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

Authors: Chenglin Li; Zhangguo Hong
Journal: Journal of Applied Mathematics
 (2012)

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## 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 parabolicLp estimates, we prove that there exists a unique global classical solution of this system.


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## Body

## 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: (1.1)∂u1∂t-d1Δu1+∇⋅(β1u1∇u2)+∇⋅(β2u1∇u3)=-au1+e2c2u1u2m2+b2u2+e3c3u1u3m3+b3u3in(0,T)×Ω,∂u2∂t-d2Δu2=r2(1-u2K2)u2-c2u1u2m2+b2u2in(0,T)×Ω,∂u3∂t-d3Δu3=r3(1-u3K3)u2-c3u1u3m3+b3u3in(0,T)×Ω,∂u1∂ν=∂u2∂ν=∂u3∂ν=0on⁡(0,T)×∂Ω,(u1(0,x),u2(0,x),u3(0,x))=(u10(x),u20(x),u30(x))≥(0,0)inΩ,
where Ω is a bounded domain in RN(N≥1 is an integer) with a smooth boundary ∂Ω; u1 and ui(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,Ki,ri,mi,ei,mi/ci,bi/ci,mi/bi(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 βi(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 that(1.2)β1=0,β2=0,foru1≥u1m.
The assumptions that β1=0 for u1≥u1m and β2=0 for u1≥u1m have a clear biological interpretation [2]: the predators stop to accumulate at given point of Ω after their density attains a certain threshold value u1m and the prey-tactic sensitivity β1 and β2 vanishes identically when u1≥u1m.Throughout this paper we also assume that(1.3)u20,u30≤K,∂Ω∈C2+α,u10(x),u20(x),u30(x)∈C2+α(Ω¯),where0<α<1,∂u10∂ν=∂u20∂ν=∂u30∂ν=0,on⁡∂Ω.
Denote by Cx,tm+α,β(QT) (m≥0 is integer, 0<α<1,0<β<1) the space of function u(x,t) with finite norm [8]:(1.4)‖u‖Cx,tm+α,β(QT)=∑|l|=0m[sup⁡QT|Dxlu|+〈Dxlu〉x,QT(α)+〈Dxlu〉t,QT(β)],
where(1.5)〈w〉x,QT(α)=∑(x,t),(y,t)∈QT|w(x,t)-w(y,t)||x-y|α,〈w〉t,QT(β)=∑(x,t),(x,τ)∈QT|w(x,t)-w(x,τ)||t-τ|β.
We denote by Cx,t2+α,1+β(QT) the space of functions u(x,t) with norm(1.6)‖u‖Cx,t2+α,β(QT)+‖ut‖Cx,tα,β(QT).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)∈C2+α,1+(α/2)(QT) of the system (1.1), where QT=(0,T)×Ω. Moreover,
(1.7)u1(x,t)≥0,0≤u2(x,t)≤K2,0≤u3(x,t)≤K3,
for any x∈Ω and t>0.This paper is organized as follows. In Section2, 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 onT by N, while we denote various constants which are independent of T by N0.Lemma 2.1.
Let(u,x)∈C2+α,1+(α/2)(QT). Then
(2.1)‖u(x,t)-u(x,0)‖C1+α,α/2(QT)≤N0η(T)‖u‖C2+α,1+(α/2)(QT),
where η(T)=max{Tα/2,T(1-α)/2}.Proof.
Using the definition of Hölder norm, we have(2.2)|u(x,t)-u(x,0)||t|(1+α)/2≤|Dtu|⋅|T|(1-α)/2,[u(x,t)-u(x,0)]C1+α,0(QT)≤N0‖Dx2u(x,t)-Dx2u(x,0)‖L∞(QT)≤N0[Dx2u]C0,α/2(QT)⋅|T|α/2≤N0[u]C2,α/2(QT)⋅|T|α/2,
