Dynamical analysis of a predator-prey system with prey vigilance and hunting cooperation in predators

Eric M. Takyi; Charles Ohanian; Margaret Cathcart; Nihal Kumar; Eric M. Takyi; Charles Ohanian; Margaret Cathcart; Nihal Kumar

doi:10.3934/mbe.2024123

Mathematical Biosciences and Engineering

2024, Volume 21, Issue 2: 2768-2786. doi: 10.3934/mbe.2024123

Previous Article Next Article

Research article

Dynamical analysis of a predator-prey system with prey vigilance and hunting cooperation in predators

1.
Department of Mathematics, Computer Science and Statistics, Ursinus College, Collegeville, PA 19426, USA
2.
Department of Mathematics and Computer Science, Muhlenberg College, Allentown, PA 18104, USA
3.
Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA
4.
Department of Mathematics, Penn State University, State College, PA 16802, USA

Academic Editor: Hao Wang

Received: 12 November 2023 Revised: 31 December 2023 Accepted: 17 January 2024 Published: 24 January 2024

In this work, we propose a predator-prey system with a Holling type Ⅱ functional response and study its dynamics when the prey exhibits vigilance behavior to avoid predation and predators exhibit cooperative hunting. We provide conditions for existence and the local and global stability of equilibria. We carry out detailed bifurcation analysis and find the system to experience Hopf, saddle-node, and transcritical bifurcations. Our results show that increased prey vigilance can stabilize the system, but when vigilance levels are too high, it causes a decrease in the population density of prey and leads to extinction. When hunting cooperation is intensive, it can destabilize the system, and can also induce bi-stability phenomenon. Furthermore, it can reduce the population density of both prey and predators and also change the stability of a coexistence state. We provide numerical experiments to validate our theoretical results and discuss ecological implications.

Keywords:

Citation: Eric M. Takyi, Charles Ohanian, Margaret Cathcart, Nihal Kumar. Dynamical analysis of a predator-prey system with prey vigilance and hunting cooperation in predators[J]. Mathematical Biosciences and Engineering, 2024, 21(2): 2768-2786. doi: 10.3934/mbe.2024123

Related Papers:

[1]	Bilal Akbar, Khuram Pervez Amber, Anila Kousar, Muhammad Waqar Aslam, Muhammad Anser Bashir, Muhammad Sajid Khan . Data-driven predictive models for daily electricity consumption of academic buildings. AIMS Energy, 2020, 8(5): 783-801. doi: 10.3934/energy.2020.5.783
[2]	Zhongjiao Ma, Zichun Yan, Mingfei He, Haikuan Zhao, Jialin Song . A review of the influencing factors of building energy consumption and the prediction and optimization of energy consumption. AIMS Energy, 2025, 13(1): 35-85. doi: 10.3934/energy.2025003
[3]	Kaovinath Appalasamy, R Mamat, Sudhakar Kumarasamy . Smart thermal management of photovoltaic systems: Innovative strategies. AIMS Energy, 2025, 13(2): 309-353. doi: 10.3934/energy.2025013
[4]	Feng Tian . A wind power prediction model with meteorological time delay and power characteristic. AIMS Energy, 2025, 13(3): 517-539. doi: 10.3934/energy.2025020
[5]	Xiaojing Tian, Weiqi Ye, Liang Xu, Anjian Yang, Langming Huang, Shenglong Jin . Optimization research on laminated cooling structure for gas turbines: A review. AIMS Energy, 2025, 13(2): 354-401. doi: 10.3934/energy.2025014
[6]	Emrah Gulay . Forecasting the total electricity production in South Africa: Comparative analysis to improve the predictive modelling accuracy. AIMS Energy, 2019, 7(1): 88-110. doi: 10.3934/energy.2019.1.88
[7]	Rahaman Abu, John Amakor, Rasaq Kazeem, Temilola Olugasa, Olusegun Ajide, Nosa Idusuyi, Tien-Chien Jen, Esther Akinlabi . Modeling influence of weather variables on energy consumption in an agricultural research institute in Ibadan, Nigeria. AIMS Energy, 2024, 12(1): 256-270. doi: 10.3934/energy.2024012
[8]	Yan Li, Yaheng Su, Qixin Zhao, Bala Wuda, Kaibo Qu, Lei Tang . An electricity price optimization model considering time-of-use and active distribution network efficiency improvements. AIMS Energy, 2025, 13(1): 13-34. doi: 10.3934/energy.2025002
[9]	Anbazhagan Geetha, S. Usha, J. Santhakumar, Surender Reddy Salkuti . Forecasting of energy consumption rate and battery stress under real-world traffic conditions using ANN model with different learning algorithms. AIMS Energy, 2025, 13(1): 125-146. doi: 10.3934/energy.2025005
[10]	Ghafi Kondi Akara, Benoit Hingray, Adama Diawara, Arona Diedhiou . Effect of weather on monthly electricity consumption in three coastal cities in West Africa. AIMS Energy, 2021, 9(3): 446-464. doi: 10.3934/energy.2021022

Abstract

1. Introduction

In the present paper, we study the following coupled nonlinear wave equations (see ^[1])

$\begin{align} u_{tt}-div\Big(\rho\Big(|\nabla u|^2\Big)\nabla u\Big)+|u_t|^{m-1}u_t+f_1(u, v)& = 0, \end{align}$

(1.1)

$\begin{align} v_{tt}-div\Big(\rho\Big(|\nabla v|^2\Big)\nabla v\Big)+|v_t|^{n-1}v_t+f_2(u, v)& = 0, \end{align}$

(1.2)

where $u_t = \frac{\partial u}{\partial t}, v_t = \frac{\partial v}{\partial t}$ , $m, n > 1$ , $\nabla$ is the gradient operator, div is the divergence operator and $f_i(., .):\mathbb{R}^2\rightarrow \mathbb{R}, i = 1, 2$ are known functions. For arbitrary solutions of (1.1) and (1.2), the function $\rho$ is supposed to satisfy one or the other of the two conditions:

A1). $0 < \rho^2(q^2)q^2\leq m_1\widetilde{\rho}(q^2),$

A2). $0 < \frac{1}{K_1} < \rho^{-1}(q^2)\leq K_2\widetilde{\rho}(q^2)\frac{1}{q^2},$

where $\widetilde{\rho}(q^2) = \int_0^{q^2}\rho(\zeta)d\zeta, \ q^2 = |\nabla u|^2, m_1, K_1, K_2 > 0.$

Quintanilla ^[2] imposed condition A1 and obtained the growth or decay estimates of the solution to the type Ⅲ heat conduction. The condition A1 can be satisfied easily, e.g., $\rho(q^2) = \frac{1}{\sqrt{1+q^2}}$ or $\rho(q^2) = (1+\frac{b}{p}q^2)^{p-1}, b > 0, 0 < p\leq1.$ The similar condition as A2 was also considered by many papers (see ^[3,4]). $\rho(q^2) = \sqrt{1+q^2}$ satisfies the condition A2.

In addition, we introduce a function $F(u, v)$ which is defined as $\frac{\partial F}{\partial u} = f_1(u, v), \ \frac{\partial F}{\partial v} = f_2(u, v),$ where $F(0, 0) = 0.$

The wave equations have attracted many attentions of scholars due to their wide application, and a large number of achievements have been made in the existence of solutions (see ^{[1,5,6,7,8,9,10,11,12]}). The fuzzy inference method is used to solve this problem. The algebraic formulation of fuzzy relation is studied in ^[13,14]. In this paper, we study the Phragmén-Lindelöf type alternative property of solutions of wave equations (1.1)–(1.2). It is proved that the solution of the equations either grows exponentially (polynomially) or decays exponentially (polynomially) when the space variable tends to infinite. In the case of decay, people usually expect a fast decay rate. The Phragmén-Lindelöf type alternative research on partial differential equations has lasted for a long time (see ^{[2,15,16,17,18,19,20,21,22,23]}).

