Analysis of Wave Solutions of an Adhenovirus-Tumor Cell System

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.

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:𝜕𝑉₁𝜕𝑡=𝜇₁Δ𝑉₁+𝑉₁𝜌()𝑉𝑡,𝑥)(1−𝑢1−¹𝐾(−()𝑉−𝑑𝑥,𝑡)𝛽𝑟𝑡,𝑥₁𝑉₃1+𝜀𝑉₃,𝑥∈Ω,𝑡>0,𝜕𝑉₂𝜕𝑡=𝜇₂Δ𝑉₂−𝑉₂[𝑎()(+()𝑉𝑡,𝑥)(1−𝑢+𝑑𝑡,𝑥)]𝛽𝑟𝑡,𝑥₁𝑉₃1+𝜀𝑉₃,𝑥∈Ω,𝑡>0,𝜕𝑉₃𝜕𝑡=𝜇₃Δ𝑉₃+𝑘(𝑡,𝑥)(1−𝑢)𝑉₂−𝑏(𝑡,𝑥)𝑉₃,𝑥∈Ω,𝑡>0,𝜕𝑉₁𝜕𝑛_𝜕Ω=0,𝜕𝑉₂𝜕𝑛_𝜕Ω=0,𝜕𝑉₂𝜕𝑛_𝜕Ω=0,𝑡>0, (1.1)(1.2)(1.3)(1.4) 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 functions 𝑣(𝑡,𝑥)=(𝑣₁(𝑡,𝑥),…,𝑣_𝑚(𝑡,𝑥)) and 𝑤(𝑡,𝑥)=(𝑤₁(𝑡,𝑥),…,𝑤_𝑚(𝑡,𝑥)), 𝑚≥1, satisfy the following conditions: (i)they are of class 𝐶² in 𝑥, 𝑥∈𝑋, and of class 𝐶¹ in (𝑡,𝑥)∈[𝑎,𝑏]×Ω, where Ω⊂ℝ^𝑛 is a bounded domain with smooth boundary;(ii)𝑣_𝑡−𝜇Δ𝑣−𝑔(𝑡,𝑥,𝑣)≤𝑤_𝑡−𝜇Δ𝑤−𝑔(𝑡,𝑥,𝑤), where (𝑡,𝑥)∈[𝑎,𝑏]×Ω, 𝜇=(𝜇₁,…,𝜇_𝑚)>0, vector function 𝑔(𝑡,𝑥,𝑣)=(𝑔₁(𝑡,𝑥,𝑣),…,𝑔_𝑚(𝑡,𝑥,𝑣)) is continuously differential and quasi-monotonically increasing with respect to 𝑢=(𝑢₁,…,𝑢_𝑚) and 𝜕𝑔_𝑖(𝑡,𝑥,𝑢₁,…,𝑢_𝑚)/𝜕𝑢_𝑗≥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]×Ω.

Lemma 2.2. Assume that 𝑇 and 𝜇 are positive real numbers, a function 𝑢(𝑡,𝑥) is continuous on [0,𝑇]×Ω, continuously differential in 𝑥∈Ω, with continuous derivatives 𝜕²𝑢/𝜕𝑥_𝑖𝜕𝑥_𝑗 and 𝜕𝑢/𝜕𝑡 on (0,𝑇]×Ω, and 𝑢(𝑡,𝑥) satisfies the following inequalities: 𝑢_𝑡−𝜇Δ𝑢−𝑐(𝑡,𝑥,𝑢)≥0,(𝑡,𝑥)∈(0,𝑇]×Ω,𝜕𝑢()∈(]𝑢()𝜕𝑛≥0,𝑡,𝑥0,𝑇×𝜕Ω,0,𝑥≥0,𝑥∈Ω, (2.1) where 𝑐(𝑡,𝑥,𝑢) is bounded on (0,𝑇]×Ω. Then 𝑢(𝑡,𝑥)≥0 on 𝑥∈(0,𝑇]×Ω. Moreover, 𝑢(𝑡,𝑥) is strictly positive on 𝑥∈(0,𝑇]×Ω if 𝑢(𝑡,𝑥) is not identically zero.

