Global Existence of Classical Solutions to a Three-Species Predator-Prey Model with Two Prey-Taxes

Abstract

We are concerned with three-species predator-prey model including two prey-taxes and Holling type II functional response under no flux boundary condition. By applying the contraction mapping principle, the parabolic Schauder estimates, and parabolic 𝐿^𝑝 estimates, we prove that there exists a unique global classical solution of this system.

1. Introduction

In addition to random diffusion of the predator and the prey, the spatial-temporal variations of the predators’ velocity are directed by prey gradient. Several field studies measuring characteristics of individual movement confirm the basis of the hypothesis about the dependence of acceleration on a stimulus [1]. Understanding spatial and temporal behaviors of interacting species in ecological system is a central problem in population ecology. Various types of mathematical models have been proposed to study problem of predator-prey. Recently, the appearance of prey-taxis in relation to ecological interactions of species was studied by many scholars, ecologists, and mathematicians [2–5].

In [2] the authors proved the existence and uniqueness of weak solutions to the two-species predator-prey model with one prey-taxis. In [3], the author extended the results of [2] to an 𝑛×𝑚 reaction-diffusion-taxis system. In [4], the author proved the existence and uniqueness of classical solutions to this model. In this paper, we deal with three-species predator-prey model with two prey-taxes including Holling type II functional response as follows: 𝜕𝑢₁𝜕𝑡−𝑑¹Δ𝑢₁+∇⋅𝛽₁𝑢₁∇𝑢₂+∇⋅𝛽₂𝑢₁∇𝑢₃=−𝑎𝑢₁+𝑒²𝑐₂𝑢₁𝑢₂𝑚₂+𝑏₂𝑢₂+𝑒³𝑐₃𝑢₁𝑢₃𝑚₃+𝑏₃𝑢₃𝜕𝑢₂in(0,𝑇)×Ω,𝜕𝑡−𝑑²Δ𝑢₂=𝑟₂1−𝐾𝑢²₂𝑢²−𝑐²𝑢₁𝑢₂𝑚₂+𝑏₂𝑢₂𝜕𝑢₃in(0,𝑇)×Ω,𝜕𝑡−𝑑³Δ𝑢₃=𝑟₃1−𝐾𝑢³₃𝑢²−𝑐³𝑢₁𝑢₃𝑚₃+𝑏₃𝑢₃𝜕𝑢₁in(0,𝑇)×Ω,𝑢𝜕𝜈=𝜕𝜕𝑢𝜈²=𝜕𝜕𝑢𝜈³=0on(0,𝑇)×𝜕Ω,₁(0,𝑥),𝑢₂(0,𝑥),𝑢₃(0,𝑥)=𝑢₁₀(𝑥),𝑢₂₀(𝑥),𝑢₃₀(𝑥)≥(0,0)inΩ,(1.1) where Ω is a bounded domain in 𝑅^𝑁(𝑁≥1 is an integer) with a smooth boundary 𝜕Ω; 𝑢₁ and 𝑢_𝑖(𝑖=2,3) represent the densities of the predator and prey, respectively; the positive constants 𝑑₁, 𝑑₂, and 𝑑₃ are the diffusion coefficient of the corresponding species; the positive constants 𝑎,𝐾_𝑖,𝑟_𝑖,𝑚_𝑖,𝑒_𝑖,𝑚_𝑖/𝑐_𝑖,𝑏_𝑖/𝑐_𝑖,𝑚_𝑖/𝑏_𝑖(𝑖=2,3) represent the death rate of the predator, the carrying capacity of prey, the prey intrinsic growth rate, the half-saturation constant, the conversion rate, the time spent by a predator to catch a prey, the manipulation time which is a saturation effect for large densities of prey, the density of prey necessary to achieve one-half the rate, respectively; the predators are attracted by the preys, and the positive constant 𝛽_𝑖(𝑖=1,2) denotes their prey-tactic sensitivity. The parts 𝛽₁𝑢₁∇𝑢₂ and 𝛽₂𝑢₁∇𝑢₃ of the flux are directed toward the increasing population density of 𝑢₂ and 𝑢₃, respectively. In this way, the predators move in the direction of higher concentration of the prey species.

The aim of this paper is to prove that there is a unique classical solution to the model (1.1). It is difficult to deal with the two prey-taxes terms. To get our goal we employ the techniques developed by [6, 7] to investigate.

Throughout this paper we assume that𝛽₁=0,𝛽₂=0,for𝑢₁≥𝑢_1𝑚.(1.2) The assumptions that 𝛽₁=0 for 𝑢₁≥𝑢_1𝑚 and 𝛽₂=0 for 𝑢₁≥𝑢_1𝑚 have a clear biological interpretation [2]: the predators stop to accumulate at given point of Ω after their density attains a certain threshold value 𝑢_1𝑚 and the prey-tactic sensitivity 𝛽₁ and 𝛽₂ vanishes identically when 𝑢₁≥𝑢_1𝑚.

