Boundedness of a predator-prey model with density-dependent motilities and stage structure for the predator

1.   Introduction

This paper deals with the predator-prey model with density-dependent motilities and stage structure for the predator

under homogeneous Neumann boundary conditions in a smooth bounded domain $\Omega\subset \mathbb{R}$ and $\partial/\partial\nu$ represents the outer unit normal vector of $\partial \Omega$, where $u = u(x, t), v = v(x, t)$ and $w = w(x, t)$ are the densities of the mature predator, immature predator and prey at position $x$ and time $t$, respectively. $d_3, a, b, c, k, r$ are positive constants and more details of the parameters can be found in ^[1,2]. The terms $(d_1(w)u)_{xx}$ and $(d_2(u)v)_{xx}$ state that the motility functions $d_1(w)$ and $d_2(u)$ have some influence on the diffusion of mature predator and immature predator.

Biological predator-prey model plays a critical role in survival and reproduction of organisms, especially the predator-prey system with stage structure of predator describes the biological predator-prey phenomenon and its irregular movement more vividly(see ^[3,4,5,6,7] and reference therein). Recently, the following stage structure of predator with taxis mechanisms model has been studied by Wang and Wang ^[2]:

In the case $n = 1$ with $\rho > 0$ and $n = 2$ with $\rho = 0$, the authors ^[2] first established that the solutions of problem (1.2) are global existence and boundedness. Secondly, the linearized stability of normal steady state and predator-free steady state are obtained by using local bifurcation and Hopf bifurcation theory. Moreover, they proved the global stability of predator-free steady state. On the other hand, many scholars have also studied the stage state for prey ^[8,9] and the different state of the predator^[10].

In order to describe the movement of species more meaningfully, we illustrate a chemotaxis system with density-dependent motility to describe the motility law of predators. At present, this kind of model is mostly used in the field of chemical signal substances. The classic model is proposed in ^[11]

where $u(x, t)$ is the densities of bacteria and $v(x, t)$ is the concentration of AHL at position $x$ and time $t$. This system describes bacteria with logistic sources whose diffusion rate depends on the motion function $\gamma(v)$, which considers the repressive effect of AHL concentration on bacteria motility by supposing $\gamma'(v) < 0$. This diffusion mechanism is called "density suppression motility" in ^[12,13]. Therefore, it is a very interesting phenomenon and has been widely studied. If $\mu > 0$, Jin et.al ^[14] proved that the problem in two dimensions possesses a global classical solution and coexistence steady state is globally asymptotically stable. Yoon and Kim ^[15] obtained a global classical solution with $\mu = 0$ and a particular form of $\gamma(v) = \frac{c_0}{v^k}, c_0 > 0, k > 0$ in any dimensions provided $c_0$ is small. Moreover, Tao and Winkler ^[16] proved that some weak solutions exist globally under high dimensional conditions and in a specific three-dimensional case, this solution is bounded and classical with $\mu = 0$. We refer the readers to ^{[17,18,19,20,21,22,23,24]} for other interesting results on density-suppressed model.

Recently, this kind of model is also studied in the predator-prey mode ^[25,26]. In ^[26], the following density-dependent model with homogeneous Neumann boundary conditions is proposed

when $b_1 = c_2, c_1 = c, b_2 = b$ and $m(x) = 1$, the model (1.4) exists the global bounded classical solution, and asymptotic behavior is derived in different parameter regimes. $d_i(w)(i = 1, 2)$ indicates the resource dependent diffusion rate of species with monotonic properties: $d'_i(w) < 0(i = 1, 2)$, which is consistent with the fact that predators reduce their random diffusion when encountering the prey observed by kareiva and odell ^[27]. The major difference between (1.1) and (1.4) is that the motility of immature predators are influenced by mature predators rather than prey and mature predators grow from immature predators. Hence, due to its biological significance, the density-dependent model has attracted the interest of many scholars.

The goal of this paper is to establish global existence and large time behavior of classical solutions to the model (1.1). We shall suppose that there exist $\eta_2 > \eta_1 > 0$ such that $d_1(w)$ and $d_2(u)$ satisfy

$(H_1) \; d_1(w) \in C^3\left([0, \infty)\right), d_1(w) > 0$ and ${d'_{1}}(w)\leq 0$ for all $w\geq0$,

$(H_2) \; d_2(u) \in C^3\left([0, \infty)\right), {\eta_1} \le d_2(u) \le {\eta_2}$ for all $u\geq0$.

