Global Stability and Bifurcations of a Diffusive Ratio-Dependent Holling-Tanner System

Abstract

The dynamics of a diffusive ratio-dependent Holling-Tanner predator-prey system subject to Neumann boundary conditions are considered. By choosing the ratio of intrinsic growth rates of predators to preys as a bifurcation parameter, the existence and stability of spatially homogeneous and nonhomogeneous Hopf bifurcations and steady state bifurcation are investigated in detail. Meanwhile, we show that Turing instability takes place at a certain critical value; that is, the stationary solution becomes unstable induced by diffusion. Particularly, the sufficient conditions of the global stability of the positive constant coexistence are given by the upper-lower solutions method.

1. Introduction

Predator-prey systems describing the interaction between two species with positive-negative feedbacks continue to arouse the interest of applied mathematicians and ecologists. The functional response is the important component depicting the predator-prey relationship [1]. In order to better characterize the ecological interaction such as lynx and hare and sparrow and sparrow hawk especially by Tanner [2] and Wollkind et al. [3], May developed the so-called Holling-Tanner predator-prey model [4] which takes the following form: 𝑑𝑈(𝑇)𝑈(𝑇)𝑑𝑇=𝑟𝑈1−𝐾−𝑚𝑈(𝑇)𝑉(𝑇)𝑢(𝑇),+𝑞𝑑𝑉(𝑇𝑉(𝑇)𝑑𝑇=𝑠𝑉)1−ℎ𝑈(𝑇), (1) where 𝑈(𝑇), 𝑉(𝑇) stand for the prey and the predator densities, respectively. And in ecological senses, 𝑟, 𝐾, 𝑚, 𝑞, 𝑠, ℎ are positive constants which denote prey intrinsic growth rate, carrying capacity, capturing rate, half capturing saturation constant, predator intrinsic growth rate, and conversion rate of prey into predator, respectively. The functional response 𝑚𝑈/(𝑈+𝑞) is of Michaelis-Menten type in enzyme-substrate kinetics, proposed by Holling [5].

Hsu and Huang [6] proved the global stability of the positive equilibrium of System (1) by applying the Dulac’s criterion and constructing Lyapunov function. Gasull et al. [7] showed that the stable positive equilibrium is surrounded by two limit cycles. Thus, the local stability does not imply the global stability.

Spatial diffusion is ubiquitous. Assuming the preys and the predators are in an isolate patch, I neglect the impact of migration, including immigration and emigration, and only consider the diffusion of the spatial domain. It has been shown that the diffusion can generate the more rich spatiotemporal complex [8–10]. A diffusive Holling-Tanner predator-prey system is as follows: 𝜕𝑈𝜕𝑇=𝐷₁𝑈_𝑥𝑥()𝑈()+𝑟𝑈𝑥,𝑇1−𝑥,𝑇𝐾−𝑚𝑈(𝑥,𝑇)𝑉(𝑥,𝑇)𝑈()()𝑥,𝑇+𝑞,𝑥∈0,𝑙𝜋,𝑇>0,𝜕𝑉𝜕𝑇=𝐷₂𝑉_𝑥𝑥()𝑉()+𝑠𝑉𝑥,𝑇1−ℎ𝑥,𝑇𝑈(),()𝑈𝑥,𝑇𝑥∈0,𝑙𝜋,𝑇>0,_𝑥(0,𝑇)=𝑈_𝑥(𝑙𝜋,𝑇)=𝑉_𝑥(0,𝑇)=𝑉_𝑥()𝑈()𝑙𝜋,𝑇=0,𝑇≥0,𝑥,0=𝑈₀(𝑥)≥0,𝑉(𝑥,0)=𝑉₀(𝑥)≥0,𝑥∈[0,𝑙𝜋]. (2) Ma and Li [11] investigated the existence of Hopf bifurcation and the steady state bifurcations of simple and double eigenvalues of System (2). Peng and Wang [12] proved the global stability of the unique positive equilibrium of System (2) under certain hypothesis by constructing Lyapunov function. Chen and Shi [13] improved the results given in [12].

Some biologists have questioned the functional response solely depending on the prey density [14–16], especially when predators have to search for food, (and therefore have to share or compete for food). Numerous fields and laboratory experiments have revealed that a more reasonable functional response should be the so-called ratio-dependent predator-prey theory [14, 15], which can be strongly stated as that the per capita predator growth rate should be the function of the ratio of predator abundance. A predator-prey system with the ratio-dependent functional response can exhibit rich dynamics [17, 18].

In this paper, we consider the following diffusive ratio-dependent Holling-Tanner predator-prey system: 𝜕𝑈𝜕𝑇=𝐷₁𝑈_𝑥𝑥()𝑈()+𝑟𝑈𝑥,𝑇1−𝑥,𝑇𝐾−𝑚𝑈(𝑥,𝑇)𝑉(𝑥,𝑇)𝑈()()()𝑥,𝑇+𝑞𝑉𝑥,𝑇,𝑥∈0,𝑙𝜋,𝑇>0,𝜕𝑉𝜕𝑇=𝐷₂𝑉_𝑥𝑥()𝑉()+𝑠𝑉𝑥,𝑇1−ℎ𝑥,𝑇𝑈(),()𝑈𝑥,𝑇𝑥∈0,𝑙𝜋,𝑇>0,_𝑥(0,𝑇)=𝑈_𝑥(𝑙𝜋,𝑇)=𝑉_𝑥(0,𝑇)=𝑉_𝑥()𝑈()𝑙𝜋,𝑇=0,𝑇≥0,𝑥,0=𝑈₀(𝑥)≥0,𝑉(𝑥,0)=𝑉₀(𝑥)≥0,𝑥∈[0,𝑙𝜋]. (3) M. Banerjee and S. Banerjee [19] examined the existence of Turing and non-Turing patterns of System (3) by numerical simulations and revealed the fact that Hopf bifurcation is essential for spatiotemporal chaos. Liang and Pan [20] established the global stability of positive equilibrium of System (3) without diffusion by constructing Lyapunov function and obtained the uniqueness of the limit cycle.

My work is the extension of [11, 19]. In addition, what is more important in ecosystem is whether the species would coexist in the long run; that is, whether the positive equilibrium is globally asymptotically stable. In this paper, we aim to study Hopf bifurcations and steady state bifurcations induced by intrinsic growth rate and Turing instability induced by diffusion. Particularly, the global stability of the unique positive equilibrium is obtained by the upper-lower solutions method under certain conditions. Simulations are carried out to strongly support the theoretical results.

The rest of the paper is organized as follows. In Section 2, the instability of the semitrivial equilibrium and the local stability of the positive coexistence of system (3) are studied by the distribution of eigenvalues. And the existence, stability, and direction of Hopf bifurcations are investigated in detail as the intrinsic growth rate 𝜆 crosses the critical value. Turing bifurcation induced by diffusion takes place at the certain critical values. In Section 3, the steady state bifurcations are considered with 𝜆 as the bifurcation parameter and the interaction of Hopf bifurcation and steady state bifurcation are discussed. In Section 4, the global stability of the unique positive coexistence is proved by the upper-lower solutions method.

