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Global boundedness of solutions in a reaction–diffusion system of Beddington–DeAngelis-type predator–prey model with nonlinear prey-taxis and random diffusion

Boundary Value Problems20182018:33

https://doi.org/10.1186/s13661-018-0952-8

  • Received: 15 February 2018
  • Accepted: 5 March 2018
  • Published:

Abstract

In this paper, a reaction–diffusion system of a predator–prey model with Beddington–DeAngelis functional response is considered. This model describes a prey-taxis mechanism that is an immediate movement of the predator u in response to a change of the prey v (which leads to the collection of u). We use some approaches to prove the global existence-boundedness of classical solutions and overcome the substantial difficulty of the existence of a nonlinear prey-taxis term.

Keywords

  • Predator–prey
  • Reaction–diffusion system
  • Beddington–DeAngelis-type
  • Boundedness

MSC

  • 35A01
  • 35K57

1 Introduction

In this paper, we investigate the following reaction–diffusion system of a Beddington–DeAngelis-type predator–prey model with nonlinear prey-taxis and random diffusion:
$$ \textstyle\begin{cases} u_{t}-d_{1}\Delta u+\nabla \cdot (u\chi (u)\nabla v)=-au+\frac{ \delta uv}{\beta u+v+\alpha }, \quad x \in \Omega , t\in (0,T),\\ v_{t}-d _{2}\Delta v=rv-\frac{r}{K}v^{2}-\frac{\gamma uv}{\beta u+v+\alpha },\quad x \in \Omega , t\in (0,T),\\ \frac{\partial u}{\partial \nu }=\frac{ \partial v}{\partial \nu }=0, \quad x \in \partial \Omega , t\in (0,T),\\ (u(x,0),v(x,0))=(u _{0}(x),v_{0}(x))\geq 0, \quad x \in \Omega , \end{cases} $$
(1.1)
where Ω is a bounded domain in \(\mathbb{R}^{N}\) (\(N=1,2,3\)) with smooth boundary Ω, \(0< T\leq +\infty \), initial condition \(u_{0}(x),v_{0}(x)\in C^{2+\alpha }(\overline{\Omega })\) compatible on Ω, constants \(d_{1},d_{2},a,K,r,\alpha ,\beta , \gamma ,\delta >0\), and ν is the outward directional derivative normal to Ω.

Differential equations are supposed to be sufficient in modeling of the countless processes in all fields of science. Many phenomena in physical sciences, chemistry and biology are naturally described by Partial Differential Equations (PDEs). In a population dynamics system, a lot of mathematical models in partial differential equations to study the relationship between predators and preys have been proposed by mathematicians, biologists and ecologists [1, 2]. In fact, there are many well-known predator–prey models such as ratio-dependent predator–prey models (so-called Michaelis–Menten-type models) [3, 4], Holling-type models [5], Holling-type II models [6], Ivlev-type models [7], Lotka–Volterra-type models [8, 9] and so on. To the best of our knowledge, this is the first study to report that the global boundedness of solutions of the Beddington–DeAngelis-type predator–prey model with nonlinear prey-taxis.

There is a Beddington–DeAngelis-type functional response contained in the model (1.1), where u and v describe the population density of the predator and the prey at time t with diffusion rates \(d_{1}\) and \(d_{2}\), respectively. The parameter a denotes the death rate of the predator u. As usual, K is called the carrying capacity of the prey v. The constant r is called the intrinsic growth rate of the prey v. The constants δ, γ are the conversion rate of the predator and the prey, respectively. The term βu measures the mutual interference between predators. The reason for the model is that if one takes the parameter β close to 0 then the model can be regarded as a Holling-type II predator–prey model,
$$ \textstyle\begin{cases} u_{t}-d_{1}\Delta u+u+\nabla \cdot (u\chi (u)\nabla v)=-au+\frac{ \delta uv}{v+\alpha }, \quad x \in \Omega , t\in (0,T),\\ v_{t}-d_{2} \Delta v=rv-\frac{r}{K}v^{2}-\frac{\gamma uv}{v+\alpha }, \quad x \in \Omega , t\in (0,T),\\ \frac{\partial u}{\partial \nu }=\frac{\partial v}{\partial \nu }=0, \quad x \in \partial \Omega , t\in (0,T),\\ (u(x,0),v(x,0))=(u _{0}(x),v_{0}(x))\geq 0, \quad x \in \Omega , \end{cases} $$
(1.2)
and also if one takes the parameter α close to 0 then the model can be thought of as a ratio-dependent predator–prey model,
$$ \textstyle\begin{cases} u_{t}-d_{1}\Delta u+u+\nabla \cdot (u\chi (u)\nabla v)=-au+\frac{ \delta uv}{\beta u+v}, \quad x \in \Omega , t\in (0,T),\\ v_{t}-d_{2} \Delta v=rv-\frac{r}{K}v^{2}-\frac{\gamma uv}{\beta u+v}, \quad x \in \Omega , t\in (0,T),\\ \frac{\partial u}{\partial \nu }=\frac{\partial v}{\partial \nu }=0,\quad x \in \partial \Omega , t\in (0,T),\\ (u(x,0),v(x,0))=(u _{0}(x),v_{0}(x))\geq 0, \quad x \in \Omega . \end{cases} $$
(1.3)

In the present paper, motivated by [1013], we will prove the global boundedness of classical solutions to (1.1). In a reaction–diffusion system of predator–prey model, the prey-taxis mechanism means a direct movement of the predator u in response to a variation of the prey v (which results in the aggregation of u), and the involved factor is assumed to satisfy \(\chi (u)\in C^{1}([0,+\infty ))\), \(\chi (u)\equiv 0\) for \(u\geq M\), and \(\vert \chi '(u_{1})-\chi '(u_{2}) \vert \leq L \vert u_{1}-u_{2} \vert \) for \(u_{1},u_{2} \in [0,+\infty )\), with \(L, M>0\). Here the assumption \(\chi (u)\equiv 0\) for \(u\geq M\) says that there is a threshold value M for the accumulation of u, over which the prey-tactic cross-diffusion \(\chi (u)\) vanishes. That is the statement that the author proves.