It is worth emphasizing that Quintanilla ^[2] considered an exterior or cone-like region. Under some appropriate conditions, the growth/decay estimates of some parabolic problems are obtained. Inspired by ^[2], we extends the research of to the nonlinear wave model in this paper. However, different from ^[2], in addition to condition A1 and condition A2, we also consider a special condition of $\rho$ . The appropriate energy function is established, and the nonlinear differential inequality about the energy function is derived. By solving this differential inequality, the Phragmén-Lindelöf type alternative results of the solution are obtained. Our model is much more complex than ^[2], so the methods used in ^[2] can not be applied to our model directly. Finally, a nonlinear system of viscoelastic type is also studied when the system is defined in an exterior or cone-like regions and the growth or decay rates are also obtained.

2. The Phragmén-Lindelöf type alternative result under A1

Letting that $\Omega(r)$ denotes a cone-like region, i.e.,

$\Omega(r) = \Big\{\mathit{\boldsymbol{x}}\Big||\mathit{\boldsymbol{x}}|^2\geq r^2, r\geq R_0 > 0\Big\},$

and that $B(r)$ denotes the exterior surface to the sphere, i.e.,

$B(r) = \Big\{\mathit{\boldsymbol{x}}\Big||\mathit{\boldsymbol{x}}|^2 = r^2, r\geq R_0 > 0\Big\},$

Equations (1.1) and (1.2) also have the following initial-boundary conditions

$\begin{align} u(\mathit{\boldsymbol{x}}, 0) = v(\mathit{\boldsymbol{x}}, 0) = 0, \ & in\ \Omega, \end{align}$

(2.1)

$\begin{align} u(\mathit{\boldsymbol{x}}, t) = g_1(\mathit{\boldsymbol{x}}, t), \ v(\mathit{\boldsymbol{x}}, t) = g_2(\mathit{\boldsymbol{x}}, t), \ & in\ B(R_0)\times(0, \tau), \end{align}$

(2.2)

where $g_1$ and $g_2$ are positive known functions, $\textbf{x} = (x_1, x_2, x_3)$ and $\tau > 0$ .

Now, we establish an energy function

$\begin{align} \mathcal{E}(r, t)& = \int_0^t\int_{B(r)}e^{-\omega\eta}\rho\Big(|\nabla u|^2\Big)\nabla u\cdot \frac{\mathit{\boldsymbol{x}}}{r}u_\eta dSd\eta +\int_0^t\int_{B(r)}e^{-\omega\eta}\rho\Big(|\nabla v|^2\Big)\nabla v\cdot\frac{\mathit{\boldsymbol{x}}}{r}v_\eta dSd\eta\\ &\doteq \mathcal{E}_1(r, t)+\mathcal{E}_2(r, t). \end{align}$

(2.3)

Let $r_0$ be a positive constant which satisfies $r > r_0\geq R_0$ . Integrating $\mathcal{E}(z, t)$ from $r_0$ to $r$ , using the divergence theorem and Eqs (1.1) and (1.2), (2.1) and (2.2), we have

$\begin{align} \mathcal{E}(r, t)&-\mathcal{E}(r_0, t) = \int_0^t\int_{r_0}^r\int_{B(\xi)}e^{-\omega \eta}\nabla\cdot\Big[\rho\Big(|\nabla u|^2\Big)\nabla uu_\eta\Big] dsd\xi d\eta \\ &+\int_0^t\int_{r_0}^r\int_{B(\xi)}e^{-\omega \eta}\nabla\cdot\Big[\rho\Big(|\nabla v|^2\Big)\nabla vv_\eta\Big] dsd\xi d\eta \\ &+\int_0^t\int_{r_0}^r\int_{B(\xi)}e^{-\omega \eta}\Big[u_{\eta\eta}+|u_\eta|^{m+1}+f_1(u, v)\Big]u_\eta dsd\xi d\eta \\ &+\int_0^t\int_{r_0}^r\int_{B(\xi)}e^{-\omega \eta}\Big[v_{\eta\eta}+|v_\eta|^{n+1}+f_2(u, v)\Big]v_\eta dsd\xi d\eta \\ &+\frac{1}{2}\int_0^t\int_{r_0}^r\int_{B(\xi)}e^{-\omega \eta}\frac{\partial}{\partial \eta}\widetilde{\rho}\Big(|\nabla u|^2\Big) dsd\xi d\eta +\frac{1}{2}\int_0^t\int_{r_0}^r\int_{B(\xi)}e^{-\omega \eta}\frac{\partial}{\partial \eta}\widetilde{\rho}\Big(|\nabla v|^2\Big) dsd\xi d\eta \\ & = \frac{1}{2}e^{-\omega t}\int_{r_0}^r\int_{B(\xi)}\Big[|u_t|^2+|v_t|^2+\widetilde{\rho}\Big(|\nabla u|^2\Big)+\widetilde{\rho}\Big(|\nabla v|^2\Big)+2F(u, v)\Big]dsd\xi \\ &+\frac{1}{2}\omega\int_0^t\int_{r_0}^r\int_{B(\xi)}e^{-\omega\eta}\Big[|u_\eta|^2+|v_\eta|^2+\widetilde{\rho}\Big(|\nabla u|^2\Big)+\widetilde{\rho}\Big(|\nabla v|^2\Big)+2F(u, v)\Big]dSd\xi d\eta \\&+\int_0^t\int_{r_0}^r\int_{B(\xi)}e^{-\omega\eta}\Big[|u_\eta|^{m+1}+|v_\eta|^{n+1}\Big]dsd\xi d\eta, \end{align}$

(2.4)

from which it follows that

$\begin{align} \frac{\partial}{\partial r}\mathcal{E}(r, t)& = \frac{1}{2}e^{-\omega t}\int_{B(r)}\Big[|u_t|^2+|v_t|^2+\widetilde{\rho}\Big(|\nabla u|^2\Big)+\widetilde{\rho}\Big(|\nabla v|^2\Big)+2F(u, v)\Big]ds \\ &+\frac{1}{2}\omega\int_0^t\int_{B(r)}e^{-\omega\eta}\Big[|u_\eta|^2+|v_\eta|^2+\widetilde{\rho}\Big(|\nabla u|^2\Big)+\widetilde{\rho}\Big(|\nabla v|^2\Big)+2F(u, v)\Big]dS d\eta \\&+\int_0^t\int_{B(r)}e^{-\omega\eta}\Big[|u_\eta|^{m+1}+|v_\eta|^{n+1}\Big]dsd\eta, \end{align}$

(2.5)

where $\omega$ is positive constant.

Now, we show how to bound $\mathcal{E}(r, t)$ by $\frac{\partial}{\partial r}\mathcal{E}(r, t)$ . We use the Hölder inequality, the Young inequality and the condition A1 to have

$\begin{align} \Big|\mathcal{E}_1(r, t)\Big|&\leq\Big[\int_0^t\int_{B(r)}e^{-\omega\eta}\rho^2\Big(|\nabla u|^2\Big)|\nabla u|^2ds d\eta\cdot\int_0^t\int_{B(r)}e^{-\omega\eta}u_\eta^2ds d\eta\Big]^\frac{1}{2}\\ &\leq\sqrt{m_1}\Big[\int_0^t\int_{B(r)}e^{-\omega\eta}\widetilde{\rho}\Big(|\nabla u|^2\Big)ds d\eta\cdot\int_0^t\int_{B(r)}e^{-\omega\eta}u_\eta^2ds d\eta\Big]^\frac{1}{2}\\ &\leq\frac{\sqrt{m_1}}{2}\Big[\int_0^t\int_{B(r)}e^{-\omega\eta}\widetilde{\rho}\Big(|\nabla u|^2\Big)ds d\eta+\int_0^t\int_{B(r)}e^{-\omega\eta}u_\eta^2ds d\eta\Big], \end{align}$

(2.6)

and

$\begin{align} \Big|\mathcal{E}_2(r, t)\Big|&\leq\frac{\sqrt{m_1}}{2}\Big[\int_0^t\int_{B(r)}e^{-\omega\eta}\widetilde{\rho}\Big(|\nabla v|^2\Big)ds d\eta+\int_0^t\int_{B(r)}e^{-\omega\eta}v_\eta^2dsd\eta\Big]. \end{align}$

(2.7)

Inserting (2.6) and (2.7) into (2.3) and combining (2.5), we have

$\begin{align} \Big|\mathcal{E}(r, t)\Big|\leq \frac{\sqrt{m_1}}{\omega}\Big[\frac{\partial}{\partial r}\mathcal{E}(r, t)\Big]. \end{align}$

(2.8)

We consider inequality (2.8) for two cases.