Consider the following logistic differential equation:𝑑𝑧()𝑑𝑡=𝑧𝑎−𝑏𝑧, (2.2) where 𝑧∈ℝ⁺; 𝑎 and 𝑏 are positive constants.

Lemma 2.3. Every solution 𝑧(𝑡,0,𝑧₀), 𝑧₀>0 of (2.2) satisfies lim_𝑡→+∞𝑧(𝑡)=𝑎𝑏. (2.3)

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×Ω 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 𝑡=𝑡(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∈Ω and 𝑡>𝑡.

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=𝑡(̂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∈Ω and 𝑡>𝑡̂.
For simplicity, for a bounded function 𝜑(𝑡,𝑥), we denote 𝜑_𝑚=inf_(𝑡,𝑥)𝜑(𝑡,𝑥) and 𝜑_𝑀=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 𝑉₁(𝑡,𝑥) be a solution of 𝜕𝑉₁𝜕𝑡−𝜇₁Δ𝑉₁+𝑉₁𝑑_𝑀−𝜌_𝑚()𝑉1−𝑢1−¹𝐾+𝜀⁻¹𝛽_𝑀𝑟_𝑀𝑉=0,₁(0,𝑥)=𝑉₀₁(𝑥),𝜕𝑉₁𝜕𝑛_𝜕Ω≥0,𝑡>0. (3.1) It holds that 𝜕𝑉₁𝜕𝑡−𝜇₁Δ𝑉₁+𝑉₁𝑑()(𝑉𝑥,𝑡−𝜌𝑡,𝑥)(1−𝑢)1−¹𝐾+𝛽𝑟(𝑡,𝑥)𝑉₁𝑉₃1+𝜀𝑉₃≤𝜕𝑉₁𝜕𝑡−𝜇₁Δ𝑉₁+𝑉₁𝑑_𝑀−𝜌_𝑚()𝑉1−𝑢1−¹𝐾+𝜀⁻¹𝛽_𝑀𝑟_𝑀, (3.2) which implies that 𝑉₁(𝑡,𝑥) is a lower solution of (1.1). By Lemma 2.2, we have 𝑉₁(𝑡,𝑥)≥0, for all 𝑡>0 and 𝑥∈Ω. In addition, since 𝑉₀₁(𝑥)≥0 (𝑉₀₁(𝑥)≠0), then 𝑉₁(𝑡,𝑥)>0, for all 𝑡>0 and 𝑥∈Ω. Thus by Lemma 2.1, 𝑉₁(𝑡,𝑥) is bounded from below by positive function 𝑉₁(𝑡,𝑥), and so 𝑉₁(𝑡,𝑥)>0.
Let 𝑉₂(𝑡,𝑥) be a solution of 𝜕𝑉₂𝜕𝑡−𝜇₂Δ𝑉₂+𝑉₂𝑎_𝑀(1−𝑢)+𝑑_𝑀𝑉=0,₂(0,𝑥)=𝑉₀₂(𝑥),𝜕𝑉₂𝜕𝑛_𝜕Ω≥0,𝑡>0. (3.3)
It holds that 𝜕𝑉₂𝜕𝑡−𝜇₂Δ𝑉₂+𝑉₂[𝑎()(−()𝑉𝑡,𝑥)(1−𝑢+𝑑𝑡,𝑥)]𝛽𝑟𝑡,𝑥₁𝑉₃1+𝜀𝑉₃≤𝜕𝑉₂𝜕𝑡−𝜇₂Δ𝑉₂+𝑉₂𝑎_𝑀(1−𝑢)+𝑑_𝑀=0, (3.4) which implies that 𝑉₂(𝑡,𝑥) is a lower solution of (1.2).
Let 𝑉₃(𝑡,𝑥) be a solution of 𝜕𝑉₃𝜕𝑡−𝜇₃Δ𝑉₃+𝑏_𝑀𝑉₃𝑉=0,₃(0,𝑥)=𝑉₀₃(𝑥),𝜕𝑉₃𝜕𝑛_𝜕Ω≥0,𝑡>0. (3.5)
It holds that 𝜕𝑉₃𝜕𝑡−𝜇₃Δ𝑉₃+𝑏(𝑡,𝑥)𝑉₃−𝑘(𝑡,𝑥)(1−𝑢)𝑉₂≤𝜕𝑉₃𝜕𝑡−𝜇₃Δ𝑉₃+𝑏_𝑀𝑉₃=0, (3.6) which implies that 𝑉₃(𝑡,𝑥) is a lower solution of (1.3).