Theorem 1.1

Suppose that \(\chi (u)\) satisfies:
  1. (i)

    \(\chi (u)\in C^{1}([0,+\infty ))\);

     
  2. (ii)

    \(\chi (u)\equiv 0\) for \(u\geq M\), with \(M>0\);

     
  3. (iii)

    \(\vert \chi '(u_{1})-\chi '(u_{2}) \vert \leq L \vert u_{1}-u_{2} \vert \) for \(u_{1},u_{2}\in [0,+\infty )\), with \(L>0\),

     
then we see that the solutions to (1.1) are global and uniformly bounded in time.

The remainder of this paper is organized as follows. In Sect. 2, we propose some preliminary results, which are essential to the proof of Theorem 1.1. Section 3 illustrates the proof of our main theorem, the global boundedness of solutions, based, mainly, on the use of the standard Moser iterative technique.

2 Preliminaries

In this section we state the following lemmas which are essential in the proofs of Theorem 1.1. The first is the global existence of classical solutions to (1.1).

Lemma 2.1

Under the assumptions for \(\chi (u)\) and initial data in the paper, there exists a unique solution \((u,v)\in (C^{2+\alpha ,1+\frac{\alpha }{2}}(\Omega \times (0,T)))^{2}\) of (1.1) for any given \(T>0\).

The proof of Lemma 2.1 is similar to the proof of Theorem 3.5 in [14]. Hence, we omit it. The second is on the boundedness of v.

Lemma 2.2

Let \((u,v)\in (C^{2+\alpha ,1+\frac{\alpha }{2}}(\Omega \times (0,T)))^{2}\) be a solution of (1.1). Then \(u\geq 0\) and \(0\leq v\leq K_{0}= \max \{ \max_{\overline{\Omega }}v_{0}(x), K \} \).

The proof of the lemma is based on the comparison principle of Ordinary Differential Equations (ODEs). Refer to the proof of Lemma 3.1 in [14] for the details.

The next lemma is the well-known classical \(L^{p}-L^{q}\) estimate for the Neumann heat semigroup on bounded domains.

Lemma 2.3

Suppose \((e^{t\Delta })_{t>0}\) is the Neumann heat semigroup in Ω, and \(\lambda_{1}>0\) denotes the first nonzero eigenvalue of −Δ in Ω under Neumann boundary conditions. Then the following \(L^{p}-L^{q}\) estimates hold with \(C_{1}, C_{2}>0\) only depending on Ω:
  1. (i)
    if \(1\leq q\leq p\leq +\infty \), then
    $$ \begin{aligned} \bigl\Vert \nabla e^{t\Delta }w \bigr\Vert _{L^{p}(\Omega )} \leq C_{1}\bigl(1+t^{- \frac{1}{2}-\frac{n}{2}(\frac{1}{q}-\frac{1}{p})}\bigr)e^{-\lambda_{1}t} \Vert w \Vert _{L^{p}(\Omega )}, \quad t>0 \end{aligned} $$
    for all \(w\in L^{q}(\Omega )\);
     
  2. (ii)
    if \(2\leq q\leq p<+\infty \), then
    $$ \begin{aligned} \bigl\Vert \nabla e^{t\Delta }w \bigr\Vert _{L^{p}(\Omega )} \leq C_{2}\bigl(1+t^{- \frac{n}{2}(\frac{1}{q}-\frac{1}{p})}\bigr)e^{-\lambda_{1}t} \Vert \nabla w \Vert _{L^{p}(\Omega )}, \quad t>0 \end{aligned} $$
    for all \(w\in W^{1,q}(\Omega )\).
     