I. If $\exists r_0 > R_0$ such that $\mathcal{E}(r_0, t)\geq0$ . From (2.5), we obtain $\mathcal{E}(r, t)\geq\mathcal{E}(r_0, t)\geq0, r\geq r_0$ . Therefore, from (2.8) we have

$\begin{align} \mathcal{E}(r, t)\leq \frac{\sqrt{m_1}}{\omega}\Big[\frac{\partial}{\partial r}\mathcal{E}(r, t)\Big], r\geq r_0. \end{align}$

(2.9)

Integrating (2.9) from $r_0$ to $r$ , we have

$\begin{align} \mathcal{E}(r, t)\geq\Big[\mathcal{E}(r_0, t)\Big]e^{\frac{\omega}{\sqrt{m_1}}(r-r_0)}, r\geq r_0. \end{align}$

(2.10)

Combing (2.4) and (2.10), we have

$\begin{align} \lim\limits_{r\rightarrow \infty}&\Big\{e^{-\frac{\omega}{\sqrt{m_1}}r}\Big[\frac{1}{2}e^{-\omega t}\int_{r_0}^r\int_{B(\xi)}\Big[|u_t|^2+|v_t|^2+\widetilde{\rho}\Big(|\nabla u|^2\Big)+\widetilde{\rho}\Big(|\nabla v|^2\Big)+2F(u, v)\Big]dsd\xi \\ &+\frac{1}{2}\omega\int_0^t\int_{r_0}^r\int_{B(\xi)}e^{-\omega\eta}\Big[|u_\eta|^2+|v_\eta|^2+\widetilde{\rho}\Big(|\nabla u|^2\Big)+\widetilde{\rho}\Big(|\nabla v|^2\Big)+2F(u, v)\Big]dsd\xi d\eta \\&+\int_0^t\int_{r_0}^r\int_{B(\xi)}e^{-\omega\eta}\Big[|u_\eta|^{m+1}+|v_\eta|^{n+1}\Big]dsd\xi d\eta\Big]\Big\} \\ &\geq \mathcal{E}(R_0, t)e^{-\frac{\omega}{\sqrt{m_1}}R_0}. \end{align}$

(2.11)

II. If $\forall r > R_0$ such that $\mathcal{E}(r, t) < 0$ . The inequality (2.8) can be rewritten as

$\begin{align} -\mathcal{E}(r, t)\leq \frac{\sqrt{m_1}}{\omega}\Big[\frac{\partial}{\partial r}\mathcal{E}(r, t)\Big], r\geq R_0. \end{align}$

(2.12)

Integrating (2.12) from $r_0$ to $r$ , we have

$\begin{align} -\mathcal{E}(r, t)\geq\Big[-\mathcal{E}(R_0, t)\Big]e^{-\frac{\omega}{\sqrt{m_1}}(r-R_0)}, r\geq R_0. \end{align}$

(2.13)

Inequality (2.13) shows that $\lim_{r\rightarrow \infty}\Big[-\mathcal{E}(r, t)\Big] = 0$ . Integrating (2.5) from $r$ to $\infty$ and combining (2.13), we obtain

$\begin{align} \frac{1}{2}&e^{-\omega t}\int_{r}^\infty\int_{B(\xi)}\Big[|u_t|^2+|v_t|^2+\widetilde{\rho}\Big(|\nabla u|^2\Big)+\widetilde{\rho}\Big(|\nabla v|^2\Big)+2F(u, v)\Big]dsd\xi \\ &+\frac{1}{2}\omega\int_0^t\int_{r}^\infty\int_{B(\xi)}e^{-\omega\eta}\Big[|u_\eta|^2+|v_\eta|^2+\widetilde{\rho}\Big(|\nabla u|^2\Big)+\widetilde{\rho}\Big(|\nabla v|^2\Big)+2F(u, v)\Big]dsd\xi d\eta \\&+\int_0^t\int_{r}^\infty\int_{B(\xi)}e^{-\omega\eta}\Big[|u_\eta|^{m+1}+|v_\eta|^{n+1}\Big]dsd\xi d\eta \\ &\leq \Big[-\mathcal{E}(r_0, t)\Big]e^{-\frac{\omega}{\sqrt{m_1}}(r-R_0)}. \end{align}$

(2.14)

We summarize the above result as the following theorem.

Theorem 2.1. Let $(u, v)$ be solution of the (1.1), (1.2), (2.1), (2.2) in $\Omega(R_0)$ , and $\rho$ satisfies condition A1. Then for fixed $t$ , $(u, v)$ either grows exponentially or decays exponentially. Specifically, either (2.11) holds or (2.14) holds.

3. The Phragmén-Lindelöf type alternative result under A2

If the function $\rho$ satisfies the condition A2, we recompute the inequality (2.6) and (2.7). Therefore

$\begin{align} \Big|\mathcal{E}_1(r, t)\Big|&\leq\Big[\int_0^t\int_{B(r)}e^{-\omega\eta}\rho^2\Big(|\nabla u|^2\Big)|\nabla u|^2ds d\eta\cdot\int_0^t\int_{B(r)}e^{-\omega\eta}u_\eta^2ds d\eta\Big]^\frac{1}{2}\\ &\leq K_1\sqrt{K_1}\Big[\int_0^t\int_{B(r)}e^{-\omega\eta}\rho^{-1}\Big(|\nabla u|^2\Big)ds d\eta\cdot\int_0^t\int_{B(r)}e^{-\omega\eta}u_\eta^2ds d\eta\Big]^\frac{1}{2}\\ &\leq\frac{K_1\sqrt{K_1K_2}}{2}\Big[\int_0^t\int_{B(r)}e^{-\omega\eta}\widetilde{\rho}\Big(|\nabla u|^2\Big)ds d\eta+\int_0^t\int_{B(r)}e^{-\omega\eta}u_\eta^2ds d\eta\Big], \end{align}$

(3.1)

and

$\begin{align} \Big|\mathcal{E}_2(r, t)\Big|&\leq\frac{K_1\sqrt{K_1K_2}}{2}\Big[\int_0^t\int_{B(r)}e^{-\omega\eta}\widetilde{\rho}\Big(|\nabla v|^2\Big)ds d\eta+\int_0^t\int_{B(r)}e^{-\omega\eta}v_\eta^2dsd\eta\Big]. \end{align}$

(3.2)

Inserting (3.1) and (3.2) into (2.3) and combining (2.5), we have

$\begin{align} \Big|\mathcal{E}(r, t)\Big|\leq \frac{K_1\sqrt{K_1K_2}}{\omega}\Big[\frac{\partial}{\partial r}\mathcal{E}(r, t)\Big]. \end{align}$

(3.3)

By following a similar method to that used in Section 2, we can obtain the Phragmén-Lindelöf type alternative result.

Theorem 3.1. Let $(u, v)$ be solution of the (1.1), (1.2), (2.1), (2.2) in $\Omega(R_0)$ , and $\rho$ satisfies condition A1. Then for fixed $t$ , $(u, v)$ either grows exponentially or decays exponentially.

Remark 3.1. Clearly, the rates of growth or decay obtained in Theorems 2.1 and 3.1 depend on $\omega$ . Because $\omega$ can be chosen large enough, the rates of growth or decay of the solutions can become large as we want.