Throughout this paper we also assume that𝑢₂₀,𝑢₃₀≤𝐾,𝜕Ω∈𝐶^2+𝛼𝜕,𝑢₁₀𝑢₁₀(𝑥),𝑢₂₀(𝑥),𝑢₃₀(𝑥)∈𝐶^2+𝛼Ω,where0<𝛼<1,𝜕𝜈=𝜕𝜕𝑢𝜈²⁰=𝜕𝜕𝑢𝜈³⁰=0,on𝜕Ω.(1.3) Denote by 𝐶^{𝑚+𝛼,𝛽𝑥,𝑡}(𝑄_𝑇) (𝑚≥0 is integer, 0<𝛼<1,0<𝛽<1) the space of function 𝑢(𝑥,𝑡) with finite norm [8]:‖𝑢‖_𝐶^{𝑚+𝛼,𝛽𝑥,𝑡}_(𝑄𝑇)=^𝑚_||_𝑙||=0sup_𝑄𝑇||𝐷^𝑙𝑥𝑢||+𝐷^𝑙𝑥𝑢^{(𝛼)𝑥,𝑄}_𝑇+𝐷^l_𝑥𝑢^{(𝛽)𝑡,𝑄}_𝑇,(1.4) where⟨𝑤⟩^{(𝛼)𝑥,𝑄}_𝑇=_{(𝑥,𝑡),(𝑦,𝑡)∈𝑄𝑇}||𝑤(𝑥,𝑡)−𝑤(𝑦,𝑡)||||𝑥−𝑦||^𝛼⟨𝑤⟩_{(𝛽)𝑡,𝑄𝑇},=_{(𝑥,𝑡),(𝑥,𝜏)∈𝑄𝑇}||𝑤(𝑥,𝑡)−𝑤(𝑥,𝜏)|||𝑡−𝜏|^𝛽.(1.5) We denote by 𝐶^{2+𝛼,1+𝛽𝑥,𝑡}(𝑄_𝑇) the space of functions 𝑢(𝑥,𝑡) with norm‖𝑢‖_𝐶^{2+𝛼,𝛽𝑥,𝑡}_(𝑄𝑇)+‖‖𝑢_𝑡‖‖_{𝐶𝛼,𝛽𝑥,𝑡(𝑄𝑇)}.(1.6)

The main result of this paper is as follows.

Theorem 1.1. Under assumptions (1.2) and (1.3), for any given 𝑇>0 there exists a unique solution 𝐔=(𝑢₁,𝑢₂,𝑢₃)∈𝐶^{2+𝛼,1+(𝛼/2)}(𝑄_𝑇) of the system (1.1), where 𝑄_𝑇=(0,𝑇)×Ω. Moreover, 𝑢₁(𝑥,𝑡)≥0,0≤𝑢₂(𝑥,𝑡)≤𝐾₂,0≤𝑢₃(𝑥,𝑡)≤𝐾₃,(1.7) for any 𝑥∈Ω and 𝑡>0.

This paper is organized as follows. In Section 2, we present some preliminary lemmas that will be used in proving later theorem. In Section 3, we prove local existence and uniqueness to system (1.1). In Section 4, we prove global existence to system (1.1).

2. Some Preliminaries

For the convenience of notations, in what follows we denote various constants which depend on 𝑇 by 𝑁, while we denote various constants which are independent of 𝑇 by 𝑁₀.

Lemma 2.1. Let (𝑢,𝑥)∈𝐶^{2+𝛼,1+(𝛼/2)}(𝑄_𝑇). Then ‖𝑢(𝑥,𝑡)−𝑢(𝑥,0)‖_𝐶^{1+𝛼,𝛼/2}_(𝑄𝑇)≤𝑁₀𝜂(𝑇)‖𝑢‖_𝐶^{2+𝛼,1+(𝛼/2)}_(𝑄𝑇),(2.1) where 𝜂(𝑇)=max{𝑇^𝛼/2,𝑇^(1−𝛼)/2}.

Proof. Using the definition of Hölder norm, we have ||𝑢(𝑥,𝑡)−𝑢(𝑥,0)|||𝑡|^(1+𝛼)/2[𝑢(𝑥,𝑡)−𝑢(𝑥,0)]_{𝐶1+𝛼,0(𝑄𝑇)}≤||𝐷_𝑡𝑢||⋅||𝑇||^(1−𝛼)/2,≤𝑁₀‖‖𝐷^2𝑥𝑢(𝑥,𝑡)−𝐷^2𝑥𝑢(𝑥,0)‖‖_{𝐿∞(𝑄𝑇)}≤𝑁₀𝐷^2𝑥𝑢_{𝐶0,𝛼/2(𝑄𝑇)}⋅||𝑇||^𝛼/2≤𝑁₀[𝑢]_𝐶^2,𝛼/2_(𝑄𝑇)⋅||𝑇||^𝛼/2,(2.2) which yields that ‖𝑢(𝑥,𝑡)−𝑢(𝑥,0)‖_𝐶^{1+𝛼,(1+𝛼)/2}_(𝑄𝑇)≤𝑁₀𝜂‖𝑢‖_𝐶^{2+𝛼,(1+𝛼)/2}_(𝑄𝑇).(2.3) Therefore, ‖𝑢(𝑥,𝑡)−𝑢(𝑥,0)‖_𝐶^{1+𝛼,𝛼/2}_(𝑄𝑇)≤𝑁₀𝜂‖𝑢‖_𝐶^{2+𝛼,1+(𝛼/2)}_(𝑄𝑇).(2.4)