Lemma 2.4

Suppose that \(T\in (0,\infty ]\) and that \(\Omega \subset \mathbb{R} ^{n}\), \(n\geq 1\), is a bounded domain, and that D, f and g comply with \(D\in C^{1}(\overline{\Omega }\times [0,T)\times [0,\infty ))\) and \(D\geq 0\), \(f\in C^{0} ( (0,T);C^{0}(\overline{\Omega })\cap C ^{1}(\Omega ) ) \) and \(g\in C^{0} ( \Omega \times (0,T) ) \) with \(f\cdot v\leq 0\) on \(\partial \Omega \times (0,T)\). Moreover, assume that \(D(x,t,s)\geq \delta s^{m-1}\), \(f\in L^{\infty } ( (0,T);L ^{q_{1}}(\Omega ) ) \) and \(g\in L^{\infty } ( (0,T);L^{q_{2}}( \Omega ) ) \) for all \(x\in \Omega \), \(t\in (0,T)\), \(\delta >0\) and \(s\geq s_{0}\) and for some \(\delta >0\), \(m\in \mathbb{R}\) and \(s_{0}\geq 1\), and some \(q_{1}>n+2\) and \(q_{2}>\frac{n+2}{2}\). Then if \(u\in C^{0}(\overline{\Omega }\times [0,T))\cap C^{2,1}(\overline{ \Omega }\times [0,T))\) is a nonnegative function satisfying
$$ \textstyle\begin{cases} u_{t}\leq \nabla \cdot (D(x,t,u)\nabla u)+\nabla \cdot f(x,t)+g(x,t),\quad x \in \Omega , t\in (0,T),\\ \partial_{\nu }u(x,t)\leq 0, \quad x \in \partial \Omega , t\in (0,T), \end{cases} $$
and if \(u\in L^{\infty } ( (0,T);L^{p_{0}}(\Omega ) ) \) is valid for some \(p_{0}\geq 1\) fulfilling
$$ p_{0}>1-m\cdot \frac{(n+1)q_{1}-(n+2)}{q_{1}-(n+2)} $$
and
$$ p_{0}>1-\frac{m}{1-\frac{nq_{2}}{(n+2)(q_{2}-1)}} $$
as well as
$$ p_{0}>\frac{n(1-m)}{2}, $$
then there exists \(C>0\), only depending on m, δ, Ω, \(\Vert f \Vert _{ L^{\infty } ( (0,T);L^{q_{1}}(\Omega ) ) }\), \(\Vert g \Vert _{ L^{\infty } ( (0,T);L^{q_{2}}(\Omega ) ) }\), \(\Vert u \Vert _{ L^{\infty } ( (0,T);L^{q_{0}}(\Omega ) ) }\) and \(\Vert u(0) \Vert _{L^{\infty }(\Omega )}\), such that
$$ \bigl\Vert u(t) \bigr\Vert _{L^{\infty }(\Omega )}\leq C $$
for all \(t\in (0,T)\).

Refer to the proof of Lemma A.1 in [15] for the details.

3 Proof of main result

In this section, we will deal with the proof of the main result of the paper.

Proof of Theorem 1.1

The proof consists of four parts.

Part 1: Boundedness of \(\Vert u \Vert _{L^{1}(\Omega )}\).

Integrate the sum of the γ times of the first equation and the δ times of the second equation in (1.1) on Ω by parts,
$$ \begin{aligned}[b] \frac{d}{dt} \int_{\Omega }\gamma u+\frac{d}{dt} \int_{\Omega }\delta v=-a \gamma \int_{\Omega }u+r\delta \int_{\Omega }v-\frac{r\delta }{K} \int_{\Omega }v^{2}. \end{aligned} $$
(3.1)
Employing Young’s inequality, we have
$$ 2r\delta \int_{\Omega }v\leq \frac{r\delta }{K} \int_{\Omega }v^{2}+Kr \delta \vert \Omega \vert . $$
Setting the last inequality into (3.1), we obtain
$$ \begin{aligned}[b] \frac{d}{dt} \int_{\Omega }\gamma u+\frac{d}{dt} \int_{\Omega }\delta v &=-a \gamma \int_{\Omega }u+r\delta \int_{\Omega }v-\frac{r\delta }{K} \int_{\Omega }v^{2} \\ &\leq -a\gamma \int_{\Omega }u-r\delta \int_{\Omega }v+Kr\delta \vert \Omega \vert . \end{aligned} $$
(3.2)
Define
$$ y_{1}(t)= \int_{\Omega }\gamma u+ \int_{\Omega }\delta v, \quad t>0. $$
Then
$$ y_{1}'(t)+k_{1}y_{1}(t) \leq k_{2} $$
for all \(t>0\) by (3.2) with \(k_{1}=\min \{a,r\}\) and \(k_{2}=Kr \delta \vert \Omega \vert \). This ensures
$$ y_{1}(t)\leq C_{1}=\max \biggl\{ y_{1}(0),\frac{k_{2}}{k_{1}} \biggr\} $$
for all \(t>0\) by the comparison principle of ordinary differential equations.

Part 2: Boundedness of \(\Vert u \Vert _{L^{p}(\Omega )}\) with \(p>2\).