Remark 3.2. The analysis in Sections 2 and 3 can be adapted to the single-wave equation

$\begin{align} u_{tt}-div\Big(\rho\Big(|\nabla u|^2\Big)\nabla u\Big)+|u_t|^{m-1}u_t+f(u) = 0 \end{align}$

(3.4)

and the heat conduction at low temperature

$\begin{align} au_{tt}+bu_t-c\Delta u+\Delta u_t = 0, \end{align}$

(3.5)

where $a, b, c > 0$ .

4. The Phragmén-Lindelöf type alternative result when $\rho(q^2)=b_1+b_2q^{2\beta}$

In this section, we suppose that $\rho$ satisfies $\rho(q^2) = b_1+b_2q^{2\beta}$ , where $b_1, b_2$ and $\beta$ are positive constants. Clearly, $\rho(q^2) = b_1+b_2q^{2\beta}$ can not satisfy A1 or A2. In this case, we define an "energy" function

$\begin{align} \mathcal{F}(r, t)& = \int_0^t\int_{B(r)}e^{-\omega\eta}\Big(b_1+b_2|\nabla u|^{2\beta}\Big)\nabla u\cdot\frac{\mathit{\boldsymbol{x}}}{r}u_\eta dsd\eta \\ &+\int_0^t\int_{B(r)}e^{-\omega\eta}\Big(b_1+b_2|\nabla v|^{2\beta}\Big)\nabla v\cdot\frac{\mathit{\boldsymbol{x}}}{r}v_\eta dsd\eta \\ &\doteq \mathcal{F}_1(r, t)+\mathcal{F}_2(r, t). \end{align}$

(4.1)

Computing as that in (2.4) and (2.5), we can get

$\begin{align} \mathcal{F}(r, t)& = \mathcal{F}(r_0, t)+\frac{1}{2}e^{-\omega t}\int_{r_0}^r\int_{B(\xi)}\Big[|u_t|^2+b_1|\nabla u|^2+\frac{1}{\beta+1}b_2|\nabla u|^{2(\beta+1)}\Big]dsd\xi\\ &+\frac{1}{2}e^{-\omega t}\int_{r_0}^r\int_{B(\xi)}\Big[|v_t|^2+b_1|\nabla v|^2+\frac{1}{\beta+1}b_2|\nabla v|^{2(\beta+1)}+F(u, v)\Big]dsd\xi\\ &+\frac{1}{2}\omega\int_0^t\int_{r_0}^r\int_{B(\xi)}e^{-\omega\eta}\Big[|u_\eta|^2+b_1|\nabla u|^2+\frac{1}{\beta+1}b_2|\nabla u|^{2(\beta+1)}\Big]ds d\xi d\eta\\ &+\frac{1}{2}\omega\int_0^t\int_{r_0}^r\int_{B(\xi)}e^{-\omega\eta}\Big[|v_\eta|^2+b_1|\nabla v|^2+\frac{1}{\beta+1}b_2|\nabla v|^{2(\beta+1)}+F(u, v)\Big]dsd\xi d\eta\\ &+\int_0^t\int_{r_0}^r\int_{B(\xi)}e^{-\omega\eta}|u_\eta|^{m+1}dsd\xi d\eta+\int_0^t\int_{B(r)}e^{-\omega\eta}|v_\eta|^{n+1}dsd\xi d\eta, \end{align}$

(4.2)

and

$\begin{align} \frac{\partial}{\partial r}\mathcal{F}(r, t)& = \frac{1}{2}e^{-\omega t}\int_{B(r)}\Big[|u_t|^2+b_1|\nabla u|^2+\frac{1}{\beta+1}b_2|\nabla u|^{2(\beta+1)}\Big]ds\\ &+\frac{1}{2}e^{-\omega t}\int_{B(r)}\Big[|v_t|^2+b_1|\nabla v|^2+\frac{1}{\beta+1}b_2|\nabla v|^{2(\beta+1)}+F(u, v)\Big]ds\\ &+\frac{1}{2}\omega\int_0^t\int_{B(r)}e^{-\omega\eta}\Big[|u_\eta|^2+b_1|\nabla u|^2+\frac{1}{\beta+1}b_2|\nabla u|^{2(\beta+1)}\Big]ds d\eta\\ &+\frac{1}{2}\omega\int_0^t\int_{B(r)}e^{-\omega\eta}\Big[|v_\eta|^2+b_1|\nabla v|^2+\frac{1}{\beta+1}b_2|\nabla v|^{2(\beta+1)}+F(u, v)\Big]ds d\eta\\ &+\int_0^t\int_{B(r)}e^{-\omega\eta}|u_\eta|^{m+1}dsd\eta+\int_0^t\int_{B(r)}e^{-\omega\eta}|v_\eta|^{n+1}ds d\eta. \end{align}$

(4.3)

Using the Hölder inequality and Young's inequality, we have

$\begin{align} \Big|\mathcal{F}_1(r, t)\Big|&\leq b_1\Big[\int_0^t\int_{B(r)}e^{-\omega\eta}|\nabla u|^2dsd\eta \int_0^t\int_{B(r)}e^{-\omega\eta}u_\eta^2 dsd\eta\Big]^\frac{1}{2} \\ &+b_2\Big[\int_0^t\int_{B(r)}e^{-\omega\eta}|\nabla u|^{2(\beta+1)}dsd\eta\Big]^\frac{2\beta+1}{2(\beta+1)} \int_0^t\int_{B(r)}e^{-\omega\eta}|u_\eta|^{m+1} dsd\eta\Big]^\frac{1}{m+1} |t2\pi r|^{\frac{1}{2(\beta+1)}-\frac{1}{m+1}} \\ &\leq \frac{\sqrt{b_1}}{R_0\omega}\Big[\frac{b_1}{2}r\omega\int_0^t\int_{B(r)}e^{-\omega\eta}|\nabla u|^2dsd\eta+ \frac{1}{2}r\omega\int_0^t\int_{B(r)}e^{-\omega\eta}u_\eta^2 dsd\eta\Big] \\ &+b_2|2t\pi|^{\frac{1}{2(\beta+1)}-\frac{1}{m+1}}R_0^{-\frac{\beta}{\beta+1}-\frac{2}{m+1}}\Big[\frac{\frac{2\beta+1}{2(\beta+1)}r}{\frac{2\beta+1}{2(\beta+1)} +\frac{1}{m+1}}\int_0^t\int_{B(r)}e^{-\omega\eta}|\nabla u|^{2(\beta+1)}dsd\eta \\ &+\frac{\frac{1}{m+1}r}{\frac{2\beta+1}{2(\beta+1)} +\frac{1}{m+1}} \int_0^t\int_{B(r)}e^{-\omega\eta}|u_\eta|^{m+1} dsd\eta\Big]^{\frac{2\beta+1}{2(\beta+1)} +\frac{1}{m+1}}, \end{align}$

(4.4)

and

$\begin{align} \Big|\mathcal{F}_2(r, t)\Big|&\leq\frac{\sqrt{b_1}}{R_0\omega}\Big[\frac{b_1}{2}r\omega\int_0^t\int_{B(r)}e^{-\omega\eta}|\nabla v|^2dsd\eta+ \frac{1}{2}r\omega\int_0^t\int_{B(r)}e^{-\omega\eta}v_\eta^2 dsd\eta\Big] \\ &+b_2|2t\pi|^{\frac{1}{2(\beta+1)}-\frac{1}{n+1}}R_0^{-\frac{\beta}{\beta+1}-\frac{2}{n+1}} \Big[\frac{\frac{2\beta+1}{2(\beta+1)}r}{\frac{2\beta+1}{2(\beta+1)} +\frac{1}{n+1}}\int_0^t\int_{B(r)}e^{-\omega\eta}|\nabla v|^{2(\beta+1)}dsd\eta \\ &+\frac{\frac{1}{n+1}r}{\frac{2\beta+1}{2(\beta+1)} +\frac{1}{n+1}} \int_0^t\int_{B(r)}e^{-\omega\eta}|v_\eta|^{n+1} dsd\eta\Big]^{\frac{2\beta+1}{2(\beta+1)} +\frac{1}{n+1}}, \end{align}$