A similar argument to 𝑉₁(𝑡,𝑥) leads to that 𝑉₂(𝑡,𝑥) and 𝑉₃(𝑡,𝑥) are bounded from below, respectively, by positive functions 𝑉₂(𝑡,𝑥) and 𝑉₃(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. Let 𝑉₁(𝑡,𝑥) be a solution of 𝜕𝑉₁𝜕𝑡−𝜇₁Δ𝑉₁−𝜌_𝑀(1−𝑢)1−𝑉₁𝐾𝑉₁=0,𝑉₁0,𝑀_𝑉1=𝑀_𝑉1, (3.7) where 𝑀_𝑉1 is such that max_𝑥∈Ω|𝑉₀₁(𝑥)|≤𝑀_𝑉1.
It holds that 0=𝜕𝑉₁𝜕𝑡−𝜇₁Δ𝑉₁+𝑉₁𝑑()()𝑉𝑥,𝑡−𝜌𝑡,𝑥)(1−𝑢1−¹𝐾+𝛽𝑟(𝑡,𝑥)𝑉₁𝑉₃1+𝜀𝑉₃≥𝜕𝑉₁𝜕𝑡−𝜇₁Δ𝑉₁−𝜌_𝑀(𝑉1−𝑢)1−¹𝐾𝑉₁. (3.8) So, we get 𝜕0=𝑉₁𝜕𝑡−𝜇₁Δ𝑉₁−𝜌_𝑀(1−𝑢)1−𝑉₁𝐾𝑉₁≥𝜕𝑉₁𝜕𝑡−𝜇₁Δ𝑉₁−𝜌_𝑀()𝑉1−𝑢1−¹𝐾𝑉₁. (3.9)
Therefore, Lemma 2.1 gives 𝑉₁(𝑡,𝑥,𝑉₀₁,𝑉₀₂,𝑉₀₃)≤𝑉₁(𝑡,𝑀_𝑉1).
Note that, according to the uniqueness theorem, the solution 𝑉₁(𝑡,𝑀_𝑉1) of (3.7) does not depend on 𝑥 for 𝑡>0, and so 𝑉₁(𝑡,𝑀_𝑉1) satisfies the ordinary differential equation: 𝜕𝑉₁𝜕𝑡−𝜌_𝑀(1−𝑢)1−𝑉₁𝐾𝑉₁=0. (3.10) By Lemma 2.3, we have 𝑉₁𝑡,𝑀_𝑉1⟶𝐾,as𝑡⟶+∞. (3.11) So, there exists 𝑁₁ and 𝑡₁>0 such that 𝑉₁(𝑡,𝑥)≤𝑁₁.