Multiplying the first equation in (1.1) by \(u^{p-1}\) and integrate on Ω by parts. Since \(v\leq K_{0}\) by Lemma 2.1, we have
$$ \int_{\Omega }u_{t}\cdot u^{p-1}- \int_{\Omega }d_{1}\Delta u\cdot u ^{p-1}=-a \int_{\Omega }u^{p}+\delta \int_{\Omega }\frac{u^{p}v}{ \beta u+v+\alpha }. $$
Now, we need to prove the following inequality:
$$ (p-1) \int_{\Omega }\chi (u)u^{p-1}\nabla u\cdot \nabla v \leq \frac{d _{1}(p-1)}{2} \int_{\Omega }u^{p-2} \vert \nabla u \vert ^{2}+\frac{p-1}{2d_{1}} \int_{\Omega }\chi^{2}(u)u^{p} \vert \nabla v \vert ^{2}. $$
By simplifying the problem, we only need to prove
$$ \chi (u)u^{p-1}\nabla u\cdot \nabla v\leq \frac{d_{1}}{2}u^{p-2} \vert \nabla u \vert ^{2}+ \frac{1}{2d_{1}}\chi^{2}(u)u^{p} \vert \nabla v \vert ^{2}. $$
Applying Young’s inequality with ε (\(ab\leq \frac{ \varepsilon }{p}a^{p}+\frac{\varepsilon^{-\frac{q}{p}}}{q}b^{q} \)) and setting \(p=q=2\), \(\varepsilon =d_{1}\), \(a=u^{\frac{p-2}{2}}\nabla u\) and \(b=\chi (u)u^{\frac{p}{2}}\nabla v\), we obtain
$$ \begin{aligned}\chi (u)u^{p-1}\nabla u\cdot \nabla v= {}&\chi (u)u^{\frac{p-2}{2}+ \frac{p}{2}}\nabla u\cdot \nabla v=\bigl(u^{\frac{p-2}{2}}\nabla u\bigr) \cdot \bigl( \chi (u)u^{\frac{p}{2}}\nabla v\bigr) \\ \leq{} &\frac{d_{1}}{2}u^{p-2} \vert \nabla u \vert ^{2}+\frac{1}{2d_{1}}\chi^{2}(u)u^{p} \vert \nabla v \vert ^{2}. \end{aligned} $$
Multiplying the inequality by \((p-1)\) and integrating on Ω by parts yield
$$ (p-1) \int_{\Omega }\chi (u)u^{p-1}\nabla u\cdot \nabla v \leq \frac{d _{1}(p-1)}{2} \int_{\Omega }u^{p-2} \vert \nabla u \vert ^{2}+\frac{p-1}{2d_{1}} \int_{\Omega }\chi^{2}(u)u^{p} \vert \nabla v \vert ^{2}. $$
Multiply the first equation of u in (1.1) by \(u^{p-1}\) and integrate on Ω by parts. Since \(v\leq K_{0}\) by Lemma 2.1, we have
$$ \begin{aligned} & \int_{\Omega }u_{t}\cdot u^{p-1}- \int_{\Omega }d_{1}\Delta u\cdot u ^{p-1}+ \int_{\Omega }\nabla \cdot \bigl(u\chi (u)\nabla v\bigr)\cdot u^{p-1} \\ &\quad =-a \int_{\Omega }u^{p}+\delta \int_{\Omega }\frac{u^{p}v}{\beta u+v+ \alpha }. \end{aligned} $$
According to
$$\begin{aligned}& \int_{\Omega }u_{t}\cdot u^{p-1}= \frac{1}{p} \int_{\Omega }pu^{p-1} \cdot u_{t}= \frac{1}{p} \int_{\Omega }\frac{d}{dt}u^{p}=\frac{1}{p} \frac{d}{dt} \int_{\Omega }u^{p}, \\& \int_{\Omega }d_{1}\cdot \nabla \cdot \bigl(\nabla u\cdot u^{p-1}\bigr)=d_{1} \int_{\Omega }\Delta u\cdot u^{p-1}+d_{1}(p-1) \int_{\Omega }u^{p-2} \vert \nabla u \vert ^{2}=0, \end{aligned}$$
and
$$ \int_{\Omega }\nabla \cdot \bigl(u\chi (u)\nabla v\bigr)\cdot u^{p-1}+(p-1) \int_{\Omega }\chi (u)u^{p-1}\nabla u\cdot \nabla v=0, $$
we have
$$ \begin{aligned} &\frac{1}{p}\frac{d}{dt} \int_{\Omega }u^{p}+d_{1}(p-1) \int_{\Omega }u ^{p-2} \vert \nabla u \vert ^{2} \\ &\quad =-a \int_{\Omega }u^{p}+\delta \int_{\Omega }\frac{u ^{p}v}{\beta u+v+\alpha }+(p-1) \int_{\Omega }\chi (u)u^{p-1}\nabla u \cdot \nabla v \\ &\quad\leq -a \int_{\Omega }u^{p}+\delta \int_{\Omega }\frac{u ^{p}v}{v+\alpha }+\frac{d_{1}(p-1)}{2} \int_{\Omega }u^{p-2} \vert \nabla u \vert ^{2}+\frac{p-1}{2d _{1}} \int_{\Omega }\chi^{2}(u)u^{p} \vert \nabla v \vert ^{2} \\ &\quad\leq \biggl( -a+\frac{ \delta K_{0}}{\alpha +K_{0}} \biggr) \int_{\Omega }u^{p}+ \frac{d_{1}(p-1)}{2} \int_{\Omega }u^{p-2} \vert \nabla u \vert ^{2}+\frac{p-1}{2d _{1}} \int_{\Omega }\chi^{2}(u)u^{p} \vert \nabla v \vert ^{2}. \end{aligned} $$
Consequently, together with \(\chi (u)\leq M_{1}\) due to \(\chi (u) \in C^{1}\) and \(\chi \equiv 0\) for \(u\geq M\), we have
$$ \begin{aligned}[b] &\frac{1}{p}\frac{d}{dt} \int_{\Omega }u^{p}+\frac{d_{1}(p-1)}{2} \int_{\Omega }u^{p-2} \vert \nabla u \vert ^{2} \\ &\quad\leq \biggl( -a+\frac{\delta K _{0}}{\alpha +K_{0}} \biggr) \int_{\Omega }u^{p}+\frac{p-1}{2d_{1}} \int_{\Omega }\chi^{2}(u)u^{p} \vert \nabla v \vert ^{2} \\ &\quad\leq \biggl( -a+\frac{ \delta K_{0}}{\alpha +K_{0}} \biggr) \int_{\Omega }u^{p}+\frac{(p-1)M _{1}^{2}M^{p}}{2d_{1}} \int_{\Omega } \vert \nabla v \vert ^{2}. \end{aligned} $$
(3.3)
Multiply the second equation of v in (1.1) by \(-\Delta v\), and integrate on Ω by parts to obtain