(4.5)

where we have chosen $\frac{1}{2(\beta+1)} > \frac{1}{n+1}$ . Inserting (4.4) and (4.5) into (4.1), we obtain

$\begin{align} \Big|\mathcal{F}(r, t)\Big|&\leq c_1\Big[r\frac{\partial}{\partial r}\mathcal{F}(r, t)\Big]+c_2\Big[r\frac{\partial}{\partial r}\mathcal{F}(r, t)\Big]^{\frac{2\beta+1}{2(\beta+1)} +\frac{1}{m+1}} +c_3\Big[r\frac{\partial}{\partial r}\mathcal{F}(r, t)\Big]^{\frac{2\beta+1}{2(\beta+1)} +\frac{1}{n+1}}, \end{align}$

(4.6)

where $c_1 = \frac{\sqrt{b_1}}{R_0\omega}, c_2 = b_2|2t\pi|^{\frac{1}{2(\beta+1)}-\frac{1}{n+1}}R_0^{-\frac{\beta}{\beta+1}-\frac{2}{m+1}}, c_3 = b_2|2t\pi|^{\frac{1}{2(\beta+1)}-\frac{1}{n+1}}R_0^{-\frac{\beta}{\beta+1}-\frac{2}{n+1}}$ .

Next, we will analyze Eq (4.6) in two cases

I. If $\exists r_0\geq R_0$ such that $\mathcal{F}(r_0, t)\geq0$ , then $\mathcal{F}(r, t)\geq\mathcal{F}(r_0, t)\geq0, r\geq r_0.$ Therefore, (4.6) can be rewritten as

$\begin{align} \mathcal{F}(r, t)&\leq c_1\Big[r\frac{\partial}{\partial r}\mathcal{F}(r, t)\Big]+c_2\Big[r\frac{\partial}{\partial r}\mathcal{F}(r, t)\Big]^{\frac{2\beta+1}{2(\beta+1)} +\frac{1}{m+1}} \\ &+c_3\Big[r\frac{\partial}{\partial r}\mathcal{F}(r, t)\Big]^{\frac{2\beta+1}{2(\beta+1)} +\frac{1}{n+1}}, \quad\quad\quad\quad\quad r\geq r_0. \end{align}$

(4.7)

Using Young's inequality, we have

$\begin{align} \Big[r\frac{\partial}{\partial r}\mathcal{F}(r, t)\Big]^{\frac{2\beta+1}{2(\beta+1)} +\frac{1}{m+1}}&\leq\Big(\frac{4\beta+3}{4(\beta+1)}+\frac{1}{2(m+1)}\Big) \Big[r\frac{\partial}{\partial r}\mathcal{F}(r, t)\Big]^\frac{1}{2} \\ &+\Big(\frac{1}{4(\beta+1)}-\frac{1}{2(m+1)}\Big) \Big[r\frac{\partial}{\partial r}\mathcal{F}(r, t)\Big], \end{align}$

(4.8)

$\begin{align} \Big[r\frac{\partial}{\partial r}\mathcal{F}(r, t)\Big]^{\frac{2\beta+1}{2(\beta+1)} +\frac{1}{n+1}}&\leq\Big(\frac{4\beta+3}{4(\beta+1)}+\frac{1}{2(n+1)}\Big) \Big[r\frac{\partial}{\partial r}\mathcal{F}(r, t)\Big]^\frac{1}{2} \\ &+\Big(\frac{1}{4(\beta+1)}-\frac{1}{2(n+1)}\Big) \Big[r\frac{\partial}{\partial r}\mathcal{F}(r, t)\Big]. \end{align}$

(4.9)

Inserting (4.8) and (4.9) into (4.7), we have

$\begin{align} \mathcal{F}(r, t)&\leq c_4\Big[r\frac{\partial}{\partial r}\mathcal{F}(r, t)\Big]^\frac{1}{2}+c_5\Big[r\frac{\partial}{\partial r}\mathcal{F}(r, t)\Big], \quad r\geq r_0. \end{align}$

(4.10)

where $c_4 = c_2\Big(\frac{4\beta+3}{4(\beta+1)}+\frac{1}{2(m+1)}\Big)+c_3\Big(\frac{4\beta+3}{4(\beta+1)}+\frac{1}{2(n+1)}\Big)$ and $c_5 = c_1+c_2\Big(\frac{1}{4(\beta+1)}-\frac{1}{2(m+1)}\Big)+c_3\Big(\frac{1}{4(\beta+1)}-\frac{1}{2(n+1)}\Big)$ . From (4.10) we have

$\begin{align} \frac{\mathcal{F}(r, t)}{c_5}&\leq \Big[\sqrt{r\frac{\partial}{\partial r}\mathcal{F}(r, t)}+\frac{c_4}{2c_5}\Big]^2 -\frac{c^2_4}{4c^2_5}, \quad r\geq r_0 \end{align}$

$\begin{align} \frac{\frac{\partial}{\partial r}\mathcal{F}(r, t)}{\Big[\sqrt{\frac{\mathcal{F}(r, t)}{c_5}+\frac{c^2_4}{4c^2_5}}-\frac{c_4}{2c_5}\Big]^2} \geq \frac{1}{r}, \quad r\geq r_0 \end{align}$

(4.11)

Integrating (4.11) from $r_0$ to $r$ , we get

$\begin{align} 2c_5\Big[ln\Big(\sqrt{\frac{\mathcal{F}(r, t)}{c_5}+\frac{c^2_4}{4c^2_5}}-\frac{c_4}{2c_5}\Big) &-ln\Big(\sqrt{\frac{\mathcal{F}(r_0, t)}{c_5}+\frac{c^2_4}{4c^2_5}}-\frac{c_4}{2c_5}\Big)\Big] \\ -c_4\Big\{\Big[\sqrt{\frac{\mathcal{F}(r, t)}{c_5}+\frac{c^2_4}{4c^2_5}}-\frac{c_4}{2c_5}\Big]^{-1} &-\Big[\sqrt{\frac{\mathcal{F}(r_0, t)}{c_5}+\frac{c^2_4}{4c^2_5}}-\frac{c_4}{2c_5}\Big]^{-1}\Big\} \\ &\geq ln\Big(\frac{r}{r_0}\Big), \quad r\geq r_0 \end{align}$

(4.12)

Dropping the third term on the left of (4.12), we have

$\begin{align} \sqrt{\frac{\mathcal{F}(r, t)}{c_5}+\frac{c^2_4}{4c^2_5}}-\frac{c_4}{2c_5}\geq Q_1(r_0, t)\Big(\frac{r}{r_0}\Big)^\frac{1}{2c_5}, \end{align}$

(4.13)

where $Q_1(r_0, t) = \Big(\sqrt{\frac{\mathcal{F}(r_0, t)}{c_5}+\frac{c^2_4}{4c^2_5}}-\frac{c_4}{2c_5}\Big)\exp\Big\{-\frac{c_4}{2c_5} \Big[\sqrt{\frac{\mathcal{F}(r_0, t)}{c_5}+\frac{c^2_4}{4c^2_5}}-\frac{c_4}{2c_5}\Big]^{-1}\Big\}$ .