We now consider the following nonlinear parabolic problem:𝜕𝑢₁𝜕𝑡−𝑑¹Δ𝑢₁+∇⋅𝛽₁𝜕𝑢𝑢₁₁∇𝑢₂+∇⋅𝛽₂𝑢₁∇𝑢₃=𝑢₁𝑓in(0,𝑇)×Ω,𝜕𝑢𝜈=0on(0,𝑇)×𝜕Ω,₁(0,𝑥)=𝑢₁₀(𝑥)inΩ.(2.5) By the parabolic maximum principle, we have 𝑢₁(𝑥,𝑡)≥0.

Lemma 2.2. Let 𝑢₂(𝑥,‖‖𝑡)𝑢,₂𝑢₃(𝑥,𝑡)∈𝐶^{2+𝛼,1+(𝛼/2)}𝑄_𝑇,𝑓(𝑥,𝑡)∈𝐶^𝛼,𝛼/2𝑄_𝑇,‖‖_{𝐶2+𝛼,1+(𝛼/2)(𝑄𝑇)},‖‖𝑢₃‖‖_{𝐶2+𝛼,1+(𝛼/2)(𝑄𝑇)},‖𝑓‖_𝐶^𝛼,𝛼/2_(𝑄𝑇)≤𝑁₀.(2.6) Then, under assumptions (1.2) and (1.3), there exists a unique nonnegative solution 𝑢₁(𝑥,𝑡)∈𝐶^{2+𝛼,1+(𝛼/2)}(𝑄_𝑇) of the nonlinear problem (2.5) for small 𝑇>0 which depends on ‖𝑢₁₀(𝑥)‖_𝐶^2+𝛼_(Ω).