$$ \begin{aligned} &\frac{d}{dt} \int_{\Omega } \vert \nabla v \vert ^{2}+2d_{2} \int_{\Omega } \vert \Delta v \vert ^{2} \\ &\quad=2r \int_{\Omega } \vert \nabla v \vert ^{2}- \frac{4r}{K} \int_{\Omega }v \vert \nabla v \vert ^{2}+2\gamma \int_{\Omega }\frac{uv}{\beta u+v+\alpha } \Delta v \\ &\quad\leq 2r \int_{\Omega } \vert \nabla v \vert ^{2}+2\gamma \int_{\Omega }\frac{uv}{\beta u+v+\alpha }\Delta v \\ &\quad\leq 2r \int_{\Omega } \vert \nabla v \vert ^{2}+2\gamma \int_{\Omega }\frac{uv}{\beta u+v+\alpha } \vert \Delta v \vert \\ &\quad\leq 2r \int_{\Omega } \vert \nabla v \vert ^{2}+2\gamma \int_{\Omega }\frac{uv}{v+ \alpha } \vert \Delta v \vert \\ &\quad\leq 2r \int_{\Omega } \vert \nabla v \vert ^{2}+ \frac{2 \gamma K_{0}}{K_{0}+\alpha } \int_{\Omega }u \vert \Delta v \vert . \end{aligned} $$
Employing Young’s inequality, we have
$$ \frac{2\gamma K_{0}}{K_{0}+\alpha } \int_{\Omega }u \vert \Delta v \vert \leq \frac{ \varepsilon }{2} \int_{\Omega } \vert \Delta v \vert ^{2}+ \frac{2\gamma^{2}K_{0} ^{2}}{\varepsilon (K_{0}+\alpha )^{2}} \int_{\Omega }u^{2}. $$
Setting \(\varepsilon =2d_{2}\), we obtain
$$ \begin{aligned}[b] \frac{d}{dt} \int_{\Omega } \vert \nabla v \vert ^{2}+d_{2} \int_{\Omega } \vert \Delta v \vert ^{2} \leq 2r \int_{\Omega } \vert \nabla v \vert ^{2} \frac{\gamma^{2}K_{0}^{2}}{d_{2}(K _{0}+\alpha )^{2}} \int_{\Omega }u^{2}. \end{aligned} $$
(3.4)
According to
$$ \begin{aligned} d_{1}(p-1) \int_{\Omega }u^{p-2} \vert \nabla u \vert ^{2}&=d_{1}(p-1) \int_{\Omega }u^{\frac{p-2}{2}\cdot 2} \vert \nabla u \vert ^{2} \\ & =\frac{4d_{1}(p-1)}{p^{2}} \int_{\Omega } \biggl( \frac{p}{2} \biggr) ^{2}u^{(\frac{p}{2}-1)\cdot 2} \vert \nabla u \vert ^{2} \\ &=\frac{4d_{1}(p-1)}{p^{2}} \int_{\Omega } \bigl\vert \nabla u ^{\frac{p}{2}} \bigr\vert ^{2}, \end{aligned} $$
for \(p>2\), we know from (3.4) and the last equality by Young’s inequality that
$$ \begin{aligned}[b] &\frac{1}{p}\frac{d}{dt} \int_{\Omega }u^{p}+\frac{d}{dt} \int_{\Omega } \vert \nabla v \vert ^{2} + \frac{2d_{1}(p-1)}{p^{2}} \int_{\Omega } \bigl\vert \nabla u^{ \frac{p}{2}} \bigr\vert ^{2}+d_{2} \int_{\Omega } \vert \Delta v \vert ^{2} \\ &\quad\leq \biggl( -a+\frac{ \delta K_{0}}{\alpha +K_{0}} \biggr) \int_{\Omega }u^{p}+\frac{(p-1)M _{1}^{2}M^{p}}{2d_{1}} \int_{\Omega } \vert \nabla v \vert ^{2} \\ &\quad\quad {} +2r \int_{\Omega } \vert \nabla v \vert ^{2}+ \frac{\gamma^{2} K_{0}^{2}}{d_{2} (K_{0}+\alpha )^{2}} \int_{\Omega }u ^{2} \\ &\quad\leq \biggl( -a+\frac{\delta K_{0}}{\alpha +K_{0}}+1 \biggr) \int_{\Omega }u^{p}+ \biggl( 2r+\frac{(p-1)M_{1}^{2}M^{p}}{2d_{1}} \biggr) \int_{\Omega } \vert \nabla v \vert ^{2}+k_{3} \end{aligned} $$
(3.5)
with \(k_{3}=\frac{\gamma^{2} K_{0}^{2}M^{2} \vert \Omega \vert }{d_{2} (K_{0}+ \alpha )^{2}}>0\).
For \(\int_{\Omega } \vert \nabla v \vert ^{2}\), applying the Sobolev interpolation inequality,
$$ \bigl\Vert D^{j}v \bigr\Vert _{p,\Omega }\leq \varepsilon \bigl\Vert D^{k}v \bigr\Vert _{p,\Omega }+C \Vert v \Vert _{p,\Omega }, $$
setting \(j=1\), \(k=2\), \(p=2\), and integrating on Ω by parts, it is easy to see that
$$ \begin{aligned}[b] \int_{\Omega } \vert \nabla v \vert ^{2} \leq \varepsilon_{1} \int_{\Omega } \vert \Delta v \vert ^{2}+k_{4} \int_{\Omega } \vert v \vert ^{2} \leq \varepsilon_{1} \int_{\Omega } \vert \Delta v \vert ^{2}+k_{4}K_{0}^{2} \vert \Omega \vert =\varepsilon_{1} \int_{\Omega } \vert \Delta v \vert ^{2}+k_{5} \end{aligned} $$
(3.6)
depending on \(\varepsilon_{1}\).
For \(\int_{\Omega }u^{p}\), by the Gagliardo–Nirenberg inequality with \(u\geq 0\), we obtain
$$ \begin{aligned}[b] \int_{\Omega }u^{p}&= \int_{\Omega } \bigl\vert u^{\frac{p}{2}} \bigr\vert ^{2} \leq k_{6} \bigl( \bigl\Vert \nabla u^{\frac{p}{2}} \bigr\Vert _{2}^{\frac{2Np-2N}{Np-N+2}}\cdot \bigl\Vert u ^{\frac{p}{2}} \bigr\Vert _{\frac{2}{p}}^{\frac{4}{Np-N+2}} + \bigl\Vert u^{\frac{p}{2}} \bigr\Vert _{\frac{2}{p}}^{2} \bigr) \\ &=k_{6} \bigl( \bigl\Vert \nabla u^{\frac{p}{2}} \bigr\Vert _{2}^{2\theta }\cdot \bigl\Vert u^{\frac{p}{2}} \bigr\Vert _{\frac{2}{p}}^{2(1-\theta )} + \bigl\Vert u^{\frac{p}{2}} \bigr\Vert _{\frac{2}{p}}^{2} \bigr) \end{aligned} $$
(3.7)
with \(k_{6}>0\) and \(0<\theta =\frac{Np-N}{Np-N+2}<1\). Applying Young’s inequality yields
$$ \bigl\Vert \nabla u^{\frac{p}{2}} \bigr\Vert _{2}^{2\theta }\cdot \bigl\Vert u^{\frac{p}{2}} \bigr\Vert _{\frac{2}{p}}^{2(1-\theta )} \leq \epsilon \theta \bigl\Vert \nabla u^{ \frac{p}{2}} \bigr\Vert _{2}^{2}+\epsilon^{\frac{\theta }{\theta -1}}(1- \theta ) \bigl\Vert u^{\frac{p}{2}} \bigr\Vert _{\frac{2}{p}}^{2} $$
with \(\epsilon >0\). Setting the last estimate into (3.7), we see that