In view of

$\sqrt{\frac{\mathcal{F}(r, t)}{c_5}+\frac{c^2_4}{4c^2_5}}\leq\sqrt{\frac{\mathcal{F}(r, t)}{c_5}}+\frac{c_4}{2c_5},$

we have from (4.13)

$\begin{align} \mathcal{F}(r, t)\geq c_5Q_1^2(r_0, t)\Big(\frac{r}{r_0}\Big)^\frac{1}{c_5}. \end{align}$

(4.14)

Combining (4.2) and (4.14), we have

$\begin{align} \lim\limits_{r\rightarrow \infty}&\Big\{r^{-\frac{1}{c_5}}\Big[\frac{1}{2}e^{-\omega t}\int_{r_0}^r\int_{B(\xi)}\Big[|u_t|^2+b_1|\nabla u|^2+\frac{1}{\beta+1}b_2|\nabla u|^{2(\beta+1)}\Big]dsd\xi\\ &+\frac{1}{2}e^{-\omega t}\int_{r_0}^r\int_{B(\xi)}\Big[|v_t|^2+b_1|\nabla v|^2+\frac{1}{\beta+1}b_2|\nabla v|^{2(\beta+1)}+F(u, v)\Big]dsd\xi \\ &+\frac{1}{2}\omega\int_0^t\int_{r_0}^r\int_{B(\xi)}e^{-\omega\eta}\Big[|u_\eta|^2+b_1|\nabla u|^2+\frac{1}{\beta+1}b_2|\nabla u|^{2(\beta+1)}\Big]ds d\xi d\eta\\ &+\frac{1}{2}\omega\int_0^t\int_{r_0}^r\int_{B(\xi)}e^{-\omega\eta}\Big[|v_\eta|^2+b_1|\nabla v|^2+\frac{1}{\beta+1}b_2|\nabla v|^{2(\beta+1)}+F(u, v)\Big]dsd\xi d\eta \\ &+\int_0^t\int_{r_0}^r\int_{B(\xi)}e^{-\omega\eta}|u_\eta|^{m+1}dsd\xi d\eta+\int_0^t\int_{B(r)}e^{-\omega\eta}|v_\eta|^{n+1}dsd\xi d\eta\Big]\Big\} \\ &\geq c_5Q_1^2(r_0, t)r_0^{-\frac{1}{c_5}}. \end{align}$

(4.15)

II. If $\forall r\geq R_0$ such that $\mathcal{F}(r, t) < 0$ , then (4.6) can be rewritten as

$\begin{align} -\mathcal{F}(r, t)&\leq c_1\Big[r\frac{\partial}{\partial r}\mathcal{F}(r, t)\Big]+c_2\Big[r\frac{\partial}{\partial r}\mathcal{F}(r, t)\Big]^{\frac{2\beta+1}{2(\beta+1)} +\frac{1}{m+1}} \\ &+c_3\Big[r\frac{\partial}{\partial r}\mathcal{F}(r, t)\Big]^{\frac{2\beta+1}{2(\beta+1)} +\frac{1}{n+1}}, \quad\quad\quad\quad\quad r\geq R_0. \end{align}$

(4.16)

Without losing generality, we suppose that $m > n > 1$ .

$\begin{align} \Big[r\frac{\partial}{\partial r}\mathcal{F}(r, t)\Big]^{\frac{2\beta+1}{2(\beta+1)} +\frac{1}{n+1}}&\leq\frac{\frac{1}{n+1}-\frac{1}{m+1}}{\frac{2\beta+1}{2(\beta+1)}-\frac{1}{m+1}} \Big[r\frac{\partial}{\partial r}\mathcal{F}(r, t)\Big] \\ &+\frac{\frac{2\beta+1}{2(\beta+1)}-\frac{1}{m+1}}{\frac{2\beta+1}{2(\beta+1)}-\frac{1}{m+1}} \Big[r\frac{\partial}{\partial r}\mathcal{F}(r, t)\Big]^{\frac{2\beta+1}{2(\beta+1)} +\frac{1}{m+1}}, \end{align}$

(4.17)

$\begin{align} \Big[r\frac{\partial}{\partial r}\mathcal{F}(r, t)\Big]&\leq\frac{2\beta(m+3)+4}{(2\beta+1)(m+2)+1} \Big[r\frac{\partial}{\partial r}\mathcal{F}(r, t)\Big]^{\frac{2\beta+1}{2(\beta+1)} +\frac{1}{m+1}} \\ &+\frac{m+1-2(\beta+1)}{(2\beta+1)(m+2)+1} \Big[r\frac{\partial}{\partial r}\mathcal{F}(r, t)\Big]^{2\Big[\frac{2\beta+1}{2(\beta+1)} +\frac{1}{m+1}\Big]}. \end{align}$

(4.18)

Inserting (4.17) and (4.18) into (4.16), we get

$\begin{align} -\mathcal{F}(r, t)&\leq c_6\Big[r\frac{\partial}{\partial r}\mathcal{F}(r, t)\Big]^{\frac{2\beta+1}{2(\beta+1)} +\frac{1}{m+1}} +c_7\Big[r\frac{\partial}{\partial r}\mathcal{F}(r, t)\Big]^{2\Big[\frac{2\beta+1}{2(\beta+1)} +\frac{1}{m+1}\Big]}, \ r\geq R_0, \end{align}$

(4.19)

where $c_6 = \Big[c_1+c_3\frac{\frac{1}{n+1}-\frac{1}{m+1}}{\frac{2\beta+1}{2(\beta+1)}-\frac{1}{m+1}}\Big] \frac{2\beta(m+3)+4}{(2\beta+1)(m+2)+1}+c_3\frac{\frac{2\beta+1}{2(\beta+1)}-\frac{1}{m+1}}{\frac{2\beta+1}{2(\beta+1)}-\frac{1}{m+1}}, c_7 = c_3\frac{m+1-2(\beta+1)}{(2\beta+1)(m+2)+1}$ . From (4.19) we obtain

$\begin{align} r\frac{\partial}{\partial r}\mathcal{F}(r, t)&\geq \Big[\sqrt{\frac{-\mathcal{F}(r, t)}{c_7}+\frac{c_6^2}{4c_7^2}}-\frac{c_6}{2c_7}\Big]^{\frac{1}{\frac{2\beta+1}{2(\beta+1)} +\frac{1}{m+1}}}, \ r\geq R_0 \end{align}$

$\begin{align} 2c_7\frac{\sqrt{\frac{-\mathcal{F}(r, t)}{c_7}+\frac{c_6^2}{4c_7^2}}}{\Big[\sqrt{\frac{-\mathcal{F}(r, t)}{c_7}+\frac{c_6^2}{4c_7^2}}-\frac{c_6}{2c_7}\Big]^{\frac{1}{\frac{2\beta+1}{2(\beta+1)} +\frac{1}{m+1}}}}d\Big\{\sqrt{\frac{-\mathcal{F}(r, t)}{c_7}+\frac{c_6^2}{4c_7^2}}-\frac{c_6}{2c_7}\Big\}&\leq-\frac{1}{r}, \ r\geq R_0 \end{align}$

(4.20)

Integrating (4.20) from $R_0$ to $r$ , we obtain

$\begin{align} &\frac{2c_7\Big(\frac{2\beta+1}{2(\beta+1)}+\frac{1}{m+1}\Big)}{\frac{\beta}{2(\beta+1)}+\frac{2}{m+1}} \Big[\sqrt{\frac{-\mathcal{F}(r, t)}{c_7}+\frac{c_6^2}{4c_7^2}}-\frac{c_6}{2c_7}\Big]^\frac{\frac{\beta}{2(\beta+1)}+\frac{2}{m+1}}{\frac{2\beta+1}{2(\beta+1)}+\frac{1}{m+1}} \\ &-\frac{2c_7\Big(\frac{2\beta+1}{2(\beta+1)}+\frac{1}{m+1}\Big)}{\frac{\beta}{2(\beta+1)}+\frac{2}{m+1}} \Big[\sqrt{\frac{-\mathcal{F}(R_0, t)}{c_7}+\frac{c_6^2}{4c_7^2}}-\frac{c_6}{2c_7}\Big]^\frac{\frac{\beta}{2(\beta+1)}+\frac{2}{m+1}}{\frac{2\beta+1}{2(\beta+1)}+\frac{1}{m+1}} \\ &-\frac{c_6\Big(\frac{2\beta+1}{2(\beta+1)}+\frac{1}{m+1}\Big)}{\frac{1}{2(\beta+1)}-\frac{1}{m+1}} \Big[\sqrt{\frac{-\mathcal{F}(r, t)}{c_7}+\frac{c_6^2}{4c_7^2}}-\frac{c_6}{2c_7}\Big]^{-\frac{\frac{1}{2(\beta+1)}-\frac{1}{m+1}}{\frac{2\beta+1}{2(\beta+1)}+\frac{1}{m+1}}} \\ &+\frac{c_6\Big(\frac{2\beta+1}{2(\beta+1)}+\frac{1}{m+1}\Big)}{\frac{1}{2(\beta+1)}-\frac{1}{m+1}} \Big[\sqrt{\frac{-\mathcal{F}(R_0, t)}{c_7}+\frac{c_6^2}{4c_7^2}}-\frac{c_6}{2c_7}\Big]^{-\frac{\frac{1}{2(\beta+1)}-\frac{1}{m+1}}{\frac{2\beta+1}{2(\beta+1)}+\frac{1}{m+1}}} \\ &\leq ln\Big(\frac{R_0}{r}\Big). \end{align}$