In this paper, the main results are stated as below. Our first result derives global boundedness of classical solution to (1.1).

Theorem 1.1. Let $\Omega\subset \mathbb{R}$ be a bounded domain with smooth boundary and the assumptions ${(H_1)}-{(H_2)}$ hold. Suppose that the parameters $a, b, c, k, r > 0$ and $({u_0}, {v_0}, {w_0}) \in {\left[{W^{1, \infty}}(\Omega)\right]^3}$ with ${u_0}, {v_0}, {w_0}\ge 0$. Then the model (1.1) has a unique nonnegative classical solution $(u, v, w)$ satisfying

where $C > 0$ is a constant. Particularly, we have $0\leq w \le M$, where

The second result is that we consider the global stability of the classical solution obtained in Theorem 1.1.

Theorem 1.2. Let $({u_0}, {v_0}, {w_0}) \in {\left[{W^{1, \infty}}(\Omega)\right]^3}$ with ${u_0}, {v_0} \ge 0(\not \equiv 0)$ and ${w_0} > 0$ in $\bar{\Omega}$. The solution $(u, v, w)$ of (1.1) obtained in Theorem 1.1 has the following properties: If the positive parameters $a, b, c, k$ and $r$ satisfy $\frac{c-abk}{kcar} > 1$, then

In the paper, for simplicity, we abbreviate $\int_{0}^{t} \int_\Omega f(\cdot, s)dxds$, $\int_\Omega f(\cdot, s)ds$, $\|\cdot\|_{{L^p}(\Omega)}$ and $\|\cdot \|_{W^{{1, p}}(\Omega)}$ as $\int_{0}^{t} \int_\Omega f$, $\int_\Omega f$, $\|\cdot\|_p$ and $\|\cdot\|_{1, p}$, respectively. Moreover, $C$ stands for a generic positive constant which may alter from line to line and is independent of time.

The organizational structure of this paper is as below. In Section 2, we show the local existence of a solution to (1.1) and some preliminary results are given. In Section 3, we establish global existence and boundedness for the model (1.1) and proof of Theorem 1.1. Section 4, we obtain the prey-only global stability to achieve Theorem 1.2.

2.   Preliminaries

We first give the existence of local solutions of (1.1) by using Amann's theorem ^[28,29](cf. also[30,Lemma 1.1] or [31,Lemma 2.6]).

Lemma 2.1. (Local existence).Let $\Omega\subset \mathbb{R}$ be a bounded domain with smooth boundary. Suppose that the parameters $a, b, c, k, r > 0$ and the assumptions ${(H_1)}-{(H_2)}$ hold. Assume that $({u_0}, {v_0}, {w_0}) \in {\left[{W^{1, \infty}}(\Omega)\right]^3}$ with ${u_0}, {v_0}, {w_0}\ge 0$. Then there exists a constant ${T_{\max }} \in (0, \infty ]$ such that the problem (1.1) has a unique nonnegative classical solution $(u, v, w)$ and satisfies

and which is such that if $T_{max} < \infty,$

Moreover, if the initial data $({u_0}, {v_0}, {w_0}) \in {\left[{W^{1, \infty}}(\Omega)\right]^3}$ with ${u_0}, {v_0} \ge 0(\not \equiv 0)$ and ${w_0} > 0$ in $\bar{\Omega}$, then the solution of (1.1) satisfies $u, v, w > 0$ in $\overline{\Omega}\times(0, T_{max})$.

Lemma 2.2. ([32,Lemma 2.2]) Let the assumptions in Lemma 2.1 hold. Then the solution $(u, v, w)$ of system (1.1) fulfills that

where $M: = \max\{a, \|w_0\|_{L^\infty}\}$, and it also founds that

In order to prove our results, we will quote the following lemma.