$$ \begin{aligned} \int_{\Omega }u^{p}&= \int_{\Omega } \bigl\vert u^{\frac{p}{2}} \bigr\vert ^{2} \leq k_{6} \bigl( \bigl\Vert \nabla u^{\frac{p}{2}} \bigr\Vert _{2}^{2\theta }\cdot \bigl\Vert u^{\frac{p}{2}} \bigr\Vert _{\frac{2}{p}}^{2(1-\theta )} + \bigl\Vert u^{\frac{p}{2}} \bigr\Vert _{\frac{2}{p}} ^{2} \bigr) \\ &\leq k_{6} \bigl( \epsilon \theta \bigl\Vert \nabla u^{ \frac{p}{2}} \bigr\Vert _{2}^{2}+\epsilon^{\frac{\theta }{\theta -1}}(1- \theta ) \bigl\Vert u^{\frac{p}{2}} \bigr\Vert _{\frac{2}{p}}^{2} + \bigl\Vert u^{\frac{p}{2}} \bigr\Vert _{ \frac{2}{p}}^{2} \bigr) \\ &=k_{6}\epsilon \theta \bigl\Vert \nabla u^{ \frac{p}{2}} \bigr\Vert _{2}^{2}+k_{6}\epsilon^{\frac{\theta }{\theta -1}}(1- \theta ) \bigl\Vert u^{\frac{p}{2}} \bigr\Vert _{\frac{2}{p}}^{2} +k_{6} \bigl\Vert u^{\frac{p}{2}} \bigr\Vert _{\frac{2}{p}}^{2} \\ &=k_{6}\epsilon \theta \bigl\Vert \nabla u^{\frac{p}{2}} \bigr\Vert _{2}^{2}+k_{6} \bigl[ \epsilon^{\frac{\theta }{\theta -1}}(1- \theta )+1 \bigr] \bigl\Vert u^{\frac{p}{2}} \bigr\Vert _{\frac{2}{p}}^{2} \\ &=\varepsilon_{2} \bigl\Vert \nabla u ^{\frac{p}{2}} \bigr\Vert _{2}^{2}+k_{7} \bigl\Vert u^{\frac{p}{2}} \bigr\Vert _{\frac{2}{p}}^{2} \\ &=\varepsilon_{2} \bigl\Vert \nabla u^{\frac{p}{2}} \bigr\Vert _{2}^{2}+k_{7} \Vert u \Vert _{1} ^{p} \end{aligned} $$
for any \(\varepsilon_{2}>0\), with \(k_{7}>0\) depending on \(\varepsilon _{2}\). Because of \(\Vert u \Vert _{1}\leq A_{1}\) by Step 1, we know that
$$ \begin{aligned}[b] \int_{\Omega }u^{p}\leq \varepsilon_{2} \bigl\Vert \nabla u^{\frac{p}{2}} \bigr\Vert _{2} ^{2}+k_{7}A_{1}^{p}= \varepsilon_{2} \bigl\Vert \nabla u^{\frac{p}{2}} \bigr\Vert _{2}^{2}+k _{8} \end{aligned} $$
(3.8)
with \(k_{8}=k_{7}A_{1}^{p}>0\).
Now, we need to consider the value of \(\varepsilon_{1}\) and \(\varepsilon_{2}\). Fix them with
$$ \biggl( 2r+\frac{(p-1)M_{1}^{2}M^{p}}{2d_{1}} \biggr) \varepsilon_{1}= \frac{d _{2}}{2} $$
and
$$ \biggl( \frac{\delta K_{0}}{\alpha +K_{0}}+1 \biggr) \varepsilon_{2}= \frac{2d _{1}(p-1)}{p^{2}}. $$
We have from (3.5), (3.6) and (3.8) that
$$ \begin{aligned} &\frac{1}{p}\frac{d}{dt} \int_{\Omega }u^{p}+\frac{d}{dt} \int_{\Omega } \vert \nabla v \vert ^{2} + \biggl( \frac{\delta K_{0}}{\alpha +K_{0}}+1 \biggr) \varepsilon_{2} \int_{\Omega } \bigl\vert \nabla u^{\frac{p}{2}} \bigr\vert ^{2} \\ & \quad\quad {} +2 \biggl( 2r+\frac{(p-1)M _{1}^{2}M^{p}}{2d_{1}} \biggr) \varepsilon_{1} \int_{\Omega } \vert \Delta v \vert ^{2} \\ &\quad\leq \biggl( -a+\frac{\delta K_{0}}{\alpha +K_{0}}+1 \biggr) \int_{\Omega }u^{p}+ \biggl( 2r+\frac{(p-1)M_{1}^{2}M^{p}}{2d_{1}} \biggr) \int_{\Omega } \vert \nabla v \vert ^{2}+k_{3} \\ &\quad\leq -a \int_{\Omega }u^{p}+ \biggl( \frac{\delta K_{0}}{\alpha +K_{0}}+1 \biggr) \varepsilon_{2} \bigl\Vert \nabla u^{\frac{p}{2}} \bigr\Vert _{2}^{2} + \biggl( \frac{\delta K_{0}}{\alpha +K _{0}}+1 \biggr) k_{8} \\ & \quad\quad {} + \biggl( 2r+\frac{(p-1)M_{1}^{2}M^{p}}{2d_{1}} \biggr) \varepsilon_{1} \int_{\Omega } \vert \Delta v \vert ^{2}+ \biggl( 2r+ \frac{(p-1)M_{1} ^{2}M^{p}}{2d_{1}} \biggr) k_{5}+k_{3}. \end{aligned} $$
Obviously,
$$\begin{aligned} &\frac{1}{p}\frac{d}{dt} \int_{\Omega }u^{p}+\frac{d}{dt} \int_{\Omega } \vert \nabla v \vert ^{2} \\ &\quad\leq -a \int_{\Omega }u^{p}- \biggl( 2r+\frac{(p-1)M _{1}^{2}M^{p}}{2d_{1}} \biggr) \varepsilon_{1} \int_{\Omega } \vert \Delta v \vert ^{2} \\ &\quad\quad {} + \biggl[ \biggl( \frac{\delta K_{0}}{\alpha +K_{0}}+1 \biggr) k_{8}+ \biggl( 2r+ \frac{(p-1)M_{1}^{2}M^{p}}{2d_{1}} \biggr) k_{5}+k_{3} \biggr] \\ &\quad=-a \int_{\Omega }u^{p}- \biggl( 2r+ \frac{(p-1)M_{1}^{2}M^{p}}{2d_{1}} \biggr) \biggl( \varepsilon_{1} \int_{\Omega } \vert \Delta v \vert ^{2}+k_{5} \biggr) \\ &\quad\quad {} + \biggl[ \biggl( \frac{ \delta K_{0}}{\alpha +K_{0}}+1 \biggr) k_{8}+2 \biggl( 2r+ \frac{(p-1)M_{1} ^{2}M^{p}}{2d_{1}} \biggr) k_{5}+k_{3} \biggr] \\ &\quad\leq -a \int_{\Omega }u ^{p}- \biggl( 2r+\frac{(p-1)M_{1}^{2}M^{p}}{2d_{1}} \biggr) \int_{\Omega } \vert \nabla v \vert ^{2}+k_{9} \end{aligned}$$
with \(k_{9}= ( \frac{\delta K_{0}}{\alpha +K_{0}}+1 ) k_{8}+2 ( 2r+\frac{(p-1)M_{1}^{2}M^{p}}{2d_{1}} ) k_{5}+k_{3}>0\). Therefore, we define the function
$$ y_{2}(t)=\frac{1}{p} \int_{\Omega }u^{p}+ \int_{\Omega } \vert \nabla v \vert ^{2}, \quad t>0, $$
satisfies
$$ y_{2}'(t)+k_{10}y_{2}(t) \leq k_{9} $$
for all \(t>0\) with \(k_{10}=\min \{2r+\frac{(p-1)M_{1}^{2}M^{p}}{2d _{1}},pa\}\). This also ensures
$$ y_{2}(t)\leq C_{2}=\max \biggl\{ y_{2}(0),\frac{k_{9}}{k_{10}} \biggr\} $$
for all \(t>0\) by the comparison principle of ordinary differential equations.