2. Hopf Bifurcations and Turing Bifurcations

For simplicity of later discussion, we suppose that 𝐷₁=𝑟. Introducing the following dimensionless variables: 𝑈𝑢=𝐾,𝑣=𝑚𝑉𝑟𝐾,𝑡=𝑟𝑇, (4) System (3) can be transformed into 𝜕𝑢𝜕𝑡=𝑢_𝑥𝑥((−𝑢()𝑣()+𝑢𝑥,𝑡)(1−𝑢𝑥,𝑡))𝑥,𝑡𝑥,𝑡𝑢()()()𝑥,𝑡+𝛼𝑣𝑥,𝑡,𝑥∈0,𝑙𝜋,𝑡>0,𝜕𝑣𝜕𝑡=𝑑𝑣_𝑥𝑥()𝑣()+𝜆𝑣𝑥,𝑡𝛽−𝑥,𝑡𝑢(),()𝑢𝑥,𝑡𝑥∈0,𝑙𝜋,𝑡>0,_𝑥(0,𝑡)=𝑢_𝑥(𝑙𝜋,𝑡)=𝑣_𝑥(0,𝑡)=𝑣_𝑥()𝑢()𝑙𝜋,𝑡=0,𝑡≥0,𝑥,0=𝑢₀(𝑥)≥0,𝑣(𝑥,0)=𝑣₀(𝑥)≥0,𝑥∈[0,𝑙𝜋], (5) where 𝑑=𝐷₂/𝐷₁, 𝛼=𝑞𝑟/𝑚, 𝜆=𝑠ℎ/𝑚, 𝛽=𝑚/ℎ𝑟.

It is easy to check that System (5) has always a boundary equilibrium 𝐸₀(1,0) and has a positive constant coexistence 𝐸(𝑢^∗,𝑣^∗) if and only if 𝛼𝛽+1>𝛽, where 𝑢^∗𝛽=1−𝛼𝛽+1>0,𝑣^∗=𝛽𝑢^∗. (6) The characteristic equations of the linearization of (5) at 𝐸₀(1,0) are given by 𝑛𝜇+²𝑙²𝑛+1𝜇+𝑑²𝑙²−𝜆𝛽=0,𝑛=0,1,2,…. (7) Thus, 𝜇₁𝑛=−²𝑙²−1,𝜇₂𝑛=𝜆𝛽−𝑑²𝑙²,𝑛=0,1,…. (8) Hence, 𝐸₀(1,0) is always unstable since (7) has at least a positive root 𝜇=𝜆𝛽. Next we study the local stability of the positive steady state 𝐸(𝑢^∗,𝑣^∗) and the existence of bifurcations.

2.1. Stability of the Positive Coexistence and Hopf Bifurcations

The linearized operator of System (5) at 𝐸(𝑢^∗,𝑣^∗) is as follows: 𝐿(𝜆)=𝜕²𝜕𝑥²+𝛼𝛽²+2𝛽𝛼𝛽+1²1−1−𝛼𝛽+1²𝜆𝛽²𝑑𝜕²𝜕𝑥²,𝐿−𝜆𝛽_𝑛(𝜆)=−𝑛²𝑙²+𝛼𝛽²+2𝛽𝛼𝛽+1²1−1−𝛼𝛽+1²𝜆𝛽²𝑛−𝑑²𝑙².−𝜆𝛽 (9) The characteristic equations corresponding to 𝐿_𝑛(𝜆) are 𝜇²−𝑇_𝑛(𝜆)𝜇+𝐷_𝑛(𝜆)=0,𝑛=0,1,2,…, (10) where 𝑇_𝑛(𝜆)()𝑛=−1+𝑑²𝑙²−𝜆𝛽+𝛼𝛽²+2𝛽𝛼𝛽+1²()𝑛−1=−1+𝑑²𝑙²−𝜆−𝜆^𝐻₀𝐷𝛽,_𝑛(𝜆)=𝑑𝑛⁴𝑙⁴+𝑛²𝑙²𝑑1−𝛼𝛽²+2𝛽𝛼𝛽+1²𝛽+𝜆𝛽+𝜆𝛽1−=1+𝛼𝛽𝑑𝑛⁴𝑙⁴+𝑛²𝑙²𝜆−𝑑𝜆^𝐻₀𝛽+𝜆𝛽𝑢^∗, (11) where 𝜆^𝐻₀=(1/𝛽)((𝛼𝛽²+2𝛽)/(𝛼𝛽+1)²−1).

Obviously, if 𝛼𝛽²+2𝛽≤(𝛼𝛽+1)², then 𝑇_𝑛(𝜆)<0, 𝐷_𝑛(𝜆)>0, which implies that all the roots of (10) are negative real-part. Thus the unique positive constant equilibrium 𝐸(𝑢^∗,𝑣^∗) is locally asymptotically stable.

Indeed, in Section 4, we can prove the global stability of the unique positive coexistence under the stronger conditions. Next, we will seek all the possible Hopf bifurcation values 𝜆₀, so we assume 𝛼𝛽²+2𝛽>(1+𝛼𝛽)² holds.

We will identify Hopf bifurcation value 𝜆₀ which satisfies 𝑇_𝑛𝜆₀=0,𝐷_𝑛𝜆₀𝑇>0,_𝑗𝜆₀≠0,𝐷_𝑗𝜆₀≠0,∀𝑗≠𝑛. (12) From (11), 𝑇_𝑛(𝜆)<0, for all 𝜆∈(𝜆^𝐻₀,+∞), which means that the possible Hopf bifurcation points belong to the interval (0,𝜆^𝐻₀]. For any Hopf bifurcation point 𝜆^∗∈(0,𝜆^𝐻₀], 𝛼(𝜆)±𝑖𝜔(𝜆) are a pair of conjugate eigenvalues of 𝐿_𝑛(𝜆) near 𝜆=𝜆^∗, where 𝛼(𝜆)=𝑇_𝑛(𝜆)2(𝜆)=1,𝜔24𝐷_𝑛(𝜆)−𝑇²_𝑛(𝜆), (13) and 𝛼^′(𝜆^∗)=−𝛽/2<0.

Hence, all Hopf bifurcation points reduce to the following set: Λ₁=𝜆∈0,𝜆^𝐻₀()}.,forsome𝑛∈𝑁,thecondition12issatisfied (14) Obviously, 𝑇_𝑛(𝜆)=0 is equivalent to 𝜆_𝑛=𝜆^𝐻₀−(1+𝑑)𝑛²𝛽𝑙²,𝐷_𝑛𝜆_𝑛𝑛=−⁴𝑙⁴−𝑛²𝑙²𝐵₀+𝜆^𝐻₀𝛽𝑢^∗, (15)(16) where 𝐵₀=2−(2𝛼𝛽²+3𝛽−𝑑𝛽)/(1+𝛼𝛽)². Since 𝐷₀(𝜆₀)=𝜆^𝐻₀𝛽𝑢^∗>0, there exists 𝑁₀∈𝑁⁺ such that 𝐷_𝑛(𝜆_𝑛)>0, 𝑛=0,1,2,…,𝑁₀, and 𝐷_𝑛(𝜆_𝑛)≤0, 𝑛=𝑁₀+1, 𝑁₀+2,….