For the infected tumor cells, we have the following inequality: 0=𝜕𝑉₂𝜕𝑡−𝜇₂Δ𝑉₂+𝑉₂𝑎(𝑡,𝑥)(1−𝑢)𝑉₂(−()𝑉+𝑑𝑡,𝑥)𝛽𝑟𝑡,𝑥₁𝑉₃1+𝜀𝑉₃≥𝜕𝑉₂𝜕𝑡−𝜇₂Δ𝑉₂+𝑉₂𝑎_𝑚(1−𝑢)+𝑑_𝑚−𝜀⁻¹𝛽_𝑀𝑟_𝑀𝑁₁. (3.12) So, we get 𝜕0=𝑉₂𝜕𝑡−𝜇₂Δ𝑉₂+𝑉₂𝑎_𝑚(1−𝑢)+𝑑_𝑚−𝜀⁻¹𝛽_𝑀𝑟_𝑀𝑁₁≥𝜕𝑉₂𝜕𝑡−𝜇₂Δ𝑉₂+𝑉₂𝑎_𝑚(1−𝑢)+𝑑_𝑚−𝜀⁻¹𝛽_𝑀𝑟_𝑀𝑁₁. (3.13) Therefore, Lemma 2.1 gives 𝑉₂(𝑡,𝑥,𝑉₀₁,𝑉₀₂,𝑉₀₃)≤𝑉₂(𝑡,𝑀_𝑉2), where 𝑉₂(𝑡,𝑀_𝑉2) satisfies the ordinary differential equation: 𝜕𝑉₂𝜕𝑡=−𝑉₂𝑎_𝑚(1−𝑢)+𝑑_𝑚+𝜀⁻¹𝛽_𝑀𝑟_𝑀𝑁₁,𝑉₂0,𝑀_𝑉2=𝑀_𝑉2. (3.14) Since 𝑉₂𝑡,𝑀_𝑉2⟶𝜀⁻¹𝛽𝑟_𝑀𝑁₁𝑎_𝑚(1−𝑢)+𝑑_𝑚as𝑡⟶+∞, (3.15) 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: 0=𝜕𝑉₃𝜕𝑡−𝜇₃Δ𝑉₃+𝑏(𝑡,𝑥)𝑉₃−𝑘(𝑡,𝑥)(1−𝑢)𝑉₂≥𝜕𝑉₃𝜕𝑡−𝜇₃Δ𝑉₃+𝑏_𝑚𝑉₃−𝑘_𝑀(1−𝑢)𝑁₂. (3.16) So, we get 𝜕0=𝑉₃𝜕𝑡−𝜇₃Δ𝑉₃+𝑏_𝑚𝑉₃−𝑘_𝑀(1−𝑢)𝑁₂≥𝜕𝑉₃𝜕𝑡−𝜇₃Δ𝑉₃+𝑏_𝑚𝑉₃−𝑘_𝑀(1−𝑢)𝑁₂. (3.17) Therefore, Lemma 2.1 gives 𝑉₃(𝑡,𝑥,𝑉₀₁,𝑉₀₂,𝑉₀₃)≤𝑉₃(𝑡,𝑀_𝑉3), where 𝑉₃(𝑡,𝑀_𝑉3) satisfies the ordinary differential equation: 𝜕𝑉₃𝜕𝑡=−𝑏_𝑚𝑉₃+𝑘_𝑀(1−𝑢)𝑁₂,𝑉₃0,𝑀_𝑉3=𝑀_𝑉3. (3.18) Since 𝑉₃𝑡,𝑀_𝑉3⟶𝑘_𝑀(1−𝑢)𝑁₂𝑏_𝑚,as𝑡⟶+∞, (3.19) thus, there exists 𝑁₃ and 𝑡₃>0 such that 𝑉₃(𝑡,𝑥)≤𝑁₃.

Theorem 3.5. Assume that conditions (𝐻) hold; in addition, if 𝑑_𝑀+𝛽𝜀⁻¹𝑟_𝑀<𝜌_𝑚(1−𝑢), then the system (1.1)–(1.4) is permanent.