(4.21)

Dropping the first and fourth terms on the left of (4.21), we obtain

$\begin{align} \sqrt{\frac{-\mathcal{F}(r, t)}{c_7}+\frac{c_6^2}{4c_7^2}} \leq\Big[c_8ln\Big(\frac{r}{R_0}\Big) -Q_2(R_0, t)\Big]^{-\frac{\frac{2\beta+1}{2(\beta+1)}+\frac{1}{m+1}}{\frac{1}{2(\beta+1)}-\frac{1}{m+1}}} +\frac{c_6}{2c_7}, \end{align}$

(4.22)

where $Q_2(R_0, t) = \frac{2c_7\Big(\frac{1}{2(\beta+1)}-\frac{1}{m+1}\Big)}{c_6\Big(\frac{\beta}{2(\beta+1)}+\frac{2}{m+1}\Big)} \Big[\sqrt{\frac{-\mathcal{F}(R_0, t)}{c_7}+\frac{c_6^2}{4c_7^2}}-\frac{c_6}{2c_7}\Big]^\frac{\frac{\beta}{2(\beta+1)}+\frac{2}{m+1}}{\frac{2\beta+1}{2(\beta+1)}+\frac{1}{m+1}}$ and $c_8 = \frac{\frac{1}{2(\beta+1)}-\frac{1}{m+1}}{c_6\Big(\frac{2\beta+1}{2(\beta+1)} +\frac{1}{m+1}\Big)}$ .

Squaring (4.22) we have

$\begin{align} -\mathcal{F}(r, t)& \leq c_7\Big[c_8ln\Big(\frac{r}{R_0}\Big) -Q_2(R_0, t)\Big]^{-\frac{\frac{2\beta+1}{\beta+1}+\frac{2}{m+1}}{\frac{1}{2(\beta+1)}-\frac{1}{m+1}}} \\ &+c_6\Big[c_8ln\Big(\frac{r}{R_0}\Big) -Q_2(R_0, t)\Big]^{-\frac{\frac{2\beta+1}{2(\beta+1)}+\frac{1}{m+1}}{\frac{1}{2(\beta+1)}-\frac{1}{m+1}}}. \end{align}$

(4.23)

From (4.15) and (4.23) we can obtain the following theorem.

Theorem 4.1. Let $(u, v)$ be the solution of (1.1), (1.2), (2.1) and (2.2) with $\rho(q^2) = b_1+b_2q^{2\beta}$ , where $\frac{1}{2(\beta+1)} > \max\Big\{\frac{1}{m+1}, \frac{1}{n+1}\Big\}$ . Then for fixed $t$ , when $r\rightarrow \infty$ , $(u, v)$ either grows algebraically or decays logarithmically. The growth rate is at least as fast as $z^\frac{1}{c_5}$ and the decay rate is at least as fast as $\Big(ln r\Big)^{-\frac{\frac{2\beta+1}{2(\beta+1)}+\frac{1}{m+1}}{\frac{1}{2(\beta+1)}-\frac{1}{m+1}}}$ .

Remark 4.1. Obviously, in this case of $\rho(q^2) = b_1+b_2q^{2\beta}$ , the decay rate obtained by Theorem 4.1 is slower than that obtained by Theorem 2.1 and Theorem 3.1.

5. A nonlinear system of viscoelastic type

In this section, we concern with a system of two coupled viscoelastic equations

$\begin{align} u_{tt}&-\Delta u+\int_0^th_1(t-\eta)\Delta u(\eta)d\eta+f_1(u, v) = 0, \end{align}$

(5.1)

$\begin{align} v_{tt}&-\Delta v+\int_0^th_2(t-\eta)\Delta v(\eta)d\eta+f_2(u, v) = 0, \end{align}$

(5.2)

which describes the interaction between two different fields arising in viscoelasticity. In (5.1) and (5.2), $0 < t < T$ and $h_1, h_2$ are differentiable functions satisfying $h_1(0), h_2(0) > 0$ and

$\begin{align} 2\Big(&\int_0^T h_1^2(T-\eta)d\eta\Big), \ \frac{4T}{h_1(0)}\Big[\int_0^T \Big(h_1'(T-\eta)-\omega h_1(T-\eta)\Big)^2d\eta\Big] \leq h_1(0), \end{align}$

(5.3)

$\begin{align} 2\Big(&\int_0^T h_2^2(T-\eta)d\eta\Big), \ \frac{4T}{h_2(0)}\Big[\int_0^T \Big(h_2'(T-\eta)-\omega h_2(T-\eta)\Big)^2d\eta\Big] \leq h_2(0). \end{align}$

(5.4)

Messaoudi and Tatar ^[24] considered the system (5.1) and (5.2) in a bounded domain and proved the uniform decay for the solution when $t\rightarrow \infty$ . For more special cases, one can refer to ^[25,26,27]. They mainly concerned the well-posedness of the solutions and proved that the solutions decayed uniformly under some suitable conditions. However, the present paper extends the previous results to Eqs (5.1) and (5.2) in an exterior region. We consider Eqs (5.1) and (5.2) with the initial-boundary conditions (2.1) and (2.2) in $\Omega$ .

We define two functions

$\begin{align} G_1(r, t)& = \int_0^t\int_{B(r)}e^{-\omega\eta}\nabla u\cdot\frac{\mathit{\boldsymbol{x}}}{r}u_\eta dsd\eta -\int_0^t\int_{B(r)}e^{-\omega\eta}\Big(\int_0^\eta h_1(\eta-s)\nabla uds\Big)\cdot\frac{\mathit{\boldsymbol{x}}}{r}u_\eta dsd\eta \\ &\doteq I_1+I_2, \end{align}$

(5.5)

$\begin{align} G_2(r, t)& = \int_0^t\int_{B(r)}e^{-\omega\eta}\nabla v\cdot\frac{\mathit{\boldsymbol{x}}}{r}v_\eta dsd\eta -\int_0^t\int_{B(r)}e^{-\omega\eta}\Big(\int_0^\eta h_2(\eta-s)\nabla vds\Big)\cdot\frac{\mathit{\boldsymbol{x}}}{r}v_\eta dsd\eta \\ &\doteq J_1+J_2. \end{align}$

(5.6)

Integrating (5.5) from $r_0$ to $r$ and using (5.1), (5.2), (2.1), (2.2) and the divergence theorem, we have

$\begin{align} G_1(r, t)& = G_1(r_0, t)+\frac{1}{2}\int_{r_0}^r\int_{B(\xi)}e^{-\omega t}\Big[|u_t|^2+|\nabla u|^2\Big]dsd\xi+\frac{1}{2}\omega\int_0^t\int_{r_0}^r\int_{B(\xi)}e^{-\omega \eta}|u_\eta|^2dsd\xi d\eta\\ &+\frac{1}{2}\omega\int_0^t\int_{r_0}^r\int_{B(\xi)}e^{-\omega \eta}|\nabla u|^2dsd\xi d\eta+\int_0^t\int_{r_0}^r\int_{B(\xi)}e^{-\omega \eta}h_1(0)|\nabla u|^2dsd\xi d\eta\\ &-\int_{r_0}^r\int_{B(\xi)}e^{-\omega t}\Big(\int_0^t h_1(t-\tau)\nabla ud\tau\Big)\cdot\nabla udsd\xi\\ &+\int_0^t\int_{r_0}^r\int_{B(\xi)}e^{-\omega \eta}\Big[\int_0^\eta \Big(h_1'(\eta-\tau)-\omega h_1(\eta-\tau)\Big)\nabla ud\tau\Big]\cdot\nabla udsd\xi d\eta\\ &+\int_0^t\int_{r_0}^r\int_{B(\xi)}e^{-\omega \eta}f_1(u, v)u_\eta dsd\xi d\eta. \end{align}$