Clearly, 𝑇_𝑖(𝜆_𝑛)=((1+𝑑)/𝑙²)(𝑛²−𝑖²)≠0, for all 𝑖≠𝑛. And for 𝑖≠𝑛, 𝐷_𝑖𝜆_𝑛=𝑑𝑖⁴𝑙⁴+𝑖²𝑙²𝜆_𝑛−𝑑𝜆^𝐻₀𝛽+𝜆_𝑛𝛽𝑢^∗. (17) We discuss 𝐷_𝑖(𝜆_𝑛)>0, for all 𝑖≠𝑛 in two cases.

Case 1. We can choose small enough 𝑑>0 such that 𝜆_𝑛−𝑑𝜆^𝐻₀>0, that is, for 𝑛=0,1,…,𝑁₀, 𝑑<𝑑₀, where 𝑑₀∶=𝛽𝑙²𝜆^𝐻₀−𝑁²₀𝛽𝑙²𝜆^𝐻₀+𝑁²₀>0. (18)

Case 2. When 𝑑>𝑑₀, 𝑑𝜆^𝐻₀−𝜆_𝑛=(𝑑−1)𝜆^𝐻₀+(1+𝑑)𝑛²𝑙²𝛽≤(𝑑−1)𝜆^𝐻₀+(1+𝑑)𝑁²₀𝑙²𝛽≜𝑀^∗,𝜆_𝑛𝑢^∗≥𝜆^𝐻₀−(1+𝑑)𝑁²₀𝑙²𝛽𝑢^∗=𝑑𝜆^𝐻₀−𝑀^∗𝑢^∗. (19) Therefore, 𝐷_𝑖𝜆_𝑛≥𝑑𝑖⁴𝑙⁴−𝑖²𝑙²𝑀^∗𝛽+𝑑𝜆^𝐻₀−𝑀^∗𝑢^∗𝑖𝛽=𝑔²𝑙². (20) In order that 𝑔(𝑦)=𝑑𝑦²−𝑀^∗𝛽𝑦+(𝑑𝜆^𝐻₀−𝑀^∗)𝑢^∗𝛽>0 for all 𝑦∈𝑅, we only need (𝑀^∗)²𝛽<4𝑑(𝑑𝜆^𝐻₀−𝑀^∗)𝑢^∗. Thus, 𝐷_𝑖(𝜆_𝑛)>0.
Clearly, 𝜆^𝐻₀=max_{𝑛∈{0,1,…,𝑁0}}𝜆_𝑛.

Summarizing the above analysis results, we draw our main conclusion in this subsection.

Theorem 1. Suppose 𝛼,𝛽,𝑑>0 and 𝛼𝛽+1>𝛽, 𝛼𝛽²+2𝛽>(1+𝛼𝛽)² hold. If there exists 𝑑₀>0 such that 0<𝑑<𝑑₀, or (𝑀^∗)²𝛽<4𝑑(𝑑𝜆^𝐻₀−𝑀^∗)𝑢^∗, where 𝑑₀ and 𝑀^∗ are defined by (18) and (19), respectively. Then for System (5), the following results are true:(1)The positive constant steady state 𝐸(𝑢^∗,𝑣^∗) is locally asymptotically stable for 𝜆∈(𝜆^𝐻₀,+∞) and unstable for 𝜆∈(0,𝜆^𝐻₀);(2)System (5) undergoes a Hopf bifurcation at 𝜆=𝜆_𝑛,𝑛=0,1,…,𝑁₀ (defined by (15)) and the bifurcating periodic solutions can be parameterized into the following forms: 𝑢(𝑠)(𝑥,𝑡)=𝑠𝑎_𝑛𝑒^{2𝜋𝑖𝑡/𝑇(𝑠)}+𝑎_𝑛𝑒^{−2𝜋𝑖𝑡/𝑇(𝑠)}cos𝑛𝑥𝑙(𝑠),𝑣(𝑠)𝑏+𝑜)(𝑥,𝑡=𝑠_𝑛𝑒^{2𝜋𝑖𝑡/𝑇(𝑠)}+𝑏_𝑛𝑒^{−2𝜋𝑖𝑡/𝑇(𝑠)}cos𝑛𝑥𝑙+𝑜(𝑠). (21)Furthermore, the periodic solutions bifurcating from 𝐸(𝑢^∗,𝑣^∗) at 𝜆=𝜆^𝐻₀ are spatially homogeneous. And bifurcating periodic solutions from 𝜆=𝜆_𝑛, (𝑛=1,…,𝑁₀) are spatially inhomogeneous.

Next we only consider the direction and stability of Hopf bifurcation at 𝜆=𝜆^𝐻₀.

Theorem 2. For the System (5), the Hopf bifurcations at 𝜆=𝜆^𝐻₀ are backward, and the bifurcating periodic solutions are locally asymptotically stable.

