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

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.


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 twospecies 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: where Ω is a bounded domain in R N N ≥ 1 is an integer with a smooth boundary ∂Ω; u 1 and u i i 2, 3 represent the densities of the predator and prey, respectively; the positive constants d 1 , d 2 , and d 3 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 onehalf 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 β 1 u 1 ∇u 2 and β 2 u 1 ∇u 3 of the flux are directed toward the increasing population density of u 2 and u 3 , 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 that The assumptions that β 1 0 for u 1 ≥ u 1m and β 2 0 for u 1 ≥ 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 β 1 and β 2 vanishes identically when u 1 ≥ u 1m .
Throughout this paper we also assume that Q T m ≥ 0 is integer, 0 < α < 1, 0 < β < 1 the space of function u x, t with finite norm 8 : where We denote by Q T the space of functions u x, t with norm 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 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 .

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 N 0 . 1 Proof.Using the definition of H ölder norm, we have We now consider the following nonlinear parabolic problem:

2.5
By the parabolic maximum principle, we have Then, under assumptions 1.2 and 1.3 , there exists a unique nonnegative solution u 1 x, t ∈ C 2 α,1 α/2 Q T of the nonlinear problem 2.5 for small T > 0 which depends on u 10 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 For any u 1 ∈ X A , we define a corresponding function u 1 Fu 1 , where u 1 satisfies the equations

2.7
By u 1 ∈ X A , we have By the parabolic Schauder theory, this yields that there exists a unique solution u to 2.7 and where M 2 A is some constant which depends only on A. For any function u 1 x, t , by Lemma 2.1 and combining 2.9 , if T is sufficiently small T depends only on A , then we have

2.10
Therefore, u 1 x, t ∈ X A and F maps X A into itself.We now prove that F is contractive.Take u 11 , u 12 in X A , and set u 11 Fu 11 , u 12 Fu 12 , v u 11 − u 12 .Then, it follows from 2.7 that v solves the following systems:

2.12
By u 11 , u 12 ∈ X A and conditions of Lemma 2.2, it is easy to check that

2.13
Using the assumption f α,α/2 ≤ N 0 and the L p -estimate, we have

2.14
For any p ≥ 1, by using Sobolev embedding

2.16
Taking T small such that N 0 T α/2 < 1/2, we conclude from 2.16 that F is contractive in X A .Therefore F has a unique fixed point u 1 , which is the unique solution to 2.5 .Moreover, we can raise the regularity of u 1 to C 2 α,1 α/2 Q T by using the parabolic Schauder estimates.

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 u 1 , u 2 , u 3 ∈ 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 u 1 , u 2 , u 3 ∈ C 2 α,1 α/2 Q T of the system 1.2 for small T > 0 which depends on 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 and a subset where

3.4
For any U ∈ X A , we define correspondingly function U HU by U u 1 , u 2 , u 3 , where U satisfies the equations By 3.5 , u 1 , u 2 , u 3 ∈ X A , assumption 1.3 , and the parabolic Schauder theory, we have that there exists a unique solution u 2 , u 3 to 3.5 and Similarly, Moreover, by parabolic maximum principle, we have 3.9 Similarly, by using Lemma 2.2, from 3.6 we can conclude that there exists a unique solution u 1 satisfying and by parabolic maximum principle we have u 1 x, t ≥ 0 in Q T .From 3.7 , 3.8 , and 3.10 , we have

3.11
For any function U x, t , using Lemma 2.1 we get

3.12
From 3.11 and 3.12 , if T is sufficiently small we have

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.

Global Existence
First we establish some a priori estimates to 1.1 .Lemma 4.1.Suppose that U u 1 , u 2 , u 3 ∈ C 2,1 Q T is a solution to the system 1.1 , then there holds 4.1 Proof.It follows from 1.1 that

4.2
Obviously, u 1 ≡ 0 is a subsolution to 4.2 .Using the maximum principle, we get u 1 ≥ 0. Similarly, we have u 2 ≥ 0 and u 3 ≥ 0. On the other hand, it follows from model 1.1 that which implies that K 2 is a subsolution to problem 4.3 .Hence we have 0 ≤ u 2 x, t ≤ K 2 .
Similarly, we get 0 ≤ u 3 x, t ≤ K 3 .This completes the proof of Lemma 4.1.
Lemma 4.2.Suppose that U u 1 , u 2 , u 3 ∈ C 2,1 Q T is a solution to the system 1.1 , then for any p > 1 there holds Proof.Multiplying the first equation of 1.1 by u p−1 1 , integrating over Q T , using the no-flux boundary condition, and noting u 1 ≥ 0, we get

4.5
By the parabolic L p -estimate, we have Using the Sobolev embedding theorem taking p > 5 , we get Similarly, we can obtain

4.15
It follows from the first equation of 1.1 that

4.17
Using the parabolic L p -estimates again, we have

4.18
This completes the proof of Lemma 4.3.
Lemma 4.4.Suppose that U u 1 , u 2 , u 3 ∈ C 2,1 Q T is a solution to the system 1.1 , then there holds

4.19
Proof.Using the Sobolev embedding theorem taking p > 5 and Lemma 4.3, we have

4.20
Using 4.20 and the Schauder estimates to the second and third equation of model 1.1 , we have

4.21
Applying the parabolic Schauder estimate to 4.16 and using 4.21 , we have 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 u 1 , u 2 , u 3 ∈ C 2 α,1 α/2 Q T of the system 1.2 for any given T > 0.Moreover,