Proof. Theorem 3.4 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∈Ω 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 min_𝑥∈Ω𝑉₀₁(𝑥)=𝑚_𝑉1>0, min_𝑥∈Ω𝑉₀₂(𝑥)=𝑚_𝑉2>0 and, min_𝑥∈Ω𝑉₀₃(𝑥)=𝑚_𝑉3>0. Let 𝑉₁(𝑡,𝑥) be a solution of 𝜕𝑉₁+𝑉𝜕𝑡¹𝑑_𝑀−𝜌_𝑚()𝑉1−𝑢1−¹𝐾+𝜀⁻¹𝛽_𝑀𝑟_𝑀𝑉=0,₁0,𝑀_𝑉1=𝑚_𝑉1. (3.20) So using the inequality 0=𝜕𝑉₁𝜕𝑡−𝜇₁Δ𝑉₁+𝑉₁𝑑()()𝑉𝑥,𝑡−𝜌𝑡,𝑥)(1−𝑢1−¹𝐾+𝛽𝑟(𝑡,𝑥)𝑉₁𝑉₃1+𝜀𝑉₃≤𝜕𝑉₁𝜕𝑡−𝜇₁Δ𝑉₁+𝑉₁𝑑_𝑀−𝜌_𝑚(𝑉1−𝑢)1−¹𝐾+𝜀⁻¹𝛽_𝑀𝑟_𝑀, (3.21) one has, by (3.20), 𝜕𝑉0=¹𝜕𝑡−𝜇₁Δ𝑉₁+𝑉₁𝑑_𝑀−𝜌_𝑚()𝑉1−𝑢1−¹𝐾+𝜀⁻¹𝛽_𝑀𝑟_𝑀≤𝜕𝑉₁𝜕𝑡−𝜇₁Δ𝑉₁+𝑉₁𝑑_𝑀−𝜌_𝑚(𝑉1−𝑢)1−¹𝐾+𝜀⁻¹𝛽_𝑀𝑟_𝑀. (3.22) Therefore, Lemma 2.1 gives 𝑉₁(𝑡,𝑥,𝑉₀₁,𝑉₀₂,𝑉₀₃)≥𝑉₁(𝑡,𝑚_𝑉1). By the condition 𝑑_𝑀+𝜀⁻¹𝛽_𝑀𝑟_𝑀<𝜌_𝑚(1−𝑢), we have 𝑉₁𝑡,𝑚_𝑉1⟶𝐾𝜌_𝑚(1−𝑢)−𝑑_𝑀−𝜀⁻¹𝛽_𝑀𝑟_𝑀𝜌_𝑚(1−𝑢),𝑡⟶+∞. (3.23) Therefore, there exists 𝜁₁>0 such that 𝑉₁(𝑡,𝑥,𝑉₀₁,𝑉₀₂,𝑉₀₃)≥𝜁₁ for 𝑡 large enough.
Let 𝑉₂(𝑡,𝑥) be a solution of 𝜕𝑉₂+𝑉𝜕𝑡²𝑎_𝑀(1−𝑢)+𝑑_𝑀−𝜀⁻¹𝛽_𝑀𝑟_𝑀𝜁₁=0,𝑉₂(0,𝑥)=𝑚_𝑉2. (3.24) Using the inequality 0=𝜕𝑉₂𝜕𝑡−𝜇₂Δ𝑉₂+𝑉₂[𝑎()(−()𝑉𝑡,𝑥)(1−𝑢+𝑑𝑡,𝑥)]𝛽𝑟𝑡,𝑥₁𝑉₃1+𝜀𝑉₃≤𝜕𝑉₂𝜕𝑡−𝜇₂Δ𝑉₂+𝑉₂𝑎_𝑀(1−𝑢)+𝑑_𝑀−𝜀⁻¹𝛽_𝑀𝑟_𝑀𝜁₁, (3.25) we have, by (3.24), 𝜕𝑉0=²𝜕𝑡−𝜇₂Δ𝑉₂+𝑉₂𝑎_𝑀(1−𝑢)+𝑑_𝑀−𝜀⁻¹𝛽_𝑀𝑟_𝑀𝜁₁≤𝜕𝑉₂𝜕𝑡−𝜇₂Δ𝑉₂+𝑉₂𝑎_𝑀(1−𝑢)+𝑑_𝑀−𝜀⁻¹𝛽_𝑀𝑟_𝑀𝜁₁. (3.26) Therefore, 𝑉₂(𝑡,𝑥,𝑉₀₁,𝑉₀₂,𝑉₀₃)≥𝑉₂(𝑡,𝑚_𝑉2)→(𝜀⁻¹𝛽_𝑀𝑟_𝑀𝜁₁)/(𝑎_𝑀(1−𝑢)+𝑑_𝑀), as 𝑡→+∞. Therefore, there exists 𝜁₂>0 such that 𝑉₂(𝑡,𝑥,𝑉₀₁,𝑉₀₂,𝑉₀₃)≥𝜁₂ for 𝑡 large enough.