Proof. This proof is similar to that of Lemma  2.1 in [4]. For reader’s convenience we include the proof here. We will prove by a fixed point argument. Let us introduce the Banach space 𝑋 of function 𝑢₁ with norm ‖𝑢₁‖_𝐶^{1+𝛼,𝛼/2}_(𝑄𝑇)(0<𝑇<1) and a subset 𝑋_𝐴={𝑢₁∈𝑋∶𝑢₁(𝑥,0)=𝑢₁₀(𝑥)and‖𝑢₁‖_𝐶^{1+𝛼,𝛼/2}_(𝑄𝑇)≤𝐴}, where 𝐴=‖𝑢₁₀(𝑥)‖_𝐶^2+𝛼_(Ω)+‖𝑢₂₀(𝑥)‖_𝐶^2+𝛼_(Ω)+‖𝑢₃₀(𝑥)‖_𝐶^2+𝛼_(Ω)+1. For any 𝑢₁∈𝑋_𝐴, we define a corresponding function 𝑢₁̃=𝐹𝑢₁, where 𝑢₁̃ satisfies the equations 𝜕𝑢₁̃𝜕𝑡−𝑑¹Δ𝑢₁̃−𝑢₁̃𝑓=−𝛽₁∇𝑢₁⋅𝜕∇𝑢𝑢₁̃₂−𝛽₂∇𝑢₁⋅∇𝑢₃−𝛽₁𝑢₁Δ𝑢₂−𝛽₂𝑢₁Δ𝑢₃in(0,𝑇)×Ω,𝜕𝑢𝜈₁̃=0on(0,𝑇)×𝜕Ω,(0,𝑥)=𝑢₁₀(𝑥)inΩ.(2.7) By 𝑢₁∈𝑋_𝐴, we have ℎ₁≜−𝛽₁∇𝑢₁⋅∇𝑢₂−𝛽₂∇𝑢₁⋅∇𝑢₃−𝛽₁𝑢₁Δ𝑢₂−𝛽₂𝑢₁Δ𝑢₃∈𝐶^𝛼,𝛼/2𝑄_𝑇.(2.8) By the parabolic Schauder theory, this yields that there exists a unique solution 𝑢̃ to (2.7) and ‖‖𝑢₁̃‖‖_{𝐶2+𝛼,1+(𝛼/2)(𝑄𝑇)}≤≤𝐴‖‖𝑢+₁̃|𝑀_𝑡=0‖‖_{𝐶2+𝛼(Ω)1}+𝑀₁(𝐴)(𝐴)≜𝑀₂(𝐴),(2.9) where 𝑀₂(A) is some constant which depends only on 𝐴. For any function 𝑢₁̃(𝑥,𝑡), by Lemma 2.1 and combining (2.9), if 𝑇 is sufficiently small (𝑇 depends only on 𝐴), then we have ‖‖𝑢₁̃(𝑥,𝑡)‖‖_{𝐶1+𝛼,𝛼/2(𝑄𝑇)}≤‖‖𝑢₁̃(𝑥,0)‖‖_{𝐶1+𝛼,𝛼/2(𝑄T)}+𝑁₀𝜂(𝑇)‖‖𝑢₁̃‖‖_{𝐶2+𝛼,1+(𝛼/2)(𝑄𝑇)}≤‖‖𝑢̃₁₀(𝑥)‖‖_𝐶2+𝛼+1≤𝐴.(2.10) Therefore, 𝑢₁̃(𝑥,𝑡)∈𝑋_𝐴 and 𝐹 maps 𝑋_𝐴 into itself. We now prove that 𝐹 is contractive. Take 𝑢₁₁,𝑢₁₂ in 𝑋_𝐴, and set 𝑢̃₁₁=𝐹𝑢₁₁,𝑢̃₁₂=𝐹𝑢₁₂,𝑣̃=𝑢̃₁₁−𝑢̃₁₂. Then, it follows from (2.7) that 𝑣̃ solves the following systems: 𝜕𝑣̃𝜕𝑡−𝑑¹𝜕𝑣Δ̃𝑣̃−𝑣𝑓̃=ℎ₂in(0,𝑇)×Ω,𝜕𝜈=𝑣̃(00,𝑥o)n=(00,𝑇in)×Ω,𝜕Ω,(2.11) where ℎ₂≜−𝛽₁∇𝑢₁₁−∇𝑢₁₂⋅∇𝑢₂−𝛽₂∇𝑢₁₁−∇𝑢₁₂⋅∇𝑢₃−𝛽₁𝑢₁₁−𝑢₁₂Δ𝑢₂−𝛽₂𝑢₁₁−𝑢₁₂Δ𝑢₃.(2.12) By 𝑢₁₁,𝑢₁₂∈𝑋_𝐴 and conditions of Lemma 2.2, it is easy to check that ‖‖ℎ₂‖‖_{𝐿∞(𝑄𝑇)}≤‖‖𝛽₁∇𝑢₁₁−∇𝑢₁₂⋅∇𝑢₂‖‖_{𝐶0(𝑄𝑇)}+‖‖𝛽₂∇𝑢₁₁−∇𝑢₁₂⋅∇𝑢₃‖‖_{𝐶0(𝑄𝑇)}+‖‖𝛽₁𝑢₁₁−𝑢₁₂Δ𝑢₂‖‖_{𝐶0(𝑄𝑇)}+‖‖𝛽₂𝑢₁₁−𝑢₁₂Δ𝑢₃‖‖_{𝐶0(𝑄𝑇)}≤𝑁₀‖‖𝑢₁₁−𝑢₁₂‖‖_𝐶1,0+𝑁₀‖‖𝑢₁₁−𝑢₁₂‖‖_{𝐶0(𝑄𝑇)}≤𝑁₀‖‖𝑢₁₁−𝑢₁₂‖‖_{𝐶1,0(𝑄𝑇)}.(2.13) Using the assumption ‖𝑓‖_𝛼,𝛼/2≤𝑁₀ and the 𝐿^𝑝-estimate, we have ‖𝑓‖_𝐿^∞_(𝑄𝑇)≤‖𝑓‖_𝐶^𝛼,𝛼/2≤𝑁₀,‖𝑣‖̃_𝑤^𝑝2,1≤𝑁₀‖‖ℎ₂‖‖_{𝐿∞(𝑄𝑇)}.(2.14) For any 𝑝≥1, by using Sobolev embedding 𝑊^𝑝2,1(𝑄_𝑇)↪𝐶^{1+𝛾,(1+𝛾)/2}(𝑄_𝑇) (𝛾=1−(5/𝑝)>𝛼 if we take 𝑝 sufficiently large), we have ‖𝑣‖̃_𝐶^{1+𝛾,(1+𝛾)/2}_(𝑄𝑇)≤𝑁₀‖‖ℎ₂‖‖_{𝐿∞(𝑄𝑇)}≤𝑁₀‖‖𝑢₁₁−𝑢₁₂‖‖_{𝐶1,0(𝑄𝑇)}.(2.15) Then, noting 𝛾>𝛼, we can easily check that [4] ‖𝑣‖̃_𝐶^{1+𝛼,𝛼/2}_(𝑄𝑇)≤𝑁₀𝑇^𝛼/2‖‖𝑢₁₁−𝑢₁₂‖‖_{𝐶1+𝛼,𝛼/2(𝑄𝑇)}.(2.16) Taking 𝑇 small such that 𝑁₀𝑇^𝛼/2<1/2, we conclude from (2.16) that 𝐹 is contractive in 𝑋_𝐴. Therefore 𝐹 has a unique fixed point 𝑢₁, which is the unique solution to (2.5). Moreover, we can raise the regularity of 𝑢₁ to 𝐶^{2+𝛼,1+(𝛼/2)}(𝑄_𝑇) by using the parabolic Schauder estimates.