Proof. Here we apply the notations and calculations in [9, 21], we put 𝑎𝑞∶=⁰𝑏₀=𝜆^𝐻₀𝛽+𝑖𝜔₀𝜆^𝐻₀𝛽²1,𝑞^∗𝑎∶=^∗0𝑏^∗₀=𝜆^𝐻₀𝛽²2𝑙𝜋𝜔₀𝑖𝜔₀−𝑖𝜆^𝐻₀𝛽2𝑙𝜋𝜔₀, (22) such that ⟨𝑞^∗,𝑞⟩=1 and ⟨𝑞^∗,𝑞⟩=0, where 𝜔₀=𝜆^𝐻₀𝛽𝑢^∗.
Recall that in our context, 𝑓(𝜆,𝑢,𝑣)=𝑢(1−𝑢)−𝑢𝑣,𝑔()𝑣𝑢+𝛼𝑣𝜆,𝑢,𝑣=𝜆𝑣𝛽−𝑢. (23)
By direct computation, it follows that 𝑐₀2=−𝛽²+2𝜔²₀𝜆^𝐻₀²𝛽⁴−2𝛼𝛼𝛽+1³𝜆^𝐻₀𝛽−4𝜔₀𝜆^𝐻₀𝛽³𝑑𝑖,₀=2𝛽,𝑒₀2=−𝛽²−2𝑢^∗𝜆^𝐻₀𝛽³+2𝛼𝛼𝛽+1³𝜆^𝐻₀𝛽,𝑓₀2=−𝛽,𝑔₀=𝛼𝜆^𝐻₀𝛽²𝑢^∗𝛼𝛽+1⁴−2𝛼𝛽−2−3𝜔₀𝜆^𝐻₀𝛽𝑖,ℎ₀=2𝑢^∗1𝛽²+3𝜔₀𝜆^𝐻₀𝛽³𝑖, (24) and ⟨𝑞^∗,𝑄_𝑞𝑞1⟩=−𝛽+𝑖2𝜆^𝐻₀𝜔₀−𝜔₀𝜆^𝐻₀𝛽²+𝛼𝛽𝛼𝛽+1³𝜔₀,⟨𝑞^∗,𝑄_𝑞𝑞⟩=⟨𝑞^∗,𝑄_𝑞𝑞1⟩=−𝛽+𝑖𝜔₀𝑢^∗𝛽−𝛼𝛽𝛼𝛽+1³,⟨𝑞^∗,𝑄_𝑞𝑞3⟩=𝛽−𝑖2𝜆^𝐻₀𝜔₀−𝜔₀𝜆^𝐻₀𝛽²+𝛼𝛽𝛼𝛽+1³𝜔₀,𝑞^∗,𝐶_𝑞𝑞𝑞=−3𝛼2𝑢^∗𝜆^𝐻₀𝛽𝛼𝛽+1⁴+2𝛽²𝑢^∗𝛼+𝑖𝜔₀𝑢^∗𝛼𝛽+1³+𝜆^𝐻₀𝜔₀𝑢^∗𝛽+3𝜔₀𝜆^𝐻₀𝛽³𝑢^∗. (25) Hence, it is direct to calculate, by (24) and (25), 𝐻₂₀=𝑐₀𝑑₀−𝑞^∗,𝑄_𝑞𝑞𝑎₀𝑏₀−⟨𝑞^∗,𝑄_𝑞𝑞⟩𝑎₀𝑏₀=00,𝐻₁₁=𝑒₀𝑓₀−𝑞^∗,𝑄_𝑞𝑞𝑎₀𝑏₀−⟨𝑞^∗,𝑄_𝑞𝑞⟩𝑎₀𝑏₀=00, (26) which implies that 𝑊₂₀=𝑊₁₁=0. Thus, ⟨𝑞^∗,𝑄_𝑊11𝑞⟩=⟨𝑞^∗,𝑄_𝑊20𝑞⟩=0. Therefore, by (25) and 𝜔²₀=𝜆^𝐻₀𝛽𝑢^∗, we can obtain that Re𝐶₁𝜆^𝐻₀𝑖=Re2𝜔₀𝑔₂₀𝑔₁₁−2𝑔₁₁²−13𝑔₀₂²+𝑔₂₁2𝑖=Re2𝜔₀𝑞^∗,𝑄_𝑞𝑞𝑞^∗,𝑄_𝑞𝑞+12𝑞^∗,𝐶_𝑞𝑞𝑞=𝑢^∗2𝛽²𝜔²₀−12𝜆^𝐻₀𝛽³+𝜆^𝐻₀𝛽𝜔²₀−1𝛽²𝑢^∗−3𝛼4𝑢^∗𝜆^𝐻₀𝛽1+𝛼𝛽⁴=−3𝛼4𝑢^∗𝜆^𝐻₀𝛽1+𝛼𝛽⁴<0. (27) Since 𝛼^′(𝜆^𝐻₀)=−𝛽/2, then the bifurcating periodic solutions are backward and exist for 𝜏<𝜏^𝐻₀. And combining 𝑇_𝑛(𝜆^𝐻₀)=−((1+𝑑)/𝑙²)𝑛²<0,𝐷_𝑛(𝜆^𝐻₀)>0, for all 𝑛=1,2,…,𝑁₀, we can obtain that the bifurcating periodic solutions are asymptotically stable since Re(𝐶₁(𝜆^𝐻₀))<0.

Next, we give an example to illustrate the theoretical results above.

Example 3. Taking 𝑑=0.2,𝛼=0.4,𝛽=1.2,𝑙=2, System (5) has a unique positive constant steady state 𝐸(0.1892,0.227). Obviously, 𝛼𝛽+1>𝛽, 𝛼𝛽²+2𝛽>(1+𝛼𝛽)². By the formula above, we obtain 𝜆^𝐻₀=0.2989. And 𝐷₀(𝜆^𝐻₀)<0,𝐷₁(𝜆₁)>0. Thus, 0<𝑑<𝑑₀=1. By Theorems 1 and 2, 𝐸(0.1892,0.227) is asymptotically stable for 𝜆∈(0.2989,+∞) as illustrated in Figures 1(a) and 1(b). When 𝜆 crosses through the critical value, 𝐸(0.1892,0.227) losses its stability and a family of periodic solutions appear and are asymptotically stable since Re(𝐶₁(𝜆^𝐻₀))=−0.9215<0 as illustrated in Figures 2(a) and 2(b).

(a)

(b)

Figure 1 

𝐸(0.1892,0.227) is asymptotically stable when 𝜆=0.32>𝜆𝐻0 and the initial function 𝑢0(𝑥)=0.5+0.2cos𝑥, 𝑣0(𝑥)=0.5+0.2cos𝑥.

(a)

(b)

Figure 2 

Spatially homogenous periodic solutions are asymptotically stable when 𝜆=0.24<𝜆𝐻0 and the initial function 𝑢0(𝑥)=0.5+0.2cos𝑥, 𝑣0(𝑥)=0.5+0.2cos𝑥.

We can compute the direction and stability of bifurcating periodic solutions by the same way as Theorem 2 and omit it here. Combining the stability theory, we have the following results.

Theorem 4. For System (5), the bifurcating periodic solutions from 𝜆=𝜆_𝑛, 𝑛=1,2,…,𝑁₀ (defined by (15)) are backward (forward) if Re(𝐶₁(𝜆_𝑛))<(>)0 and the spatially inhomogeneous Hopf bifurcations are unstable.

2.2. Turing Instability Induced by Diffusion

In this subsection, we consider Turing instability with diffusion effect. As mentioned earlier, when 𝛼𝛽²+2𝛽≤(𝛼𝛽+1)², 𝑇_𝑛(𝜆)<0,𝐷_𝑛(𝜆)>0, for all 𝑛∈{0,1,2,…}. Hence, no Turing instability occurs. Next we always confine 𝛼𝛽²+2𝛽>(𝛼𝛽+1)².

When System (5) has no diffusion, the characteristic equation (10) at 𝐸(𝑢^∗,𝑣^∗) can be transformed into 𝜇²−𝜆^𝐻₀−𝜆𝛽𝜇+𝜆𝛽𝑢^∗=0. (28) Then all the roots of (28) are negative real part if 𝜆>𝜆^𝐻₀. Combining the stability theory [22], we easily know that the unique positive equilibrium 𝐸(𝑢^∗,𝑣^∗) of System (5) without diffusion is asymptotically stable for 𝜆∈(𝜆^𝐻₀,+∞) and unstable for 𝜆∈[0,𝜆^𝐻₀). In the following, we mainly consider the occurrence of Turing instability induced by diffusion. Hence we confine 𝜆∈(𝜆^𝐻₀,+∞). That is, 𝐸(𝑢^∗,𝑣^∗) is locally asymptotically stable without diffusion effect.

