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 estimates, we prove that there exists a unique global classical solution of this system.
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 . 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  the authors proved the existence and uniqueness of weak solutions to the two-species predator-prey model with one prey-taxis. In , the author extended the results of  to an reaction-diffusion-taxis system. In , 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 is a bounded domain in is an integer) with a smooth boundary ; and represent the densities of the predator and prey, respectively; the positive constants , , and are the diffusion coefficient of the corresponding species; the positive constants 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 denotes their prey-tactic sensitivity. The parts and of the flux are directed toward the increasing population density of and , 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 for and for have a clear biological interpretation : the predators stop to accumulate at given point of after their density attains a certain threshold value and the prey-tactic sensitivity and vanishes identically when .
Throughout this paper we also assume that Denote by ( is integer, ) the space of function with finite norm : where We denote by the space of functions with norm
The main result of this paper is as follows.
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 by , while we denote various constants which are independent of by .
Lemma 2.1. Let . Then where .
Proof. Using the definition of Hölder norm, we have which yields that Therefore,
We now consider the following nonlinear parabolic problem: By the parabolic maximum principle, we have .
Proof. This proof is similar to that of Lemma 2.1 in . For reader’s convenience we include the proof here. We will prove by a fixed point argument. Let us introduce the Banach space of function with norm and a subset , where . For any , we define a corresponding function , where satisfies the equations By , we have By the parabolic Schauder theory, this yields that there exists a unique solution to (2.7) and where is some constant which depends only on . For any function , by Lemma 2.1 and combining (2.9), if is sufficiently small ( depends only on ), then we have Therefore, and maps into itself. We now prove that is contractive. Take in , and set . Then, it follows from (2.7) that solves the following systems: where By and conditions of Lemma 2.2, it is easy to check that Using the assumption and the -estimate, we have For any , by using Sobolev embedding ( if we take sufficiently large), we have Then, noting , we can easily check that  Taking small such that , we conclude from (2.16) that is contractive in . Therefore has a unique fixed point , which is the unique solution to (2.5). Moreover, we can raise the regularity of to by using the parabolic Schauder estimates.
3. Local Existence and Uniqueness of Solutions
Proof. We will prove the local existence by a fixed point argument again. Introducing the Banach space of the function , we define the norm
and a subset
For any , we define correspondingly function by , where satisfies the equations
By (3.5), , assumption (1.3), and the parabolic Schauder theory, we have that there exists a unique solution to (3.5) and
Moreover, by parabolic maximum principle, we have
Similarly, by using Lemma 2.2, from (3.6) we can conclude that there exists a unique solution satisfying
and by parabolic maximum principle we have in . From (3.7), (3.8), and (3.10), we have
For any function , using Lemma 2.1 we get
From (3.11) and (3.12), if is sufficiently small we have
which yields . Therefore, maps into itself.
Next, we can prove that is contractive as done in the proof of Lemma 2.2 in if we take sufficiently small. By the contraction mapping theorem has a unique fixed point , which is the unique solution of (1.1). Moreover, we can raise the regularity of to by using the parabolic Schauder estimates.
4. Global Existence
First we establish some a priori estimates to (1.1).
Lemma 4.1. Suppose that is a solution to the system (1.1), then there holds
Proof. It follows from (1.1) that
Obviously, is a subsolution to (4.2). Using the maximum principle, we get . Similarly, we have and .
On the other hand, it follows from model (1.1) that which implies that is a subsolution to problem (4.3). Hence we have . Similarly, we get . This completes the proof of Lemma 4.1.
Lemma 4.2. Suppose that is a solution to the system (1.1), then for any there holds
Proof. Multiplying the first equation of (1.1) by , integrating over , using the no-flux boundary condition, and noting , we get For , we get Therefore Using Gronwall’s Lemma, we have Therefore, for , we have Obviously, we have This completes the proof of Lemma 4.2.
Lemma 4.3. Suppose that is a solution to the system (1.1), then for any there holds
Proof. Note that the second equation of (1.1) can be rewritten as follows:
By the parabolic -estimate, we have Using the Sobolev embedding theorem (taking ), we get Similarly, we can obtain It follows from the first equation of (1.1) that where Using the parabolic -estimates again, we have This completes the proof of Lemma 4.3.
Lemma 4.4. Suppose that is a solution to the system (1.1), then there holds
Proof. Using the Sobolev embedding theorem (taking ) and Lemma 4.3, we have Using (4.20) and the Schauder estimates to the second and third equation of model (1.1), we have Applying the parabolic Schauder estimate to (4.16) and using (4.21), we have This completes the proof of Lemma 4.4.
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).
A. Okubo and H. C. Chiang, Analysis of the Kinematics of Anarete Pritchardi Kim, vol. 10, Springer, Berlin, Germany, 1980.