Now, let 𝑉₃(𝑡,𝑥) be a solution of 𝜕𝑉₃𝜕𝑡+𝑏_𝑀𝑉₃−𝑘_𝑀(1−𝑢)𝜁₂=0,𝑉₃(0,𝑥)=𝑚_𝑉3. (3.27) By the following inequality, 0=𝜕𝑉₃𝜕𝑡−𝜇₃Δ𝑉₃+𝑏(𝑡,𝑥)𝑉₃−𝑘(𝑡,𝑥)(1−𝑢)𝑉₂≤𝜕𝑉₃𝜕𝑡−𝜇₃Δ𝑉₃+𝑏_𝑀𝑉₃−𝑘_𝑀(1−𝑢)𝜁₂, (3.28) we have 𝜕𝑉₃𝜕𝑡−𝜇₃Δ𝑉₃+𝑏_𝑀𝑉₃−𝑘_𝑀(1−𝑢)𝜁₂≤𝜕𝑉₃𝜕𝑡−𝜇₃Δ𝑉₃+𝑏_𝑀𝑉₃−𝑘_𝑀(1−𝑢)𝜁₂. (3.29) Therefore, 𝑉₃(𝑡,𝑥,𝑉₀₁,𝑉₀₂,𝑉₀₃)≥𝑉₃(𝑡,𝑚_𝑉3)→(𝑘_𝑀(1−𝑢)𝜁₂)/(𝑏_𝑀), 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 2𝜌_𝑀(1−𝑢)1−2𝜁𝐾−𝑑_𝑚−𝜀⁻¹𝛽_𝑀𝑟_𝑀𝛽<0,_𝑀𝜀𝑟_𝑀+𝑘_𝑀−4𝜁𝑎_𝑀(1−𝑢)+2𝑑_𝑀𝑘<0,_𝑀−2𝑏_𝑚<0, (4.1) then the system has a unique globally asymptotic stable strictly positive 𝜔-periodic solution.

Proof. For convenience, we denote 𝑎=𝑎(𝑡,𝑥) and similar meaning to 𝑏, 𝑑, 𝑘, 𝑟, and 𝜌. Let (𝑉¹₁(𝑡,𝑥),𝑉¹₂(𝑡,𝑥),𝑉¹₃(𝑡,𝑥)) and (𝑉¹₂(𝑡,𝑥),𝑉²₂(𝑡,𝑥),𝑉²₃(𝑡,𝑥)) be two solutions of system bounded by constants 𝜁 and 𝜂 for below and above. Consider the function 𝐿(𝑡)=_Ω𝑉¹₁(𝑡,𝑥)−𝑉²₁(𝑡,𝑥)²+𝑉¹₂(𝑡,𝑥)−𝑉²₂(𝑡,𝑥)²+𝑉¹₃(𝑡,𝑥)−𝑉²₃(𝑡,𝑥)²𝑑𝑥. (4.2) So, 𝑑𝐿(𝑡)𝑑𝑡=2_Ω𝑉¹₁−𝑉²₁𝜕𝑉¹₁−𝜕𝑡𝜕𝑉²₁𝜕𝑡𝑑𝑥+2_Ω𝑉¹₂−𝑉²₂𝜕𝑉¹₂−𝜕𝑡𝜕𝑉²₂𝜕𝑡𝑑𝑥+2_Ω𝑉¹₃−𝑉²₃𝜕𝑉¹₃−𝜕𝑡𝜕𝑉²₃𝜕𝑡𝑑𝑥=2𝜇_1Ω𝑉¹₁−𝑉²₁Δ𝑉¹₁−𝑉²₁𝑑𝑥+2𝜇_2Ω𝑉¹₂−𝑉²₂Δ𝑉¹₂−𝑉²₂𝑑𝑥+2𝜇_3Ω𝑉¹₃−𝑉²₃Δ𝑉¹₃−𝑉²₃𝑑𝑥+2_Ω𝑉¹₁−𝑉²₁²𝜌()