which yields that
(2.3)‖u(x,t)-u(x,0)‖C1+α,(1+α)/2(QT)≤N0η‖u‖C2+α,(1+α)/2(QT).
Therefore,
(2.4)‖u(x,t)-u(x,0)‖C1+α,α/2(QT)≤N0η‖u‖C2+α,1+(α/2)(QT).We now consider the following nonlinear parabolic problem:(2.5)∂u1∂t-d1Δu1+∇⋅(β1u1∇u2)+∇⋅(β2u1∇u3)=u1fin(0,T)×Ω,∂u1∂ν=0on⁡(0,T)×∂Ω,u1(0,x)=u10(x)inΩ.
By the parabolic maximum principle, we have u1(x,t)≥0.Lemma 2.2.
Let(2.6)u2(x,t),u3(x,t)∈C2+α,1+(α/2)(QT),f(x,t)∈Cα,α/2(QT),‖u2‖C2+α,1+(α/2)(QT),‖u3‖C2+α,1+(α/2)(QT),‖f‖Cα,α/2(QT)≤N0.
Then, under assumptions (1.2) and (1.3), there exists a unique nonnegative solution u1(x,t)∈C2+α,1+(α/2)(QT) of the nonlinear problem (2.5) for small T>0 which depends on ∥u10(x)∥C2+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∥C1+α,α/2(QT)(0<T<1) and a subset XA={u1∈X:u1(x,0)=u10(x)and∥u1∥C1+α,α/2(QT)≤A}, where A=∥u10(x)∥C2+α(Ω)+∥u20(x)∥C2+α(Ω)+∥u30(x)∥C2+α(Ω)+1. For any u1∈XA, we define a corresponding function ũ1=Fu1, where ũ1 satisfies the equations
(2.7)∂ũ1∂t-d1Δũ1-ũ1f=-β1∇u1⋅∇u2-β2∇u1⋅∇u3-β1u1Δu2-β2u1Δu3in(0,T)×Ω,∂ũ1∂ν=0on⁡(0,T)×∂Ω,ũ1(0,x)=u10(x)inΩ.
By u1∈XA, we have
(2.8)h1≜-β1∇u1⋅∇u2-β2∇u1⋅∇u3-β1u1Δu2-β2u1Δu3∈Cα,α/2(QT).
By the parabolic Schauder theory, this yields that there exists a unique solution ũ to (2.7) and
(2.9)‖ũ1‖C2+α,1+(α/2)(QT)≤‖ũ1|t=0‖C2+α(Ω)+M1(A)≤A+M1(A)≜M2(A),
where M2(A) is some constant which depends only on A. For any function ũ1(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)‖ũ1(x,t)‖C1+α,α/2(QT)≤‖ũ1(x,0)‖C1+α,α/2(QT)+N0η(T)‖ũ1‖C2+α,1+(α/2)(QT)≤‖ũ10(x)‖C2+α+1≤A.
Therefore, ũ1(x,t)∈XA and F maps XA into itself. We now prove that F is contractive. Take u11,u12 in XA, and set ũ11=Fu11,ũ12=Fu12,ṽ=ũ11-ũ12. Then, it follows from (2.7) that ṽ solves the following systems:
(2.11)∂ṽ∂t-d1Δṽ-ṽf=h2in(0,T)×Ω,∂ṽ∂ν=0on⁡(0,T)×∂Ω,ṽ(0,x)=0inΩ,
where
(2.12)h2≜-β1(∇u11-∇u12)⋅∇u2-β2(∇u11-∇u12)⋅∇u3-β1(u11-u12)Δu2-β2(u11-u12)Δu3.
By u11,u12∈XA and conditions of Lemma 2.2, it is easy to check that
(2.13)‖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).
Using the assumption ∥f∥α,α/2≤N0 and the Lp-estimate, we have
(2.14)‖f‖L∞(QT)≤‖f‖Cα,α/2≤N0,‖ṽ‖wp2,1≤N0‖h2‖L∞(QT).
For any p≥1, by using Sobolev embedding Wp2,1(QT)↪C1+γ,(1+γ)/2(QT) (γ=1-(5/p)>α if we take p sufficiently large), we have
(2.15)‖ṽ‖C1+γ,(1+γ)/2(QT)≤N0‖h2‖L∞(QT)≤N0‖u11-u12‖C1,0(QT).
Then, noting γ>α, we can easily check that [4]
(2.16)‖ṽ‖C1+α,α/2(QT)≤N0Tα/2‖u11-u12‖C1+α,α/2(QT).
Taking T small such that N0Tα/2<1/2, we conclude from (2.16) that F is contractive in XA. 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 C2+α,1+(α/2)(QT) by using the parabolic Schauder estimates.