It is well-known that 𝐸(𝑢^∗,𝑣^∗) is unstable if the characteristic equation (10) has at least a positive real-part root. Notice when 𝜆>𝜆^𝐻₀, 𝑇_𝑛(𝜆)<0, for all 𝑛=0,1,2,…. The necessary condition of the instability of the equlibirium 𝐸(𝑢^∗,𝑣^∗) of System (5) is 𝐷_𝑛(𝜆)<0 for some 𝑛∈𝑁⁺={1,2,…}.

Recall 𝐷_𝑛(𝜆)=𝑑𝑛⁴𝑙⁴+𝑛²𝑙²𝜆−𝑑𝜆^𝐻₀𝛽+𝜆𝛽𝑢^∗𝑛≜𝑔²𝑙², (29) where 𝑔(𝑥)=𝑑𝑥²−(𝑑𝜆^𝐻₀−𝜆)𝛽𝑥+𝜆𝛽𝑢^∗.

Hence we need 𝜆∈(𝜆^𝐻₀,𝑑𝜆^𝐻₀),𝑑>1 and Δ≜(𝑑𝜆^𝐻₀−𝜆)²𝛽²−4𝑑𝜆𝛽𝑢^∗>0. That is, (𝑑𝜆^𝐻₀−𝜆)𝛽>2𝑑𝜆𝛽𝑢^∗. Otherwise, 𝐷_𝑖(𝜆)≥0 for all 𝑖∈𝑁.

Obviously, 𝑔(𝑥)=0 has two positive roots 𝑥₁,𝑥₂ since 𝑥₁+𝑥₂=𝑑𝜆^𝐻₀−𝜆𝛽>0,𝑥₁𝑥₂>0, (30) where 𝑥₁=12𝑑𝑑𝜆^𝐻₀−𝜆𝛽−𝑑𝜆^𝐻₀−𝜆²𝛽²−4𝑑𝜆𝛽𝑢^∗,𝑥₂=12𝑑𝑑𝜆^𝐻₀−𝜆𝛽+𝑑𝜆^𝐻₀−𝜆²𝛽²−4𝑑𝜆𝛽𝑢^∗. (31) If we can find some 𝑗∈𝑁⁺ such that 𝑥₁<𝑗²/𝑙²<𝑥₂. Then 𝐷_𝑗(𝜆)<0. That is, √𝑥₁𝑙<𝑗<√𝑥₂𝑙. The constant positive coexistence 𝐸(𝑢^∗,𝑣^∗) is unstable. Hence, we draw the following conclusions.

Theorem 5. Assume that 𝛼, 𝛽>0, 𝑑>1, and 𝛼𝛽+1>𝛽, 𝛼𝛽²+2𝛽>(𝛼𝛽+1)² hold. Then 𝐸(𝑢^∗,𝑣^∗) is unstable for 𝜆∈(𝜆^𝐻₀,𝑑𝜆^𝐻₀). That is, Turing instability occurs if 𝑑𝜆^𝐻₀𝛽−𝜆𝛽>2𝑑𝜆𝛽𝑢^∗,√𝑥₁𝑙<𝑗<√𝑥₂𝑙𝑓𝑜𝑟𝑠𝑜𝑚𝑒𝑗∈𝑁⁺. (32)

3. Steady State Bifurcations

In this section, we still choose 𝜆 as the main bifurcation parameter with 𝛼,𝛽,𝑑 positive constants and study the existence of the steady state bifurcation points. The steady state solutions of System (5) satisfy the following elliptic system: 𝑢_𝑥𝑥+𝑢(1−𝑢)−𝑢𝑣[],𝑢+𝛼𝑣=0,𝑥∈0,𝑙𝜋𝑑𝑣_𝑥𝑥𝑣+𝜆𝑣𝛽−𝑢[],𝑢=0,𝑥∈0,𝑙𝜋_𝑥(0,𝑡)=𝑢_𝑥(𝑙𝜋,𝑡)=𝑣_𝑥(0,𝑡)=𝑣_𝑥()𝑢()𝑙𝜋,𝑡=0,𝑡≥0,𝑥,0=𝑢₀(𝑥)𝑣()≥0,𝑥,0=𝑣₀(𝑥)≥0. (33) System (33) has a unique positive constant coexistence 𝐸(𝑢^∗,𝑣^∗) if and only if 𝛼𝛽+1>𝛽. By the same way as Section 2.1, the characteristic equation of the linearization of (33) at 𝐸(𝑢^∗,𝑣^∗) is 𝜇²−𝑇_𝑛(𝜆)𝜇+𝐷_𝑛(𝜆)=0,𝑛=0,1,2,…, (34) where 𝑇_𝑛(𝜆),𝐷_𝑛(𝜆) are defined by (11).

Next we identify all the possible steady state bifurcation points by [9] which satisfy the following conditions: 𝐷_𝑛𝜆₀=0,𝑇_𝑛𝜆₀𝐷≠0,_𝑚𝜆₀≠0,𝑇_𝑚𝜆₀≠0,∀𝑚≠𝑛,𝑑𝐷_𝑛𝜆₀𝑑𝜆≠0. (35) For convenience, we rewrite 𝐷_𝑛(𝜆)=𝑑𝑝²+𝑝𝜆−𝑑𝜆^𝐻₀𝛽+𝜆𝛽𝑢^∗≜𝑔𝑝, (36) where 𝑝=𝑛²/𝑙².

Obviously, 𝐷_𝑛(𝜆)>0, for all 𝑛=0,1,2,… for 𝜆∈[𝑑𝜆^𝐻₀,+∞). Hence the potential steady state bifurcation points set is as follows: ∧₁=𝜆∈0,𝑑𝜆^𝐻₀∣(35)aresatisfied (37) with 𝛼, 𝛽, 𝑙, 𝑑 fixed.

Thus 𝐷_𝑛(𝜆)=0 has two roots as 𝑝_±(𝜆)=12𝑑𝑑𝜆^𝐻₀−𝜆𝛽±𝑑𝜆^𝐻₀−𝜆²𝛽²−4𝑑𝜆𝛽𝑢^∗. (38) Let ℎ(𝜆)=𝑑𝜆^𝐻₀−𝜆²𝛽²−4𝑑𝜆𝛽𝑢^∗=𝜆²𝛽²−2𝑑𝜆𝛽𝜆^𝐻₀𝛽+2𝑢^∗+𝑑²𝛽²𝜆^𝐻₀². (39) Since Δ≜4𝑑²𝛽²(𝜆^𝐻₀𝛽+2𝑢^∗)²−4𝛽⁴𝑑²(𝜆^𝐻₀)²=16𝑑²𝛽²𝑢^∗(𝜆^𝐻₀𝛽+𝑢^∗)>0, ℎ(𝜆)=0 has two positive roots as 𝜆=𝑑𝛽𝜆^𝐻₀𝛽+2𝑢^∗−2𝑢^∗𝜆^𝐻₀𝛽+𝑢^∗,𝑑𝜆=𝛽𝜆^𝐻₀𝛽+2𝑢^∗+2𝑢^∗𝜆^𝐻₀𝛽+𝑢^∗. (40) By ℎ(𝑑𝜆^𝐻₀)=−4𝑑𝜆𝛽𝑢^∗<0, hence 𝜆<𝑑𝜆^𝐻₀<𝜆. That is, when 𝜆∈(𝜆,𝑑𝜆^𝐻₀), 𝐷_𝑛(𝜆)>0, for all 𝑛∈𝑁. No steady state bifurcation occurs. Then the potential steady state bifurcation points reduce to the following set: ∧₂=𝜆∈0,𝜆∣(35)aresatisfied, (41) with 𝛼, 𝛽, 𝑙, 𝑑 fixed.