𝑉1−𝑢1−¹¹+𝑉²₁𝐾−𝑑𝑑𝑥−2_Ω𝑉¹₁−𝑉²₁𝛽𝑟𝑉¹₁𝑉¹₃1+𝜀𝑉¹₃𝑑𝑥+2_Ω𝑉¹₁−𝑉²₁𝛽𝑟𝑉²₁𝑉²₃1+𝜀𝑉²₃𝑑𝑥−2_Ω𝑉¹₂−𝑉²₂²𝑎(1−𝑢)𝑉¹₁+𝑉²₁+𝑑𝑑𝑥+2_Ω𝑉¹₂−𝑉²₂𝛽𝑟𝑉¹₁𝑉¹₃1+𝜀𝑉¹₃𝑑𝑥−2_Ω𝑉¹₂−𝑉²₂𝛽𝑟𝑉²₁𝑉²₃1+𝜀𝑉²₃𝑑𝑥+2_Ω𝑘(1−𝑢)𝑉¹₂−𝑉²₂𝑉¹₃−𝑉²₃𝑑𝑥−2_Ω𝑏𝑉¹₃−𝑉²₃²𝑑𝑥∶=𝐼₁+⋯+𝐼₁₁. (4.3) It follows from the boundary condition (1.4) that 𝐼₁+𝐼₂+𝐼₃=−2_Ω𝜇₁∇𝑉¹₁−𝑉²₁²+𝜇₂∇𝑉¹₂−𝑉²₂²+2𝜇₃∇𝑉¹₃−𝑉²₃²𝑑𝑥≤0 (4.4) The terms 𝐼₄+𝐼₇+𝐼₁₀+𝐼₁₁≤2_Ω𝜌_𝑀(1−𝑢)1−2𝜁𝐾−𝑑_𝑚𝑉¹₁−𝑉²₁²−𝑑𝑥_Ω4𝜁𝑎_𝑀(1−𝑢)+2𝑑_𝑀𝑉¹₂−𝑉²₂²𝑑𝑥−2_Ω𝑏_𝑚𝑉¹₃−𝑉²₃²𝑑𝑥+_Ω𝑘_𝑀𝑉¹₂−𝑉²₂²+𝑉¹₃−𝑉²₃²𝐼𝑑𝑥,₅+𝐼₆+𝐼₈+𝐼₉=2_Ω𝛽𝑟𝑉¹₁𝑉¹₃1+𝜀𝑉¹₃𝑉¹₂−𝑉²₂−𝑉¹₁−𝑉²₁𝑑𝑥+2_Ω𝛽𝑟𝑉²₁𝑉²₃1+𝜀𝑉²₃𝑉¹₁−𝑉²₁−𝑉¹₂−𝑉²₂𝑑𝑥≤2_Ω𝜀⁻¹𝛽_𝑀𝑟_𝑀𝑉¹₁𝑉¹₂−𝑉²₂−𝑉¹₁−𝑉²₁𝑑𝑥+2_Ω𝜀⁻¹𝛽_𝑀𝑟_𝑀𝑉²₁𝑉¹₁−𝑉²₁−𝑉¹₂−𝑉²₂𝑑𝑥≤−2_Ω𝜀⁻¹𝛽_𝑀𝑟_𝑀𝑉¹₁−𝑉²₁²𝑑𝑥+2_Ω𝜀⁻¹𝛽_𝑀𝑟_𝑀𝑉¹₁−𝑉²₁𝑉¹₂−𝑉²₂𝑑𝑥≤−_Ω𝜀⁻¹𝛽_𝑀𝑟_𝑀𝑉¹₁−𝑉²₁²−𝑉¹₂−𝑉²₂²𝑑𝑥. (4.5) Thus, we have𝑑𝐿(𝑡)≤𝑑𝑡_Ω2𝜌_𝑀(1−𝑢)1−2𝜁𝐾−𝑑_𝑚−𝜀⁻¹𝛽_𝑀𝑟_𝑀𝑉¹₁−𝑉²₁²+𝑑𝑥_Ω𝜀⁻¹𝛽_𝑀𝑟_𝑀+𝑘_𝑀−4𝜁𝑎_𝑀(1−𝑢)+2𝑑_𝑀𝑉¹₂−𝑉²₂²+𝑑𝑥_Ω𝑘_𝑀−2𝑏_𝑚𝑉¹₃−𝑉²₃²≤𝑉𝑑𝑥₁₁−𝑉²₁,𝑉¹₂−𝑉²₂,𝑉¹₃−𝑉²₃𝑃𝑉¹₁−𝑉²₁,𝑉¹₂−𝑉²₂,𝑉¹₃−𝑉²₃^𝑇≤𝜆_𝑀(𝑃)𝑉¹₁−𝑉²₁²_𝐿2(Ω)+𝑉¹₂−𝑉²₂²_𝐿2(Ω)+𝑉¹₃−𝑉²₃²_𝐿2(Ω), (4.6) where 𝜆_𝑀(𝑃) is the maximal eigenvalue of the diagonal matrix 𝑃=diag(𝑃₁,𝑃₂,𝑃₃) with 