## 3. Local Existence and Uniqueness of Solutions

In this section, we will prove Theorem3.1 which show that system (1.1) has a unique solution U(x,t)=(u1,u2,u3)∈C2+α,1+(α/2)(QT) 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)∈C2+α,1+(α/2)(QT) of the system (1.2) for small T>0 which depends on
(3.1)‖U0(x)‖C2+α(Ω)≜‖u10(x)‖C2+α(Ω)+‖u20(x)‖C2+α(Ω)+‖u30(x)‖C2+α(Ω).
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 spaceX of the function U, we define the norm
(3.2)‖U‖Cα,α/2(QT)=‖u1‖Cα,α/2(QT)+‖u2‖Cα,α/2(QT)+‖u3‖Cα,α/2(QT)(0<T<1),
and a subset
(3.3)XA={U∈X:u1,u2,u3≥0,‖U‖Cα,α/2(QT)≤A},
where
(3.4)U(x,0)=(u10(x),u20(x),u30(x)),A=‖u10(x)‖C2+α(Ω)+‖u20(x)‖C2+α(Ω)+‖u30(x)‖C2+α(Ω)+1.
For any U∈XA, we define correspondingly function U¯=HU by U¯=(u¯1,u¯2,u¯3), where U¯ satisfies the equations
(3.5)∂u¯2∂t-d2Δu¯2=[r2(1-u2K2)-c2u1m2+b2u2]u¯2in(0,T)×Ω,∂u¯3∂t-d3Δu¯3=[r3(1-u3K3)-c3u1m3+b3u3]u¯3in(0,T)×Ω,∂u¯2∂ν=∂u¯3∂ν=0on⁡(0,T)×∂Ω,u¯2(x,0)=u20(x),u¯3(x,0)=u30(x),x∈Ω,(3.6)∂u¯1∂t-d1Δu¯1+∇⋅(β1u¯1∇u¯2)+∇⋅(β2u¯1∇u¯3)=-au¯1+e2c2u¯1u2m2+b2u2+e3c3u¯1u3m3+b3u3in(0,T)×Ω,∂u¯1∂ν=0on⁡(0,T)×∂Ω,u¯1(x,0)=u10(x),x∈Ω.
By (3.5), (u1,u2,u3)∈XA, assumption (1.3), and the parabolic Schauder theory, we have that there exists a unique solution u¯2,u¯3 to (3.5) and
(3.7)‖u¯2‖C2+α,1+(α/2)(QT)≤‖u¯2|t=0‖C2+α+M3(A)≤A+M3(A)≜M4(A).
Similarly,
(3.8)‖u¯3‖C2+α,1+(α/2)(QT)≤‖u¯3|t=0‖C2+α+M5(A)≤A+M5(A)≜M6(A).
Moreover, by parabolic maximum principle, we have
(3.9)u¯2(x,t)≥0inQT,u¯3(x,t)≥0inQT.
Similarly, by using Lemma 2.2, from (3.6) we can conclude that there exists a unique solution u¯1 satisfying
(3.10)‖u¯1‖C2+α,1+(α/2)QT≤M7(A),
and by parabolic maximum principle we have u¯1(x,t)≥0 in QT. From (3.7), (3.8), and (3.10), we have
(3.11)‖U¯‖C2+α,1+(α/2)(QT)≤M8(A).
For any function U¯(x,t), using Lemma 2.1 we get
(3.12)‖U¯(x,t)-U¯(x,0)‖Cα,α/2(QT)≤N0η(T)‖U‖C2+α,1+(α/2)(QT).
From (3.11) and (3.12), if T is sufficiently small we have
(3.13)‖U¯(x,t)‖Cα,α/2(QT)≤‖U¯(x,0)‖Cα,α/2+N0η(T)M8(A)≤‖U0(x)‖C2+α(Ω)+1≡A,
which yields U¯∈XA. Therefore, H maps XA into itself.
Next, we can prove thatH is contractive as done in the proof of Lemma 2.2 in XA 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 C2+α,1+(α/2)(QT) by using the parabolic Schauder estimates.