Part 3: Boundedness of \(\Vert \nabla v \Vert _{L^{p}(\Omega )}\) with \(p>2\).

We can define \(f(u,v)=rv-\frac{r}{K}v^{2}-\frac{\gamma uv}{\beta u+v+ \alpha }\). It follows from Part 2 and Lemma 2.3 that there exists \(C_{3}>0\) such that
$$ \sup_{t>0} \bigl\Vert f(u,v) \bigr\Vert _{L^{p}(\Omega )}\leq C_{3}< +\infty . $$
By the variation-of-constants formula for v, we have
$$ v(\cdot , t)=e^{d_{2}t\Delta }v_{0}+ \int_{0}^{t}e^{d_{2}(t-s)\Delta }f\bigl(u(s),v(s) \bigr)\,ds, \quad t>0. $$
Because of Lemma 2.3, we can draw the conclusion that
$$\begin{aligned} & \Vert \nabla v \Vert _{L^{p}(\Omega )} \\ &\quad= \biggl\Vert \nabla e^{d_{2}t\Delta }v_{0}+ \int_{0}^{t}\nabla e^{d_{2}(t-s)\Delta }f\bigl(u(s),v(s) \bigr)\,ds \biggr\Vert _{L^{p}( \Omega )} \\ &\quad\leq \bigl\Vert \nabla e^{d_{2}t\Delta }v_{0} \bigr\Vert _{L^{p}(\Omega )}+ \biggl\Vert \int_{0}^{t}\nabla e^{d_{2}(t-s)\Delta }f\bigl(u(s),v(s) \bigr)\,ds \biggr\Vert _{L^{p}( \Omega )} \\ &\quad\leq C_{2}\bigl(1+d_{2}t^{-\frac{n}{2}(\frac{1}{q}- \frac{1}{p})} \bigr)e^{-\lambda_{1}d_{2}t} \Vert \nabla v_{0} \Vert _{L^{p}(\Omega )} + \int_{0}^{t} \bigl\Vert \nabla e^{d_{2}(t-s)\Delta }f \bigl(u(s),v(s)\bigr) \bigr\Vert _{L^{p}( \Omega )}\,ds \\ &\quad\leq 2C_{2}e^{-\lambda '_{1}t} \Vert \nabla v_{0} \Vert _{L^{p}( \Omega )}+C_{1} \int_{0}^{t}\bigl(1+d_{2}^{-\frac{1}{2}}(t-s)^{-\frac{1}{2}} \bigr)e ^{-\lambda '_{1}(t-s)} \bigl\Vert f\bigl(u(s),v(s)\bigr) \bigr\Vert _{L^{p}(\Omega )}\,ds \\ &\quad\leq 2C _{2}e^{-\lambda '_{1}t} \Vert \nabla v_{0} \Vert _{L^{p}(\Omega )}+C_{1}C_{3} \int_{0}^{t}\bigl(1+d_{2}^{-\frac{1}{2}}s^{-\frac{1}{2}} \bigr)e^{-\lambda '_{1}s}\,ds \\ &\quad\leq 2C_{2} \Vert \nabla v_{0} \Vert _{L^{p}(\Omega )}+C_{1}C_{3} \biggl( \frac{1}{ \lambda '_{1}}+d_{2}^{-\frac{1}{2}} \biggl( 2+\frac{1}{\lambda '_{1}} \biggr) \biggr) \end{aligned}$$
for all \(t>0\). Therefore, \(\Vert \nabla v \Vert _{L^{p}(\Omega )}\) is global bounded.