3. Local Existence and Uniqueness of Solutions

In this section, we will prove Theorem 3.1 which show that system (1.1) has a unique solution 𝐔(𝑥,𝑡)=(𝑢₁,𝑢₂,𝑢₃)∈𝐶^{2+𝛼,1+(𝛼/2)}(𝑄_𝑇) as done in [6, 7].

Theorem 3.1. Assume that (1.2) and (1.3) hold, then there exists a unique solution 𝐔(𝑥,𝑡)=(𝑢₁,𝑢₂,𝑢₃)∈𝐶^{2+𝛼,1+(𝛼/2)}(𝑄_𝑇) of the system (1.2) for small 𝑇>0 which depends on ‖‖𝐔₀(𝑥)‖‖_{𝐶2+𝛼(Ω)}≜‖‖𝑢₁₀(𝑥)‖‖_{𝐶2+𝛼(Ω)}+‖‖𝑢₂₀(𝑥)‖‖_{𝐶2+𝛼(Ω)}+‖‖𝑢₃₀(𝑥)‖‖_{𝐶2+𝛼(Ω)}.(3.1) Furthermore, 𝑢₁(𝑥,𝑡)≥0,𝑢₂(𝑥,𝑡)≥0,𝑢₃(𝑥,𝑡)≥0.

Proof. We will prove the local existence by a fixed point argument again. Introducing the Banach space 𝑋 of the function 𝐔, we define the norm ‖𝐔‖_𝐶^𝛼,𝛼/2_(𝑄𝑇)=‖‖𝑢₁‖‖_{𝐶𝛼,𝛼/2(𝑄𝑇)}+‖‖𝑢₂‖‖_{𝐶𝛼,𝛼/2(𝑄𝑇)}+‖‖𝑢₃‖‖_{𝐶𝛼,𝛼/2(𝑄𝑇)}(0<𝑇<1),(3.2) and a subset 𝑋_𝐴=𝐔∈𝑋∶𝑢₁,𝑢₂,𝑢₃≥0,‖𝐔‖_𝐶^𝛼,𝛼/2_(𝑄𝑇)≤𝐴,(3.3) where 𝐴=‖‖𝑢₁₀𝐔(𝑥,0)=𝑢₁₀(𝑥),𝑢₂₀(𝑥),𝑢₃₀(𝑥),(𝑥)‖‖_{𝐶2+𝛼(Ω)}+‖‖𝑢₂₀(𝑥)‖‖_{𝐶2+𝛼(Ω)}+‖‖𝑢₃₀(𝑥)‖‖_{𝐶2+𝛼(Ω)}+1.(3.4) For any 𝐔∈𝑋_𝐴, we define correspondingly function 𝐔=𝐇𝐔 by 𝐔=(𝑢₁,𝑢₂,𝑢₃), where 𝐔 satisfies the equations 𝜕𝑢₂𝜕𝑡−𝑑²Δ𝑢₂=𝑟₂1−𝐾𝑢²₂−𝑐²𝑢₁𝑚₂+𝑏₂𝑢₂𝜕𝑢₂in(0,𝑇)×Ω,𝑢₃𝜕𝑡−𝑑³Δ𝑢₃=𝑟₃1−𝐾𝑢³₃−𝑐³𝑢₁𝑚₃+𝑏₃𝑢₃𝜕𝑢₃in(0,𝑇)×Ω,𝑢₂𝜕𝜈=𝜕𝑢₃𝜕𝜈=0on(0,𝑇)×𝜕Ω,𝜕𝑢₂(𝑥,0)=𝑢₂₀(𝑥),𝑢₃(𝑥,0)=𝑢₃₀(𝑥),𝑥∈Ω,(3.5)𝑢₁𝜕𝑡−𝑑¹Δ𝑢₁+∇⋅𝛽₁𝑢₁∇𝑢₂+∇⋅𝛽₂𝑢₁∇𝑢₃=−𝑎𝑢₁+𝑒²𝑐₂𝑢₁𝑢₂𝑚₂+𝑏₂𝑢₂+𝑒³𝑐₃𝑢₁𝑢₃𝑚₃+𝑏₃𝑢₃𝜕in(0,T)×Ω,𝑢₁𝜕𝜈=0on(0,𝑇)×𝜕Ω,𝑢₁(𝑥,0)=𝑢₁₀(𝑥),𝑥∈Ω.(3.6) By (3.5), (𝑢₁,𝑢₂,𝑢₃)∈𝑋_𝐴, assumption (1.3), and the parabolic Schauder theory, we have that there exists a unique solution 𝑢₂,𝑢₃ to (3.5) and ‖‖𝑢₂‖‖_{𝐶2+𝛼,1+(𝛼/2)(𝑄𝑇)}≤‖‖𝑢₂|_𝑡=0‖‖_𝐶2+𝛼+𝑀₃(𝐴)≤𝐴+𝑀₃(𝐴)≜𝑀₄(𝐴).(3.7) Similarly, ‖‖𝑢₃‖‖_{𝐶2+𝛼,1+(𝛼/2)(𝑄𝑇)}≤‖‖𝑢₃|_𝑡=0‖‖_𝐶2+𝛼+𝑀₅(𝐴)≤𝐴+𝑀₅(𝐴)≜𝑀₆(𝐴).(3.8) Moreover, by parabolic maximum principle, we have 𝑢₂(𝑥,𝑡)≥0in𝑄_𝑇,𝑢₃(𝑥,𝑡)≥0in𝑄_𝑇.(3.9) Similarly, by using Lemma 2.2, from (3.6) we can conclude that there exists a unique solution 𝑢₁ satisfying ‖‖𝑢₁‖‖_{𝐶2+𝛼,1+(𝛼/2)𝑄𝑇}≤𝑀₇(𝐴),(3.10) and by parabolic maximum principle we have 𝑢₁(𝑥,𝑡)≥0 in 𝑄_𝑇. From (3.7), (3.8), and (3.10), we have ‖‖𝐔‖‖_{𝐶2+𝛼,1+(𝛼/2)(𝑄𝑇)}≤𝑀₈(𝐴).(3.11) For any function 𝐔(𝑥,𝑡), using Lemma 2.1 we get ‖‖𝐔(𝑥,𝑡)−𝐔(𝑥,0)‖‖_{𝐶𝛼,𝛼/2(𝑄𝑇)}≤𝑁₀𝜂(𝑇)‖𝐔‖_𝐶^{2+𝛼,1+(𝛼/2)}_(𝑄𝑇).(3.12) From (3.11) and (3.12), if 𝑇 is sufficiently small we have ‖‖𝐔(𝑥,𝑡)‖‖_{𝐶𝛼,𝛼/2(𝑄𝑇)}≤≤‖‖‖‖𝐔𝐔(𝑥,0)‖‖_{𝐶𝛼,𝛼/2}+𝑁₀𝜂(𝑇)𝑀₈(𝐴)₀(𝑥)‖‖_{𝐶2+𝛼(Ω)}+1≡𝐴,(3.13) which yields 𝐔∈𝑋_𝐴. Therefore, 𝐻 maps 𝑋_𝐴 into itself.
Next, we can prove that 𝐻 is contractive as done in the proof of Lemma 2.2 in 𝑋_𝐴 if we take 𝑇 sufficiently small. By the contraction mapping theorem 𝐻 has a unique fixed point 𝐔, which is the unique solution of (1.1). Moreover, we can raise the regularity of 𝐔 to 𝐶^{2+𝛼,1+(𝛼/2)}(𝑄_𝑇) by using the parabolic Schauder estimates.