𝑃₁=2𝜌_𝑀(1−𝑢)1−2𝜁𝐾−𝑑_𝑚−𝜀⁻¹𝛽_𝑀𝑟_𝑀𝑃₂=𝜀⁻¹𝛽_𝑀𝑟_𝑀+𝑘_𝑀−4𝜁𝑎_𝑀(1−𝑢)+2𝑑_𝑀𝑃₃=𝑘_𝑀−2𝑏_𝑚. (4.7) So, we deduce that 𝐿(𝑡)≤𝐿(0)𝑒^{𝜆𝑀(𝑃)𝑡}⟶0,as𝑡⟶+∞. (4.8) Thus, ‖𝑉¹₁−𝑉²₁‖_𝐿2(Ω)→0, ‖𝑉¹₂−𝑉²₂‖_𝐿2(Ω)→0, and ‖𝑉¹₃−𝑉²₃‖_𝐿2(Ω)→0, as 𝑡→+∞. By Theorem 3.3, solutions of system (1.1)–(1.4) are bounded in the space 𝐶^1+𝜈(Ω,ℝ³, where 0<𝜈<2𝑙−1−𝑛/𝑝 and 1/2+𝑛/(2𝑝)<𝑙<1. Therefore, lim_𝑡→+∞sup_𝑥∈Ω𝑉¹₁(𝑡,𝑥)−𝑉²₁(𝑡,𝑥)=0,lim_𝑡→+∞sup_𝑥∈Ω𝑉¹₂(𝑡,𝑥)−𝑉²₂(𝑡,𝑥)=0,lim_𝑡→+∞sup_𝑥∈Ω𝑉¹₃(𝑡,𝑥)−𝑉²₃(𝑡,𝑥)=0. (4.9) Consider the sequence 𝑉₁𝑞𝜔,𝑥,𝑉₀₁,𝑉₀₂,𝑉₀₃;𝑉₂𝑞𝜔,𝑥,𝑉₀₁,𝑉₀₂,𝑉₀₃;𝑉₃𝑞𝜔,𝑥,𝑉₀₁,𝑉₀₂,𝑉₀₃=𝑊𝑞𝜔,𝑊₀. (4.10) Then {𝑊(𝑞𝜔,𝑊₀),𝑞∈ℕ} is compact in the space 𝐶(Ω)³. Let 𝑊 be a limit point of this sequence. It follows, from 𝑊𝜔,𝑊−𝑊_𝐶≤𝑊𝜔,𝑊−𝑊𝜔,𝑊𝑞_𝑛𝜔,𝑊_0𝐶+𝑊𝜔,𝑊𝑞_𝑛𝜔,𝑊₀−𝑊𝑞_𝑛𝜔,𝑊_0𝐶+𝑊𝑞_𝑛𝜔,𝑊₀−𝑊_𝐶⟶0as𝑛⟶+∞, (4.11) that 𝑊(𝜔,𝑊)=𝑊.
Next, let 𝑊 and 𝑊 be two limit points of the sequence {𝑊(𝑞𝜔,𝑊₀),𝑞∈ℕ}. Using (4.9) and 𝑊=𝑊(𝑞_𝑛𝜔,𝑊), we have 𝑊−𝑊_𝐶≤𝑊−𝑊𝑞_𝑛𝜔,𝑊_0𝐶+𝑊𝑞_𝑛𝜔,𝑊₀−𝑊_𝐶⟶0,as𝑛⟶+∞. (4.12) Thus, 𝑊=𝑊, and so (𝑉₁(𝑡,𝑥,𝑉₁,𝑉₂,𝑉₃);𝑉₂(𝑡,𝑥,𝑉₁,𝑉₂,𝑉₃);𝑉₃(𝑡,𝑥,𝑉₁,𝑉₂,𝑉₃)) is the unique periodic solutions of system (1.1)–(1.4). By (4.9), it is asymptotically stable.