Part 4: Global boundedness.

Based on Part 2, Part 3 and Lemma 2.4, the global boundedness of solutions can be proved by using the standard Moser iterative technique. The proof is complete. □

If we take the parameter β close to 0 then the model can be regarded as a Holling-type II predator–prey model (1.2). Therefore, we can obtain the following corollary.

Corollary 3.1

Under the assumptions for \(\chi (u)\) and initial data described above, the unique nonnegative classical solution of (1.2) is globally bounded.

On the other hand, if we take the parameter α close to 0 then the model can be though of a ratio-dependent predator–prey model (1.3). That is the statement of the corollary we are trying to illustrate.

Corollary 3.2

Under the assumptions for \(\chi (u)\) and initial data described above, the unique nonnegative classical solution of (1.3) is globally bounded.

Remark 3.3

It is well known that the global boundedness of solutions in a reaction–diffusion system is a comparatively effortless outcome to the corresponding predator–prey model. The existence of a prey-taxis term in (1.1) makes it massively hard to obtain the global boundedness, and even the global existence of solutions. In addition, the prey-taxis term \(\nabla \cdot (u\chi (u)\nabla v)\) contained in the corresponding predator–prey model means that \(\chi (u)\equiv 0\) whenever \(u\geq M\), where the maximal density M serves as a switch to repulsion at high densities of the predator population, extremely similar to the prevention of overcrowding for chemotaxis or volume-filling effect. Therefore, the global boundedness of solutions in a reaction–diffusion system of Beddington–DeAngelis-type predator–prey model with prey-taxis established by way of Theorem 1.1 should be reasonable and natural.

Declarations

Acknowledgements

The author is grateful to the referee for careful reading of the paper and for his or her useful reports which help them to improve the paper.

Funding

This work was supported in part by the National Natural Science Foundation of China under Grant No. 61673121, in part by the Natural Science Foundation of Guangdong Province under Grant No. 2014A030313507, and in part by the Projects of Science and Technology of Guangzhou under Grant No. 201508010008.

Author’s contributions

DML participated in the design of the study, conceived of the study, and drafted the manuscript. The author read and approved the final manuscript.

Competing interests

The author declares to have no competing interests.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
School of Applied Mathematics, Guangdong University of Technology, Guangzhou, P.R. China

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