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:
()
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
()
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
()
Denote by
(
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
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 U = (u1, u2, u3) ∈ C2+α,1+(α/2)(QT) of the system (1.1), where QT = (0, T) × Ω. Moreover,
()
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) ∈ C2+α, 1+(α/2)(QT). Then
()
where
η(
T) = max {
Tα/2,
T(1−α)/2}.
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
u1(
x,
t) ≥ 0.
Lemma 2.2. Let
()
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
.
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 and a subset , where . For any u1 ∈ XA, we define a corresponding function , where satisfies the equations
()
By
u1 ∈
XA, we have
()
By the parabolic Schauder theory, this yields that there exists a unique solution
to (
2.7) and
()
where
M2(A) is some constant which depends only on
A. For any function
, by Lemma
2.1 and combining (
2.9), if
T is sufficiently small (
T depends only on
A), then we have
()
Therefore,
and
F maps
XA into itself. We now prove that
F is contractive. Take
u11,
u12 in
XA, and set
. Then, it follows from (
2.7) that
solves the following systems:
()
where
()
By
u11,
u12 ∈
XA and conditions of Lemma
2.2, it is easy to check that
()
Using the assumption ∥
f∥
α,α/2 ≤
N0 and the
Lp-estimate, we have
()
For any
p ≥ 1, by using Sobolev embedding
(
γ = 1 − (5/
p) >
α if we take
p sufficiently large), we have
()
Then, noting
γ >
α, we can easily check that [
4]
()
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 Theorem 3.1 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
()
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
()
and a subset
()
where
()
For any
U ∈
XA, we define correspondingly function
by
, where
satisfies the equations
()
()
By (
3.5), (
u1,
u2,
u3) ∈
XA, assumption (
1.3), and the parabolic Schauder theory, we have that there exists a unique solution
to (
3.5) and
()
Similarly,
()
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
QT. 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
T is sufficiently small we have
()
which yields
. Therefore,
H maps
XA into itself.
Next, we can prove that H 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 that U = (u1, u2, u3) ∈ C2,1(QT) 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
u1 ≥ 0. Similarly, we have
u2 ≥ 0 and
u3 ≥ 0.
On the other hand, it follows from model (1.1) that
()
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) ∈ C2,1(QT) is a solution to the system (1.1), then for any p > 1 there holds
()
Proof. Multiplying the first equation of (1.1) by , integrating over QT, using the no-flux boundary condition, and noting u1 ≥ 0, we get
()
For
u1 ≥
u1m, we get
()
Therefore
()
Using Gronwall’s Lemma, we have
()
Therefore, for
u1 <
u1m, we have
()
Obviously, we have
()
This completes the proof of Lemma
4.2.
Lemma 4.3. Suppose that U = (u1, u2, u3) ∈ C2,1(QT) is a solution to the system (1.1), then for any p > 5 there holds
()
Proof. Note that the second equation of (1.1) can be rewritten as follows:
()
where
.
By the parabolic Lp-estimate, we have
()
Using the Sobolev embedding theorem (taking
p > 5), we get
()
Similarly, we can obtain
()
It follows from the first equation of (
1.1) that
()
where
()
Using the parabolic
Lp-estimates again, we have
()
This completes the proof of Lemma
4.3.
Lemma 4.4. Suppose that U = (u1, u2, u3) ∈ C2,1(QT) is a solution to the system (1.1), then there holds
()
Proof. Using the Sobolev embedding theorem (taking p > 5) 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.
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) ∈ C2+α,1+(α/2)(QT) of the system (1.2) for any given T > 0. Moreover,
()
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).