Next, the properties of 𝑃_±(𝜆),𝜆∈(0,𝜆] are discussed and direct computation can obtain the following results.

Lemma 6. Suppose that 𝛼𝛽>1+𝛽, 𝛼𝛽²+2𝛽>(𝛼𝛽+1)² hold. When 𝜆∈(0,𝜆], 𝑃₊(𝜆) is decreasing on 𝜆 and 𝑃₋(𝜆) is increasing on 𝜆. Furthermore, lim_𝜆→0+𝑃₊(𝜆)=𝜆^𝐻₀𝑃𝛽,₊𝜆⁻=𝑃₋𝜆⁻=𝑑𝜆^𝐻₀−𝜆𝛽,2𝑑lim_𝜆→0+𝑃₋(𝜆)=0,lim_{𝜆→𝜆−}𝑃^′₊(𝜆)=−∞,lim_{𝜆→𝜆−}𝑃^′₋(𝜆)=+∞. (42)

From the properties of 𝑃_±(𝜆), if 0<𝑝=𝑛²/𝑙²<𝜆^𝐻₀𝛽, then there exists a unique 𝜆^𝑆_𝑛=𝑝𝛽𝑑𝜆^𝐻₀−𝑑𝑝²𝑝𝛽+𝛽𝑢^∗∈0,𝜆, (43) such that 𝑃₊𝜆^𝑆_𝑛=𝑛²𝑙²or𝑃₋𝜆^𝑆_𝑛=𝑛²𝑙². (44) Thus, 𝐷_𝑛(𝜆^𝑆_𝑛)=0.

Define 𝑙_𝑛̃=𝑛/𝜆^𝐻₀𝛽. Then for any 𝑙>𝑙_𝑛̃, there exists a unique 𝜆^𝑆_𝑖∈(0,𝜆] such that 𝑖²/𝑙²=𝑃₊(𝜆^𝑆_𝑖) or 𝑖²/𝑙²=𝑃₋(𝜆^𝑆_𝑖).

𝜆^𝑆_𝑛∈(0,𝜆] are potential steady state bifurcation values. But it is possible that for some 𝑖,𝑗∈𝑁 and 𝑖≠𝑗 such that 𝑃₊(𝜆^𝑆_𝑖)=𝑃₋(𝜆^𝑆_𝑗). In this case, 0 is a double eigenvalue of 𝐿(𝜆) for 𝜆=𝜆^𝑆_𝑖=𝜆^𝑆_𝑗. In addition, it is also possible that 𝜆^𝑆_𝑖=𝜆^𝐻_𝑗. Hence, the dimension of center manifold at 𝐸(𝑢^∗,𝑣^∗) can be as high as 3.

Next we will prove that there are only countable many 𝑙>0, in fact only finitely many 𝑙∈(0,𝑀) for any given 𝑀>0, such that 𝜆^𝑆_𝑖=𝜆^𝑆_𝑗, 𝜆^𝑆_𝑖=𝜆^𝐻_𝑗 for some 𝑖,𝑗∈𝑁. Let 𝐸_𝑛(𝜆,𝑙)=𝑙²𝑇_𝑛(𝜆), 𝐹_𝑛(𝜆,𝑙)=𝑙⁴𝐷_𝑛(𝜆) be polynomials of 𝜆 and 𝑙. Hence, on (𝜆,𝑙)-plane, 𝑞_𝑛={(𝜆,𝑙)∶𝐸_𝑛(𝜆,𝑙)=0}, 𝑝_𝑛={(𝜆,𝑙)∶𝐹_𝑛(𝜆,𝑙)=0} are finitely many analytic curves. We fix 𝜆∈(0,𝜆], so for any 𝑀>0, there are only finitely many 𝑖,𝑗∈𝑁 such that 𝑞_𝑖⋂((0,𝜆]×[0,𝑀])≠∅,𝑝_𝑗⋂((0,𝜆]×[0,𝑀])≠∅. So 𝑝_𝑗,𝑞_𝑖 have at most only finitely many intersection points. Thus the intersection points of different 𝑞_𝑖,𝑝_𝑗 in (0,𝜆]×[0,+∞] are at most countable. Define 𝐿^𝐸=𝑙>0∣𝐹_𝑖(𝜆,𝑙)=𝐹_𝑗(𝜆,𝑙)or𝐸_𝑖(𝜆,𝑙)=𝐹_𝑗(𝜆,𝑙)forsome𝜆∈0,𝜆,and𝑖,𝑗∈𝑁. (45)

We assume that 𝑙∈𝑅/𝐿^𝐸, and thus 0 is a simple eigenvalue. And 𝑑𝐷𝜆^𝑆_𝑛𝑑𝜆=𝑝𝛽+𝛽𝑢^∗>0. (46) Obviously, 𝑇_𝑚(𝜆^𝑆_𝑛)≠0,𝐷_𝑚(𝜆^𝑆_𝑛)≠0, for all 𝑚≠𝑛.

Summarizing the above, we can obtain the main results in this section.

Theorem 7. Suppose that 𝛼,𝛽,𝑑 are positive constants and 𝛼𝛽>1+𝛽, 𝛼𝛽²+2𝛽>(𝛼𝛽+1)² hold. 𝑃_±(𝜆) are defined as (38) and 𝑙_𝑛̃=𝑛/𝜆^𝐻₀𝛽.
If for some 𝑛∈𝑁, 𝑙∈(𝑙_𝑛̃,+∞)⧵𝐿^𝐸, there exactly exists one point 𝜆^𝑆_𝑛=(𝛽𝑑𝜆^𝐻₀𝑝−𝑑𝑝²)/(𝛽𝑝+𝛽𝑢^∗)∈(0,𝜆), such that 𝑝₊(𝜆^𝑆_𝑛)=𝑛²/𝑙² or 𝑃₋(𝜆^𝑆_𝑛)=𝑛²/𝑙², where 𝑝=𝑛²/𝑙². Then there is a smooth curve Γ_𝑛 of positive solutions of (33) bifurcating from (𝜆^𝑆_𝑛,𝑢^∗,𝑣^∗). Moreover, near (𝜆^𝑆_𝑛,𝑢^∗,𝑣^∗), Γ_𝑛={𝜆(𝑠),𝑢(𝑠),𝑣(𝑠)∶𝑠∈(−𝜖,+𝜖)}, where 𝑢(𝑠)=𝑢^∗+𝑠𝑎_𝑛cos𝑛𝑥𝑙+𝑠𝜓₁(𝑠),𝑣(𝑠)=𝑣^∗+𝑠𝑏_𝑛cos𝑛𝑥𝑙+𝑠𝜓₂(𝑠), (47) for 𝑠∈(−𝜖,+𝜖) for some 𝐶^∞ smooth function 𝜆,𝜓₁,𝜓₂ such that 𝜆(0)=𝜆^𝑆_𝑛 and 𝜓₁(0)=𝜓₂(0)=0. Here 𝑎_𝑛,𝑏_𝑛 satisfy 𝐿_𝑛(𝜆^𝑆_𝑛)(𝑎_𝑛,𝑏_𝑛)^𝑇=(0,0)^𝑇.