4. Global Existence

First we establish some a priori estimates to (1.1).

Lemma 4.1. Suppose that 𝐔=(𝑢₁,𝑢₂,𝑢₃)∈𝐶^2,1(𝑄_𝑇) is a solution to the system (1.1), then there holds 𝑢₁≥0,0≤𝑢₂≤𝐾₂,0≤𝑢₃≤𝐾₃.(4.1)

Proof. It follows from (1.1) that 𝜕𝑢₁𝜕𝑡−𝑑¹Δ𝑢₁+𝛽₁∇𝑢₂+𝛽₂∇𝑢₃⋅∇𝑢₁+𝛽₁Δ𝑢₂+𝛽₂Δ𝑢₃+𝑎−𝑒²𝑐₂𝑢₂𝑚₂+𝑏₂𝑢₂−𝑒³𝑐₃𝑢₃𝑚₃+𝑏₃𝑢₃𝑢¹𝜕𝑢₁=0in(0,𝑇)×Ω,𝑢𝜕𝜈=0on(0,𝑇)×𝜕Ω,₁(0,𝑥)=𝑢₁₀(𝑥)≥0inΩ.(4.2) Obviously, 𝑢₁≡0 is a subsolution to (4.2). Using the maximum principle, we get 𝑢₁≥0. Similarly, we have 𝑢₂≥0 and 𝑢₃≥0.
On the other hand, it follows from model (1.1) that𝜕𝑢₂𝜕𝑡−𝑑²Δ𝑢₂−𝑟₂1−𝐾𝑢²₂𝑢²+𝑐²𝑢₁𝑢₂𝑚₂+𝑏₂𝑢₂=0≤𝑐²𝐾₂𝑢₁𝑚₂+𝑏₂𝐾₂𝜕𝑢₂in(0,𝑇)×Ω,𝜕𝑢𝜈=0on(0,𝑇)×𝜕Ω,₂(0,𝑥)=𝑢₂₀(𝑥)inΩ,(4.3) which implies that 𝐾₂ is a subsolution to problem (4.3). Hence we have 0≤𝑢₂(𝑥,𝑡)≤𝐾₂. Similarly, we get 0≤𝑢₃(𝑥,𝑡)≤𝐾₃. This completes the proof of Lemma 4.1.