## 4. Global Existence

First we establish some a priori estimates to (1.1).Lemma 4.1.
Suppose thatU=(u1,u2,u3)∈C2,1(QT) is a solution to the system (1.1), then there holds
(4.1)u1≥0,0≤u2≤K2,0≤u3≤K3.Proof.
It follows from (1.1) that
(4.2)∂u1∂t-d1Δu1+(β1∇u2+β2∇u3)⋅∇u1+(β1Δu2+β2Δu3+a-e2c2u2m2+b2u2-e3c3u3m3+b3u3)u1=0in(0,T)×Ω,∂u1∂ν=0on⁡(0,T)×∂Ω,u1(0,x)=u10(x)≥0inΩ.
Obviously, u̲1≡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(4.3)∂u2∂t-d2Δu2-r2(1-u2K2)u2+c2u1u2m2+b2u2=0≤c2K2u1m2+b2K2in(0,T)×Ω,∂u2∂ν=0on⁡(0,T)×∂Ω,u2(0,x)=u20(x)inΩ,
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 thatU=(u1,u2,u3)∈C2,1(QT) is a solution to the system (1.1), then for any p>1 there holds
(4.4)‖u1‖Lp(QT)≤N,‖u2‖Lp(QT)≤N,‖u2‖Lp(QT)≤N.Proof.
Multiplying the first equation of (1.1) by u1p-1, integrating over QT, using the no-flux boundary condition, and noting u1≥0, we get
(4.5)1p∫Ωu1p(x,t)dx-1p∫Ωu0p(x,t)dx+(p-1)d1∫0t∫Ωu1p-2|∇u1|2dxdt≤(p-1)∫0t∫Ωβ1u1p-1∇u1⋅∇u2dxdt+(p-1)∫0t∫Ωβ2u1p-1∇u1⋅∇u3dxdt+e2c2b2∫0t∫Ωu1pdxdt+e3c3b3∫0t∫Ωu1pdxdt.
For u1≥u1m, we get
(4.6)1p∫Ωu1p(x,t)dx-1p∫Ωu0p(x,t)dx+(p-1)d1∫0t∫Ωu1p-2|∇u1|2dxdt≤(e2c2b2+e3c3b3)∫0t∫Ωu1pdxdt.
Therefore
(4.7)∫Ωu1p(x,t)dt≤N0+N0∫0t∫Ωu1pdxdt.
Using Gronwall’s Lemma, we have
(4.8)∫0t∫Ωu1p(x,t)dt≤N.
Therefore, for u1<u1m, we have
(4.9)∫0t∫Ωu1p(x,t)dt≤∫0t∫Ωu1mp(x,t)dt≤N.
Obviously, we have
(4.10)∫0t∫Ωu2p(x,t)dt≤∫0t∫ΩK2p(x,t)dt≤N,∫0t∫Ωu3p(x,t)dt≤∫0t∫ΩK3p(x,t)dt≤N.
This completes the proof of Lemma 4.2.Lemma 4.3.
Suppose thatU=(u1,u2,u3)∈C2,1(QT) is a solution to the system (1.1), then for any p>5 there holds
(4.11)‖u1‖wp2,1(QT)≤N,‖u2‖wp2,1(QT)≤N,‖u3‖wp2,1(QT)≤N.Proof.
Note that the second equation of (1.1) can be rewritten as follows:
(4.12)∂u2∂t-d2Δu2-(r2-r2K2u2-c2u1m2+b2u2)u2=0,
where ∥r2-(r2/K2)u2-(c2u1/(m2+b2u2))∥Lp(QT)≤N.
By the parabolicLp-estimate, we have(4.13)‖u2‖wp2,1(QT)≤N.
Using the Sobolev embedding theorem (taking p>5), we get
(4.14)‖∇u2‖L∞(QT)≤N.
Similarly, we can obtain
(4.15)‖u3‖wp2,1(QT)≤N,‖∇u3‖L∞(QT)≤N.
It follows from the first equation of (1.1) that
(4.16)∂u1∂t-d1Δu1+(β1∇u2+β2∇u3)⋅∇u1=-(β1Δu2+β2Δu3+a-e2c2u2m2+b2u2-e3c3u3m3+b3u3)u1in(0,T)×Ω,∂u1∂ν=0on⁡(0,T)×∂Ω,u1(0,x)=u10(x)≥0inΩ,
where
(4.17)‖-(β1Δu2+β2Δu3+a-e2c2u2m2+b2u2-e3c3u3m3+b3u3)u1‖Lp(QT)≤N.
Using the parabolic Lp-estimates again, we have
(4.18)‖u1‖wp2,1(QT)≤N.
This completes the proof of Lemma 4.3.Lemma 4.4.
Suppose thatU=(u1,u2,u3)∈C2,1(QT) is a solution to the system (1.1), then there holds
(4.19)‖u1‖C2+α,1+(α/2)(QT)≤N,‖u2‖C2+α,1+(α/2)(QT)≤N,‖u3‖C2+α,1+(α/2)(QT)≤N.Proof.
Using the Sobolev embedding theorem (takingp>5) and Lemma 4.3, we have
(4.20)‖u1‖Cα,α/2(QT)≤N,‖u2‖Cα,α/2(QT)≤N,‖u3‖Cα,α/2(QT)≤N.
Using (4.20) and the Schauder estimates to the second and third equation of model (1.1), we have
(4.21)‖u2‖C2+α,1+(α/2)(QT)≤N,‖u3‖C2+α,1+(α/2)(QT)≤N.
Applying the parabolic Schauder estimate to (4.16) and using (4.21), we have
(4.22)‖u1‖C2+α,1+(α/2)(QT)≤N.
This completes the proof of Lemma 4.4.Therefore, we can extend the local solution established in Theorem3.1 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)∈C2+α,1+(α/2)(QT) of the system (1.2) for any given T>0. Moreover,
(4.23)u1(x,t)≥0,0≤u2≤K2,0≤u3≤K3.
for any x∈Ω and t>0.



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*Source: 702603-2012-01-11.xml*