Lemma 2.3. ([33,Lemma 2.3]) Let $T > 0$ and $\tau \in (0, T)$, assume that $a, b > 0$, and $y : [0, T)\rightarrow [0, \infty)$ is absolutely continuous and satisfies

with some nonnegative function $b(t) \in L^1_{loc}([0, T))$ fulfilling

Then

Lemma 2.4. ([34,Lemma 2.4]) Let $T > 0$ and $\tau \in (0, T)$, assume that $\alpha, \beta > 0$, and $y :[0, T) \rightarrow [0, \infty)$ is absolutely continuous and satisfies

with the nonnegative functions $a(t), b(t), c(t) \in L^{1}_{loc}([0, T))$ fulfilling

and

Moreover, there also exists a positive constant $\rho$ satisfies

Then

Lemma 5. Under the assumptions in Theorem 1.1, the solution $(u, v, w)$ of (1.1) fulfills

where $C > 0$ is a constant.

Proof. The first equation of (1.1) adds the second equation of (1.1) multiplied by $b+1$ and adds the third equation of (1.1) multiplied by $k(b+1)$, then integrating we have

Using Gronwall's inequality to (2.4), we obtain (2.3) immediately.

Next, we shall obtain $W^{1, p}$ bound for the prey $w(\cdot, t)$.

Lemma 2.6. Under the assumptions in Theorem 1.1 and $(u, v, w)$ is a solution of (1.1), for any $p > 1$, there exists a constant $C > 0$ such that

Proof. By the variation-of-constants method, $w$ can be written as

using (2.1) and (2.3), then there exists a constant $c_1 > 0$ satisfies

According to standard $L^p-L^q$ estimates in [35,Lemma 1.3], there exist $\lambda > 0$ and some constants $c_i > 0(i = 2, 3)$ such that

for all $t\in (0, T_{max})$. Hence, the proof of (2.5) is completed.

Next, we apply the method of [25,Lemma 2.3] to obtain the following estimates.

Lemma 2.7. Under the conditions in Theorem 1.1 and $(u, v, w)$ is a solution of (1.1). Then there exists a constant $C > 0$ such that

where $\tau = \min\left\{1, \frac{T_{max}}{2}\right\}$.

Proof. Let $\mathcal{A}$ represents the self-adjoint realization of $-\Delta + \delta$ ([36,Lemma 3.1]) under homogeneous Neumann boundary conditions in $L^2(\Omega)$ and

where $\eta_2 > 0$ is from $(H_2)$ and

due to $(H_1)$ and Lemma 2.2. Since $\delta > 0$, $\mathcal{A}$ has an order-preserving bounded inverse $\mathcal{A}^{-1}$ on $L^2(\Omega)$, then there exists a constant $c_1 > 0$ such that

and

From (1.1), we have

which can be rewritten as

Noting the facts (2.1), (2.8) and $(H_1)-(H_2)$, one can find $c_2: = kM(b+1)(\delta d_3+a) > 0$ such that

Substituting (2.12) into (2.11), one has

hence, multiplying the above inequality by $\mathcal{A}^{-1}(u + (b+1)v+ k(b+1)w) \geq 0$, we have

which together with the fact $(H_1)-(H_2)$, we can find $d_1(M) = \min\limits_{0\leq w \leq M} d_1(w)$ and $c_3 : = \min\{d_1(M), \eta_1, d_3\}$ such that

By (2.9) and (2.10), we can obtain that

Therefore, combining with (2.13), and denoting $y_1(t) : = \int_{\Omega}|\mathcal{A}^{-\frac{1}{2}}(u + (b+1)v+ k(b+1)w)|^2$, one has

Then using Gronwall's inequality implies $y_1(t) \leq c_4$ with some constant $c_4 > 0$, thus

because $\tau\leq 1$. Similarly, we have

Hence, we can obtain (2.7).

In addition, as the result of Lemma 2.7, we can deduce the following results.

Lemma 2.8. Under the conditions in Theorem 1.1 and $(u, v, w)$ is a solution of (1.1). Then there exists a constant $C > 0$ such that

where $\tau = \min\{1, \frac{T_{max}}{2}\}$.