Lemma 4.2. Suppose that 𝐔=(𝑢₁,𝑢₂,𝑢₃)∈𝐶^2,1(𝑄_𝑇) is a solution to the system (1.1), then for any 𝑝>1 there holds ‖‖𝑢₁‖‖_{𝐿𝑝(𝑄𝑇)}≤𝑁,‖‖𝑢₂‖‖_{𝐿𝑝(𝑄𝑇)}≤𝑁,‖‖𝑢₂‖‖_{𝐿𝑝(𝑄𝑇)}≤𝑁.(4.4)

Proof. Multiplying the first equation of (1.1) by 𝑢^1𝑝−1, integrating over 𝑄_𝑇, using the no-flux boundary condition, and noting 𝑢₁≥0, we get 1𝑝_Ω𝑢^𝑝1(𝑥,𝑡)𝑑𝑥−1𝑝_Ω𝑢^𝑝0(𝑥,𝑡)𝑑𝑥+(𝑝−1)𝑑₁^𝑡0_Ω≤(𝑝−1)_𝑡0𝑢^1𝑝−2||∇𝑢₁||²𝑑𝑥𝑑𝑡_Ω𝛽₁𝑢^1𝑝−1∇𝑢₁⋅∇𝑢₂𝑑𝑥𝑑𝑡+(𝑝−1)^𝑡0_Ω+𝑒𝛽₂𝑢^1𝑝−1∇𝑢₁⋅∇𝑢₃𝑑𝑥𝑑𝑡₂𝑐₂𝑏₂^𝑡0_Ω𝑢^𝑝1𝑑𝑥𝑑𝑡+𝑒³𝑐₃𝑏₃^𝑡0_Ω𝑢^𝑝1𝑑𝑥𝑑𝑡.(4.5) For 𝑢₁≥𝑢_1𝑚, we get 1𝑝_Ω𝑢^𝑝1(𝑥,𝑡)𝑑𝑥−1𝑝_Ω𝑢^𝑝0(𝑥,𝑡)𝑑𝑥+(𝑝−1)𝑑₁^𝑡0_Ω≤𝑒𝑢^1𝑝−2||∇𝑢₁||²𝑑𝑥𝑑𝑡₂𝑐₂𝑏₂+𝑒³𝑐₃𝑏₃^𝑡0_Ω𝑢^𝑝1𝑑𝑥𝑑𝑡.(4.6) Therefore _Ω𝑢^𝑝1(𝑥,𝑡)𝑑𝑡≤𝑁₀+𝑁₀^𝑡0_Ω𝑢^𝑝1𝑑𝑥𝑑𝑡.(4.7) Using Gronwall’s Lemma, we have ^𝑡0_Ω𝑢^𝑝1(𝑥,𝑡)𝑑𝑡≤𝑁.(4.8) Therefore, for 𝑢₁<𝑢_1𝑚, we have ^𝑡0_Ω𝑢^𝑝1(𝑥,𝑡)𝑑𝑡≤^𝑡0_Ω𝑢^𝑝1𝑚(𝑥,𝑡)𝑑𝑡≤𝑁.(4.9) Obviously, we have ^𝑡0_Ω𝑢^𝑝2(𝑥,𝑡)𝑑t≤^𝑡0_Ω𝐾^𝑝2(𝑥,𝑡)𝑑𝑡≤𝑁,_𝑡0_Ω𝑢^𝑝3(𝑥,𝑡)𝑑𝑡≤^𝑡0_Ω𝐾^𝑝3(𝑥,𝑡)𝑑𝑡≤𝑁.(4.10) This completes the proof of Lemma 4.2.

Lemma 4.3. Suppose that 𝐔=(𝑢₁,𝑢₂,𝑢₃)∈𝐶^2,1(Q_𝑇) is a solution to the system (1.1), then for any 𝑝>5 there holds ‖‖𝑢₁‖‖_{𝑤𝑝2,1(𝑄𝑇)}≤𝑁,‖‖𝑢₂‖‖_{𝑤𝑝2,1(𝑄𝑇)}≤𝑁,‖‖𝑢₃‖‖_{𝑤𝑝2,1(𝑄𝑇)}≤𝑁.(4.11)