In the following, we will discuss the interaction between the Hopf and steady state bifurcation points. All possible Hopf bifurcation points 𝜆^𝐻_𝑖 are identified in Section 2.1 and all possible steady state bifurcation points 𝜆^𝑆_𝑗 can be identified in Section 3. And it has previously shown that at most countable 𝑙∈𝐿^𝐸 such that 𝜆^𝐻_𝑗=𝜆^𝑆_𝑖 or 𝜆^𝑆_𝑖=𝜆^𝑆_𝑗 for some 𝑖,𝑗∈𝑁. Hence high-dimensional center manifold from (𝜆,𝑢^∗,𝑣^∗) is still possible. We do not consider it here. We assume 𝑙∉𝐿^𝐸, Hopf bifurcations, and steady state bifurcation points may occur at this points in the following four cases.

Case 1. Neither 𝜆^𝐻_𝑖 nor 𝜆^𝑆_𝑗 exists, and then there is no bifurcation points;

Case 2. 𝜆^𝑆_𝑗 exists but not 𝜆^𝐻_𝑖, and then there is one steady state bifurcation point and no Hopf bifurcation point.

Case 3. Both 𝜆^𝑆_𝑗 and 𝜆^𝐻_𝑖 exist, and then(a)𝜆^𝐻_𝑖<𝜆^𝑆_𝑗; then there is one steady state bifurcation point and no Hopf bifurcation since 𝐷_𝑖(𝜆^𝐻_𝑖)<0;(b)𝜆^𝑆_𝑗<𝜆^𝐻_𝑖; then there is one steady state bifurcation point and one Hopf bifurcation.

Case 4. 𝜆^𝐻_𝑖 exists but no 𝜆^𝑆_𝑗, then there is no steady state bifurcation point and one Hopf bifurcation.

Example 8. In Figure 3, for 𝑛≥18 and 𝑛∈𝑁, Case 1 occurs, and there are no bifurcation points. For 15≤𝑛≤17, Case 2 occurs, and there exists 3 steady state bifurcation points and no Hopf bifurcation points. For 𝑛=14, Case 3 (a) occurs, and there is one steady state bifurcation point 𝜆^𝑆₁₄=0.0314 and no Hopf bifurcation points since 𝐷₁₄(𝜆^𝐻₁₄)<0. For 1≤𝑛≤13, Case 3 (b) occurs, and there exists 13 Hopf bifurcation points and 13 steady state bifurcation points. For 𝑛=0, Case 4 occurs, and there exists one Hopf bifurcation point and no steady state bifurcation points.

Figure 3 

Graph of 𝐸𝑛(𝜆,𝑝)𝐹𝑛(𝜆,𝑝)=0. Here 𝑑=0.5, 𝛼=0.4, 𝛽=1.2, 𝑙=30, 𝑢∗=0.1892, 𝑣∗=0.227, and 𝜆𝐻0=0.2989. The horizontal lines are 𝑝=𝑛2/𝑙2, where 𝑛=0,1,2,….

4. Global Stability of the Positive Coexistence

In this section, we will prove the global stability of the positive constant coexistence by the upper and lower solutions method in Pao [23, 24] and Chen and Shi [13].

Theorem 9. Assume that 𝑑,𝛽,𝜆 are positive constants and 𝛼>1. Then for system (5), the positive constant equilibrium 𝐸(𝑢^∗,𝑣^∗) is globally asymptotically stable if 𝛼𝛽<1 or 𝛼𝛽>1+𝛽. That is, for any initial values 𝑢₀(𝑥)>0,𝑣₀(𝑥)≥0, lim_𝑡→∞𝑢(𝑥,𝑡)=𝑢^∗,lim_𝑡→∞𝑣(𝑥,𝑡)=𝑣^∗,𝑢𝑛𝑖𝑓𝑜𝑟𝑚𝑙𝑦𝑖𝑛𝑥∈[0,𝑙𝜋] (48) if 𝛼𝛽<1or𝛼𝛽>1+𝛽. (49)