Proof. Testing the third equation of (1.1) by $-w_{xx}$, using Young's inequality and (2.1), we have

where $c_1: = \frac{2M^2|\Omega|(a^2+M^2)}{d_3}$, which yields

By the Gagliardo-Nirenberg inequality and the fact $\|w\|_2 \leq M|\Omega|^\frac{1}{2}$, there exist some constants $c_2, c_3 > 0$ such that

Combining (2.15) and (2.16), let $c_4: = 2c_1+c_3$, then we have

Let $y(t) : = \int _{\Omega}w_x^2$ and $b(t) : = \frac{4M^2}{d_3}\int _{\Omega}u^2+\frac{4r^2M^2}{d_3}\int _{\Omega}v^2+c_4$. From (2.17) we have

by Lemma 2.7 implies there exists a constant $c_5 > 0$ such that $\int_{t}^{t+\tau}\int _{\Omega}(u^2+v^2)\leq c_5$, therefore, we have

because $\tau\leq1$. Using (2.18) and Lemma 2.3 to ensure that

Therefore, an integration of (2.18) over $(t, t + \tau)$ yields

for all $t\in (0, T_{max}-\tau)$, which in view of the nonnegativity of $y$ implies (2.14).

3.   Boundedness of solutions

In the first, we will obtain a priori $L^2-estimate$ of the predator $u$.

Lemma 3.1. Let the assumptions in Theorem 1.1 hold, then there exists a constant $C > 0$ such that

Proof. Testing the first equation of (1.1) by $u$, integrating the result by part and using Young's inequality, we obtain

which yields

by Lemma 2.6 implies $\|w_x\|_2 \leq c_1$ with some $c_1 > 0$, thus using $(H_1)$ and (2.1), we have from (3.2) that

where $K_1: = \frac{\mathop {\max }\limits_{0 \le w \le M}|d_{1}'(w)|}{\sqrt {d_{1}(M)}}$. By the Gagliardo-Nirenberg inequality, Young's inequality and (2.3), there exist constants $c_i > 0(i = 2, 3)$ satisfy

This together with (3.3), one has

Using (2.7) and Lemma 2.3, we derive (3.1).

We are now in the position to derive some estimates for $u$.

Lemma 3.2. Let the assumptions in Theorem 1.1 hold and $(u, v, w)$ be a solution of (1.1). Then there exists a constant $C > 0$ such that

and

where $\tau = \min\{1, \frac{T_{max}}{2}\}$.

Proof. Testing the first equation of (1.1) by $-u_{xx}$ and using Young's inequality, we have

From Lemma 2.6, we choose $p = 2, 4$, then there exist $c_1, c_2 > 0$ such that $\|w_x\|_2 \leq c_1, \|w_x\|_4 \leq c_2$, we obtain

where $K_2: = \frac{\mathop {\max }\limits_{0 \le w \le M}|d_{1}''(w)|}{\sqrt {d_{1}(M)}}$. Using the Gagliardo-Nirenberg inequality and Lemma 3.1, for each $\varepsilon > 0$ one can find some $c_\varepsilon > 0$ and $c_i > 0(i = 3, 4, 5)$ such that

and

Using Lemma 3.1 again, for some $c_6 > 0$, we have $\|u(\cdot, t)\|_{2} \leq c_6$. Substituting (3.9)-(3.10) into (3.8), we obtain

where $c_7: = c_5+4K_2^2c_2^4c_\varepsilon+\frac{4c^2c_6^2}{d_1(M)}$. Using (3.1), for some $c_8, c_9 > 0$, we obtain

Combining it with (3.11), there exists $c_{10} > 0$ satisfying

From Lemma 2.8, one has $\int_{t}^{t+\tau}\int_{\Omega} w_{xx}^{2} \leq c_{11}$ with some $c_{11} > 0$. Let $a(t): = 1, b(t): = 4K_{1}^{2}\varepsilon\|w_{xx}\|_2^2$ and $c(t) : = 4K_{1}^{2}c_\varepsilon\|w_{xx}\|_2^2+\frac{4b^2}{d_1(M)}\int_{\Omega}{v^2} +c_9$, choosing $\varepsilon = \frac{\tau}{8K_{1}^{2}c_{11}} > 0$ such that $\int_{t}^{t+\tau}a(s)ds - \int_{t}^{t+\tau}b(s)ds = \frac{\tau}{2} > 0$. Hence, using Lemma 2.4, we can derive the boundedness of $\int_\Omega u_x^2(\cdot, t)$ for all $t\in (0, T_{max})$. Furthermore, (3.6) can be obtained upon an integration in time for (3.12). Finally, using the boundedness of $\int_\Omega u_x^2(\cdot, t)$ and (3.9), which implies (3.7).