Proof. Note that the second equation of (1.1) can be rewritten as follows: 𝜕𝑢₂𝜕𝑡−𝑑²Δ𝑢₂−𝑟₂−𝑟²𝐾₂𝑢₂−𝑐²𝑢₁𝑚₂+𝑏₂𝑢₂𝑢²=0,(4.12) where ‖𝑟₂−(𝑟₂/𝐾₂)𝑢₂−(𝑐₂𝑢₁/(𝑚₂+𝑏₂𝑢₂))‖_𝐿^𝑝_(𝑄𝑇)≤𝑁.
By the parabolic 𝐿^𝑝-estimate, we have‖‖𝑢₂‖‖_{𝑤𝑝2,1(𝑄𝑇)}≤𝑁.(4.13) Using the Sobolev embedding theorem (taking 𝑝>5), we get ‖‖∇𝑢₂‖‖_{𝐿∞(𝑄𝑇)}≤𝑁.(4.14) Similarly, we can obtain ‖‖‖‖∇𝑢𝑢₃‖‖_{𝑤𝑝2,1(𝑄𝑇)3}≤𝑁,‖‖_{𝐿∞(𝑄𝑇)}≤𝑁.(4.15) It follows from the first equation of (1.1) that 𝜕𝑢₁𝜕𝑡−𝑑¹Δ𝑢₁+𝛽₁∇𝑢₂+𝛽₂∇𝑢₃⋅∇𝑢₁=−𝛽₁Δ𝑢₂+𝛽₂Δ𝑢₃+𝑎−𝑒²𝑐₂𝑢₂𝑚₂+𝑏₂𝑢₂−𝑒³𝑐₃𝑢₃𝑚₃+𝑏₃𝑢₃𝑢¹𝜕𝑢₁in(0,𝑇)×Ω,𝑢𝜕𝜈=0on(0,𝑇)×𝜕Ω,₁(0,𝑥)=𝑢₁₀(𝑥)≥0inΩ,(4.16) where ‖‖‖−𝛽₁Δ𝑢₂+𝛽₂Δ𝑢₃+𝑎−𝑒²𝑐₂𝑢₂𝑚₂+𝑏₂𝑢₂−𝑒³𝑐₃𝑢₃𝑚₃+𝑏₃𝑢₃𝑢¹‖‖‖_{𝐿𝑝(𝑄𝑇)}≤𝑁.(4.17) Using the parabolic 𝐿^𝑝-estimates again, we have ‖‖𝑢₁‖‖_{𝑤𝑝2,1(𝑄𝑇)}≤𝑁.(4.18) This completes the proof of Lemma 4.3.

Lemma 4.4. Suppose that 𝐔=(𝑢₁,𝑢₂,𝑢₃)∈𝐶^2,1(𝑄_𝑇) is a solution to the system (1.1), then there holds ‖‖𝑢₁‖‖_{𝐶2+𝛼,1+(𝛼/2)(𝑄𝑇)}≤𝑁,‖‖𝑢₂‖‖_{𝐶2+𝛼,1+(𝛼/2)(𝑄𝑇)}≤𝑁,‖‖𝑢₃‖‖_{𝐶2+𝛼,1+(𝛼/2)(𝑄𝑇)}≤𝑁.(4.19)

Proof. Using the Sobolev embedding theorem (taking 𝑝>5) and Lemma 4.3, we have ‖‖𝑢₁‖‖_{𝐶𝛼,𝛼/2(𝑄𝑇)}≤𝑁,‖‖𝑢₂‖‖_{𝐶𝛼,𝛼/2(𝑄𝑇)}≤𝑁,‖‖𝑢₃‖‖_{𝐶𝛼,𝛼/2(𝑄𝑇)}≤𝑁.(4.20) Using (4.20) and the Schauder estimates to the second and third equation of model (1.1), we have ‖‖𝑢₂‖‖_{𝐶2+𝛼,1+(𝛼/2)(𝑄𝑇)}≤𝑁,‖‖𝑢₃‖‖_{𝐶2+𝛼,1+(𝛼/2)(Q𝑇)}≤𝑁.(4.21) Applying the parabolic Schauder estimate to (4.16) and using (4.21), we have ‖‖𝑢₁‖‖_{𝐶2+𝛼,1+(𝛼/2)(𝑄𝑇)}≤𝑁.(4.22) This completes the proof of Lemma 4.4.

Therefore, we can extend the local solution established in Theorem 3.1 to all 𝑡>0, as done in [6, 7]. Namely, we have the following.

Theorem 4.5. Under assumptions (1.2) and (1.3), there exists a unique solution 𝐔=(𝑢₁,𝑢₂,𝑢₃)∈𝐶^{2+𝛼,1+(𝛼/2)}(𝑄_𝑇) of the system (1.2) for any given 𝑇>0. Moreover, 𝑢₁(𝑥,𝑡)≥0,0≤𝑢₂≤𝐾₂,0≤𝑢₃≤𝐾₃.(4.23) for any 𝑥∈Ω and 𝑡>0.

Acknowledgments

The authors are grateful to the referees for their helpful comments and suggestions. This work is supported by the Fundamental Research Funds for the Central Universities (no. XDJK2009C152).