condition (1.4) that
(4.4)I1+I2+I3=-2∫Ω[μ1|∇(V11-V12)|2+μ2|∇(V21-V22)|2+2μ3|∇(V31-V32)|2]dx≤0
The terms
(4.5)I4+I7+I10+I11≤2∫Ω(ρM(1-u)(1-2ζK)-dm)(V11-V12)2dx-∫Ω(4ζaM(1-u)+2dM)(V21-V22)2dx-2∫Ωbm(V31-V32)2dx+∫ΩkM[(V21-V22)2+(V31-V32)2]dx,I5+I6+I8+I9=2∫ΩβrV11V311+εV31[(V21-V22)-(V11-V12)]dx+2∫ΩβrV12V321+εV32[(V11-V12)-(V21-V22)]dx≤2∫Ωε-1βMrMV11[(V21-V22)-(V11-V12)]dx+2∫Ωε-1βMrMV12[(V11-V12)-(V21-V22)]dx≤-2∫Ωε-1βMrM(V11-V12)2dx+2∫Ωε-1βMrM(V11-V12)(V21-V22)dx≤-∫Ωε-1βMrM[(V11-V12)2-(V21-V22)2]dx.
Thus, we have(4.6)dL(t)dt≤∫Ω[2(ρM(1-u)(1-2ζK)-dm)-ε-1βMrM](V11-V12)2dx+∫Ω[ε-1βMrM+kM-4ζaM(1-u)+2dM](V21-V22)2dx+∫Ω[kM-2bm](V31-V32)2dx≤(V11-V12,V21-V22,V31-V32)P(V11-V12,V21-V22,V31-V32)T≤λM(P)[‖V11-V12‖L2(Ω)2+‖V21-V22‖L2(Ω)2+‖V31-V32‖L2(Ω)2],
where λM(P) is the maximal eigenvalue of the diagonal matrix P=diag(P1,P2,P3) with
(4.7)P1=2(ρM(1-u)(1-2ζK)-dm)-ε-1βMrMP2=ε-1βMrM+kM-4ζaM(1-u)+2dMP3=kM-2bm.
So, we deduce that
(4.8)L(t)≤L(0)eλM(P)t⟶0,ast⟶+∞.
Thus, ∥V11-V12∥L2(Ω)→0, ∥V21-V22∥L2(Ω)→0, and ∥V31-V32∥L2(Ω)→0, as t→+∞. By Theorem 3.3, solutions of system (1.1)–(1.4) are bounded in the space C1+ν(Ω̅,ℝ3, where 0<ν<2l-1-n/p and 1/2+n/(2p)<l<1. Therefore,
(4.9)lim⁡t→+∞sup⁡x∈Ω|V11(t,x)-V12(t,x)|=0,lim⁡t→+∞sup⁡x∈Ω|V21(t,x)-V22(t,x)|=0,lim⁡t→+∞sup⁡x∈Ω|V31(t,x)-V32(t,x)|=0.
Consider the sequence
(4.10)(V1(qω,x,V01,V02,V03;V2)(qω,x,V01,V02,V03);V3(qω,x,V01,V02,V03))=W(qω,W0).
Then {W(qω,W0),q∈ℕ} is compact in the space C(Ω̅)3. Let W̅ be a limit point of this sequence. It follows, from
(4.11)‖W(ω,W¯)-W¯‖C≤‖W(ω,W¯)-W(ω,W(qnω,W0))‖C+‖W(ω,W(qnω,W0))-W(qnω,W0)‖C+‖W(qnω,W0)-W¯‖C⟶0asn⟶+∞,
that W(ω,W¯)=W¯.
Next, letW¯ and Ŵ be two limit points of the sequence {W(qω,W0),q∈ℕ}. Using (4.9) and Ŵ=W(qnω,Ŵ), we have
(4.12)‖W¯-Ŵ‖C≤‖W¯-W(qnω,W0)‖C+‖W(qnω,W0)-Ŵ‖C⟶0,asn⟶+∞.
Thus, W¯=Ŵ, and so (V1(t,x,V̅1,V̅2,V̅3);V2(t,x,V̅1,V̅2,V̅3);V3(t,x,V̅1,V̅2,V̅3)) is the unique periodic solutions of system (1.1)–(1.4). By (4.9), it is asymptotically stable.



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*Source: 590326-2012-05-07.xml*