Now we establish some estimates of $v$.

Lemma 3.3. Let the assumptions in Theorem 1.1 hold, then there exists a constant $C > 0$ such that

and

where $\tau = \min\left\{1, \frac{T_{max}}{2}\right\}$.

Proof. Testing the second equation of (1.1) by $v$, integrating and using Young's inequality, we have

From (3.7), we can find a constant $u^* > 0$ such that $0\leq u\leq$ ess $\sup_\Omega u = \|u\|_{\infty}\leq u^*$. Using $(H_2)$, which yields

where $K_3: = \frac{\mathop {\max }\limits_{0 \le u \le u^*}|d_{2}'(u)|}{\sqrt {\eta_1}}$. Using the Gagliardo-Nirenberg inequality, there exist some constants $c_i > 0(i = 1, 2, 3)$ such that

and

where we use Young's inequality and the Cauchy-Schwarz inequality. Using (3.5), there exists a constant $c_4 > 0$ such that

By Lemma 3.1, there exists $c_5 > 0$ such that $\|u(\cdot, t)\|_2 \leq c_5$. Substituting (3.16) and (3.18) into (3.15) and using Young's inequality, we have

where $c_6: = k^2M^2c_5^2+\frac{2K_3^2c_2c_4\eta_1+4K_3^4c_2^2c_4^2+K_3^4c_2^2c_4^2\eta_1}{\eta_1},$ which yields

Using (3.6) and Lemma 2.3, we derive (3.13) and (3.14).

Finally, we shall establish the estimate of $\|v(\cdot, t)\|_{\infty}$.

Lemma 3.4. Let the assumptions in Theorem 1.1 hold, then there exists a constant $C > 0$ such that

Proof. Testing the second equation of (1.1) by $-v_{xx}$ and using Young's inequality yields

From Lemmata 2.5, 3.2 and 3.3, there exists $c_i > 0(i = 1, 2, 3)$ satisfies $\|v(\cdot, t)\|_1\leq c_1$, $\|u_x(\cdot, t)\|_2^2\leq c_2$, $\|v(\cdot, t)\|_2^2\leq c_3$. Using the Gagliardo-Nirenberg inequality, for each $\varepsilon > 0$ one can find some $c_\varepsilon > 0$ and $c_i > 0(i = 4, 5, 6, 7, 8)$ such that

and

as well as

From Lemma 3.1, for some $c_9 > 0$, one has $\|u(\cdot, t)\|_2\leq c_9$. Combining with the above inequalities and using $(H_2)$, we conclude

where $K_4: = \frac{\mathop {\max }\limits_{0 \le u \le u^*}|d_{2}''(u)|}{\sqrt {\eta_1}}$ and $c_{10}: = 16K_3^{2}c_2c_6+4K_4^{2}c_3c_8+\frac{4k^2M^2c_9^2}{\eta_1}$. From (3.6), one has $\int_{t}^{t+\tau}\int_{\Omega} u_{xx}^{2} \leq c_{11}$ with some $c_{11} > 0$. Using Lemma 2.4, denoting $a(t): = 2$, $b(t): = 4K_3^{2}\varepsilon\|u_{xx}\|_2^2$ and $c(t): = \left(4K_3^{2}c_{\varepsilon}+4K_4^{2}c_3c_8\right)\|u_{xx}\|_2^2+c_{10}$, choosing $\varepsilon = \frac{\tau}{4K_3^2c_{11}}$ such that $\int_{t}^{t+\tau}a(s)ds - \int_{t}^{t+\tau}b(s)ds = \tau > 0$, therefore, we derive the boundedness of $\int_\Omega v_x^2(\cdot, t)$. Finally, using the boundedness of $\int_\Omega v_x^2(\cdot, t)$ and the Gagliardo-Nirenberg inequality, we obtain (3.20).