Proof. It is well known that if 𝑐>0 and 𝑤(𝑥,𝑡)>0 satisfies 𝜕𝑤()𝜕𝑡=𝐷Δ𝑤+𝑤𝑐−𝑤,𝑥∈Ω,𝑡>0,𝜕𝑤𝑤()≥(≢)𝜕𝜈=0,𝑥∈𝜕Ω,𝑥,00,𝑥∈Ω, (50) then lim_𝑡→∞𝑤(𝑥,𝑡)=𝑐 uniformly in 𝑥∈Ω.
We can choose 𝜖₀ satisfying 0<𝜖₀<𝛽(1−1/𝛼)1+𝛽,since𝛼>1. (51) By the first equation of (5), we have 𝜕𝑢𝜕𝑡≤𝑢_𝑥𝑥+𝑢(𝑥,𝑡)(1−𝑢(𝑥,𝑡)). (52) By the comparison principle of the parabolic equation, we obtain limsup_𝑡→+∞max_{𝑥∈[0,𝑙𝜋]}𝑢(𝑥,𝑡)≤1. (53) Then there exists 𝑡₀>0 such that 𝑢(𝑥,𝑡)<1+𝜖₀≜𝑢, for all 𝑡>𝑡₀, 𝑥∈[0,𝑙𝜋]. Thus, by the second equation of (5), 𝜕𝑣𝜕𝑡≤𝑑𝑣_𝑥𝑥𝑣+𝜆𝑣𝛽−1+𝜖₀,∀𝑡>𝑡₀. (54) By the comparison principle again, limsup_𝑡→+∞max_{𝑥∈[0,𝑙𝜋]}𝑣(𝑥,𝑡)≤𝛽(1+𝜖₀). There exists 𝑡₁>𝑡₀ such that 𝑣(𝑥,𝑡)<𝛽(1+𝜖₀)+𝜖₀≜𝑣, for all 𝑡>𝑡₁,𝑥∈[0,𝑙𝜋].
On the other hand, 𝜕𝑢𝜕𝑡=𝑢_𝑥𝑥𝑣+𝑢1−𝑢−𝑢+𝛼𝑣≥𝑢_𝑥𝑥1+𝑢1−𝛼−𝑢. (55) By the comparison principle, liminf_𝑡→+∞min_{𝑥∈[0,𝑙𝜋]}𝑢(𝑥,𝑡)≥1−1/𝛼>0, since 𝛼>1. Thus, there exists 𝑡₂>𝑡₁ such that 𝑢(𝑥,𝑡)>1−1/𝛼−𝜖₀≜𝑢>0, for all 𝑡>𝑡₂. Again in turn this implies 𝜕𝑣𝜕𝑡≥𝑑𝑣_𝑥𝑥𝑣+𝜆𝑣𝛽−𝑢,𝑡>𝑡₂. (56) Hence, liminf_𝑡→+∞min_{𝑥∈[0,𝑙𝜋]}𝑣(𝑥,𝑡)≥𝛽(1−1/𝛼−𝜖₀). There exists 𝑡₃>𝑡₂ such that 𝑣()1𝑥,𝑡>𝛽1−𝛼−𝜖₀−𝜖₀≜𝑣>0,∀𝑡>𝑡₃, (57) since (51) holds. Therefore, for 𝑡>𝑡₃,𝑢≤𝑢(𝑥,𝑡)≤𝑢,  𝑣≤𝑣(𝑥,𝑡)≤𝑣, for all 𝑥∈[0,𝑙𝜋] and 𝑢,𝑢,𝑣,𝑣 satisfy 1−𝑣𝑢−𝑢+𝛼𝑣<0,𝛽−𝑣𝑢<0,1−𝑢−𝑣𝑢+𝛼𝑣𝑣>0,𝛽−𝑢>0. (58) Equation (58) implies that (𝑢,𝑣) and (𝑢,𝑣) are a pair of upper and lower solutions of system (5), respectively (see [23, 24]). It is obvious that there exists 𝑀>0 such that for any (𝑢,𝑣)≤(𝑢₁,𝑣₁), (𝑢₂,𝑣₂)≤(𝑢,𝑣), 𝑢₁1−𝑢₁−𝑢₁𝑣₁𝑢₁+𝛼𝑣₁−𝑢₂1−𝑢₂+𝑢₂𝑣₂𝑢₂+𝛼𝑣₂≤𝑀𝑢₁−𝑢₂+𝑣₁−𝑣₂,𝜆𝑣₁𝑣𝛽−¹𝑢₁−𝜆𝑣₂𝑣𝛽−²𝑢₂≤𝑀𝑢₁−𝑢₂+𝑣₁−𝑣₂. (59) We construct the iteration sequences (𝑢^(𝑚),𝑣^(𝑚)) and (𝑢^(𝑚),𝑣^(𝑚)), 𝑚≥1 satisfy 𝑢^(𝑚)=𝑢^(𝑚−1)+1𝑀𝑢^(𝑚−1)×1−𝑢^(𝑚−1)−𝑣^(𝑚−1)𝑢^(𝑚−1)+𝛼𝑣^(𝑚−1),𝑣^(𝑚)=𝑣^(𝑚−1)+1𝑀𝜆𝑣^(𝑚−1)𝛽−𝑣^(𝑚−1)𝑢^(𝑚−1),𝑢^(𝑚)=𝑢^(𝑚−1)+1𝑀𝑢^(𝑚−1)×1−𝑢^(𝑚−1)−𝑣^(𝑚−1)𝑢^(𝑚−1)+𝛼𝑣^(𝑚−1),𝑣^(𝑚)=𝑣^(𝑚−1)+1𝑀𝜆𝑣^(𝑚−1)𝑣𝛽−^(𝑚−1)𝑢^(𝑚−1), (60) where (𝑢⁽⁰⁾,𝑣⁽⁰⁾)=(𝑢,𝑣),  (𝑢⁽⁰⁾,𝑣⁽⁰⁾)=(𝑢,𝑣). Then by (58) we can obtain 𝑢,𝑣≤𝑢^(𝑚),𝑣^(𝑚)≤𝑢^(𝑚+1),𝑣^(𝑚+1)≤⋯≤𝑢^(𝑚+1),𝑣^(𝑚+1)≤𝑢^(𝑚),𝑣^(𝑚)≤(𝑢,𝑣). (61) Then there exist (𝑢,𝑣)̃ and ̃(𝑢,𝑣) such that lim_𝑚→∞𝑢^(𝑚)=𝑢,lim_𝑚→∞𝑣^(𝑚)=𝑣,̃lim_𝑚→∞𝑢^(𝑚)=𝑢,lim_𝑚→∞𝑣̃^(𝑚)̃=𝑣, (62) and (𝑢,𝑣̃)≤(𝑢,𝑣)≤(𝑢,𝑣)̃≤(𝑢,𝑣). And 𝑣1−𝑢−̃̃𝑣̃𝑢+𝛼𝑣=0,𝛽−𝑢−𝑣̃=0,1−𝑢𝑢𝑣̃𝑣̃+𝛼=0,𝛽−𝑢=0. (63) According to (63), we have (𝑢),𝑢𝑢1−𝑢+𝛼𝛽𝑢=𝛽𝑢1−𝑢+𝛼𝛽=𝛽𝑢. (64) By (64), we can obtain 𝑢−𝑢1−𝛼𝛽+𝛽−𝑢−𝑢=0. (65) If we assume that 𝑢≠𝑢, then 1−𝛼𝛽+𝛽−𝑢−𝑢=0. (66) Substituting (66) into (64) and simplifying, we can obtain 𝛼𝛽−1𝑢²+𝑢1−2𝛼𝛽+𝛼²𝛽²−𝛼𝛽²()𝑢+𝛽+𝛽𝛼−11−𝛼𝛽+𝛽=0,𝛼𝛽−1²+𝑢1−2𝛼𝛽+𝛼²𝛽²−𝛼𝛽²()+𝛽+𝛽𝛼−11−𝛼𝛽+𝛽=0. (67) Thus the quadratic equation 𝛼𝛽−1𝑥²+𝑥1−2𝛼𝛽+𝛼²𝛽²−𝛼𝛽²()+𝛽+𝛽𝛼−11−𝛼𝛽+𝛽=0 (68) has two different positive roots 𝑢 and 𝑢.
On the other hand, when 𝛼𝛽<1 or 𝛼𝛽>1+𝛽, 𝑥₁𝑥₂=𝛽(𝛼−1)1−𝛼𝛽+𝛽𝛼𝛽−1<0, (69) where 𝑥₁,𝑥₂ are two roots of (68). Thus (68) cannot have two positive roots.
When 𝛼𝛽=1, (68) becomes a linear equation and has at most one real root. This contradicts the above. Hence, 𝑢=𝑢. Similarly, ̃̃𝑣=𝑣. Then from the results in [23, 24], the solution (𝑢(𝑥,𝑡),𝑣(𝑥,𝑡)) of System (5) satisfies lim_𝑡→+∞𝑢(𝑥,𝑡)=𝑢^∗,lim_𝑡→+∞𝑣(𝑥,𝑡)=𝑣^∗,uniformlyin𝑥∈[0,𝑙𝜋]. (70)

Acknowledgments

The author wish to express his special gratitude to the reviewers and the editor for their valuable comments and suggestions that led to a truly significant improvement of the paper. This research is partially supported by Shandong Provincial Natural Science Foundation, China (no. ZR2011AQ017), and by the Fundamental Research Funds for the Central Universities (nos. 13CX02011A and 12CX04081A).