We can now easily prove Theorem 1.1.

Proof of Theorem 1.1. From Lemmata 3.1 and 3.3, there exists a constant $C > 0$ satisfies $\|u(\cdot, t)\|_2+\|v(\cdot, t)\|_2\leq C$ for all $t\in(0, T_{max})$, then we have $\|w(\cdot, t)\|_{1, \infty}\leq C$ ([32,Lemma 3.1]), combining Lemmata 3.2, 3.4 and 2.1, we can obtain Theorem 1.1 immediately.

4.   Global stability

In this section, we shall construct appropriate Lyapunov functional to derive the global stability in Theorem 1.2.

Lemma 4.1. ([26,Lemma 3.6]) Let the assumptions in Theorem 1.2 hold, then there exist $\theta \in (0, 1)$ and $C > 0$ such that

Lemma 4.2. Let $a, b, c, k > 0$. Then

and

for all $t > 0.$

Proof. Integrating the three equations of (1.1), respectively, we obtain (4.2)-(4.4).

Lemma 4.3. Let $a, r, d_3 \geq 0$. Then

for all $t > 0.$

Proof. By the third equation in (1.1), we make use of the positivity of $w$ in $\bar{\Omega} \times (0, \infty)$ to see that

for all $t > 0$.

Combining Lemmata 4.2 and 4.3, when $\frac{c-abk}{kcar} > 1$, we have the following lemma.

Lemma 4.4. Under the assumptions in Theorem 1.2, there exists $C > 0$ such that

and

as well as

for all $t > 0.$

Proof. Using Lemmata 4.2 and 4.3, we have

for all $t > 0$. Since $\frac{c-abk}{kcar} > 1$, integrating (4.10) on $[ 0, t)$ to obtain

for all $t > 0$. Due to $\ln w \leq w$ for all $w > 0$, one has

for all $t > 0$, which implies (4.8) and (4.9) hold. Integrating (4.2) on $[0, t)$ to obtain

Using (4.8), we can obtain (4.7). The proof is completed.

Lemma 4.5. Under the assumptions in Theorem 1.2, there exists $C > 0$ satisfies

and

for all $t > 0.$

Proof. By (3.7) and (3.20), for some $c_1, c_2 > 0$, we have $\|u(\cdot, t)\|_{\infty} \leq c_1, \|v(\cdot, t)\|_{\infty} \leq c_2$. Testing the first and second equations in (1.1) by $u$ and $v$, respectively, using Young's inequality and integrating to see that

and

Since $(H_1)-(H_2)$, (2.1) and (3.7), which yields

Then using (4.7) and (4.9) imply (4.13) and (4.14).

Lemma 4.6. Let the assumptions in Theorem 1.2 hold, the solution of (1.1) satisfies

and

Proof. Suppose that (4.17) is false, for some $c_1 > 0$, there exist $(x_i)_{i\in N} \subset \Omega$ and $(t_i)_{i\in N} \subset (1, \infty)$ satisfying $t_i \rightarrow \infty$ as $i \rightarrow \infty$ such that

From Lemma 4.1, we know that $u$ is uniformly continuous in $\Omega \times (1, \infty)$, therefore, for any $i\in N$, we can find some $r_1 > 0$ and $\tau_1 > 0$ such that

where $L_{r_1}(x_i)$ denotes a line segment with $x_i$ as the center, $r_1$ as the radius and $2r_1$ in total length and hence

where $c_2 : = \inf_{i\in N}|L_{r_1}(x_i)\cap \Omega|$ is positive due to smoothness of $\partial\Omega$. By Lemma 4.5, we have

Together with (4.20), this leads to a contradiction, thus (4.17) is established. Similarly, we can obtain (4.18) and (4.19) immediately.

Proof of Theorem 1.2. Lemma 4.6 derives the conclusions of Theorem 1.2.

Acknowledgments

The authors are very grateful to the anonymous reviewers for their carefully reading and valuable suggestions which greatly improved this work. L. Wang is supported by Natural Science Foundation of Chongqing (No. cstc2021jcyj-msxmX0412) and China Scholarship Council (202108500085).

Conflict of interest

The authors declare there is no conflicts of interest.