(5.7)

From (5.7) it follows that

$\begin{align} \frac{\partial}{\partial z}G_1(r, t)& = \frac{1}{2}\int_{B(r)}e^{-\omega t}\Big[|u_t|^2+|\nabla u|^2\Big]ds+\frac{1}{2}\omega\int_0^t\int_{B(r)}e^{-\omega \eta}|u_\eta|^2dsd\eta \\ &+\frac{1}{2}\omega\int_0^t\int_{B(r)}e^{-\omega \eta}|\nabla u|^2dsd\eta+\int_0^t\int_{B(r)}e^{-\omega \eta}h_1(0)|\nabla u|^2dsd\eta \\ &-\int_{B(r)}e^{-\omega t}\Big(\int_0^t h_1(t-\tau)\nabla ud\tau\Big)\cdot\nabla uds\\ &+\int_0^t\int_{B(r)}e^{-\omega \eta}\Big[\int_0^\eta \Big(h_1'(\eta-\tau)-\omega h_1(\eta-\tau)\Big)\nabla ud\tau\Big]\cdot\nabla uds d\eta\\ &+\int_0^t\int_{B(r)}e^{-\omega \eta}f_1(u, v)u_\eta dsd\eta. \end{align}$

(5.8)

By the Young inequality and the Hölder inequality, we have

$\begin{align} \Big|&-\int_{B(r)}e^{-\omega t}\Big(\int_0^t h_1(\eta-\tau)\nabla ud\tau\Big)\cdot\nabla uds\Big|\\ &\leq\int_{B(r)}e^{-\omega t}\Big[\Big(\int_0^t h_1(t-s)\nabla u(s)ds\Big)^2+\frac{1}{4}|\nabla u|^2\Big]ds\\ &\leq\Big(\int_0^t h_1^2(t-\tau)d\tau\Big)\int_0^t\int_{B(r)}e^{-\omega \eta}|\nabla u|^2dsd\eta+\frac{1}{4}\int_{B(r)}e^{-\omega t}|\nabla u|^2ds, \end{align}$

(5.9)

and

$\begin{align} \Big|&\int_0^t\int_{B(r)}e^{-\omega \eta}\Big[\int_0^\eta \Big(h_1'(\eta-\tau)-\omega h_1(\eta-\tau)\Big)\nabla ud\tau\Big]\cdot\nabla uds d\eta\Big|\\ &\leq \frac{t}{h_1(0)}\Big[\int_0^t \Big(h_1'(\eta-\tau)-\omega h_1(\eta-\tau)\Big)^2d\tau\Big]\int_0^t\int_{B(r)}e^{-\omega \eta}|\nabla u|^2dsd\eta\\ &+\frac{1}{4}\int_0^t\int_{B(r)}e^{-\omega \eta}h_1(0)|\nabla u|^2dsd\eta. \end{align}$

(5.10)

Inserting (5.9) and (5.10) into (5.8) and using (5.3), we have

$\begin{align} \frac{\partial}{\partial z}G_1(r, t)&\geq\frac{1}{2}\int_{B(r)}e^{-\omega t}\Big[|u_t|^2+\frac{1}{2}|\nabla u|^2\Big]ds+\frac{1}{2}\omega\int_0^t\int_{B(r)}e^{-\omega \eta}|u_\eta|^2ds d\eta\\ &+\frac{1}{2}\omega\int_0^t\int_{B(r)}e^{-\omega \eta}|\nabla u|^2dsd\eta+\int_0^t\int_{B(r)}e^{-\omega \eta}f_1(u, v)u_\eta ds d\eta. \end{align}$

(5.11)

Similar to (5.11), we also have for $G_2(r, t)$

$\begin{align} \frac{\partial}{\partial z}G_2(r, t)&\geq\frac{1}{2}\int_{B(r)}e^{-\omega t}\Big[|v_t|^2+\frac{1}{2}|\nabla v|^2\Big]ds+\frac{1}{2}\omega\int_0^t\int_{B(r)}e^{-\omega \eta}|v_\eta|^2dsd\eta\\ &+\frac{1}{2}\omega\int_0^t\int_{B(r)}e^{-\omega \eta}|\nabla v|^2dsd\eta+\int_0^t\int_{B(r)}e^{-\omega \eta}f_2(u, v)v_\eta dsd\eta. \end{align}$

(5.12)

If we define

$G(r, t) = G_1(r, t)+G_2(r, t),$

then by (5.11) and (5.12) we have

$\begin{align} \frac{\partial}{\partial z}G(r, t)&\geq\frac{1}{2}\int_{B(r)}e^{-\omega t}\Big[|u_t|^2+|v_t|^2+\frac{1}{2}|\nabla u|^2+\frac{1}{2}|\nabla v|^2+2F(u, v)\Big]ds \\ &+\frac{1}{2}\omega\int_0^t\int_{B(r)}e^{-\omega \eta}\Big[|u_\eta|^2+|v_\eta|^2+|\nabla u|^2+|\nabla v|^2+2F(u, v)\Big]dsd\eta. \end{align}$

(5.13)

On the other hand, we bound $G(r, t)$ by $\frac{\partial}{\partial r}G(r, t)$ . Using the Hölder inequality, the AG mean inequality, (5.3) and combining (5.13), we have

$\begin{align} \Big|I_1\Big|+\Big|J_1\Big|&\leq\Big(\int_0^t\int_{B(r)}e^{-\omega\eta}|\nabla u|^2 dAd\eta\cdot\int_0^t\int_{B(r)}e^{-\omega\eta}|u_\eta|^2 dsd\eta\Big)^\frac{1}{2} \\ &+\Big(\int_0^t\int_{B(r)}e^{-\omega\eta}|\nabla v|^2 dAd\eta\cdot\int_0^t\int_{B(r)}e^{-\omega\eta}|v_\eta|^2 dsd\eta\Big)^\frac{1}{2} \\ &\leq\frac{1}{\omega}\Big[\frac{\partial}{\partial r}G(r, t)\Big], \end{align}$

(5.14)

and

$\begin{align} \Big|I_2\Big|&\leq\Big(\int_0^t\int_{B(r)}e^{-\omega\eta}\Big|\int_0^\eta h_1(\eta-\tau)\nabla ud\tau\Big|^2 dsd\eta\cdot\int_0^t\int_{B(r)}e^{-\omega\eta}|u_\eta|^2dsd\eta\Big)^\frac{1}{2} \\ &\leq\Big(\int_0^t\int_{B(r)}e^{-\omega\eta}\Big(\int_0^\eta h^2_1(\eta-\tau)d\tau\Big)\Big(\int_0^\eta |\nabla u|^2d\tau\Big) dsd\eta\cdot\int_0^t\int_{B(r)}e^{-\omega\eta}|u_\eta|^2dsd\eta\Big)^\frac{1}{2} \\ &\leq t\Big(\int_0^t h^2_1(\eta-\tau)d\tau\Big)^\frac{1}{2}\Big(\int_0^t\int_{B(r)}e^{-\omega\eta} |\nabla u|^2 dsd\eta\cdot\int_0^t\int_{B(r)}e^{-\omega\eta}|u_\eta|^2dsd\eta\Big)^\frac{1}{2} \\ &\leq\frac{t}{2}\Big(\int_0^t h^2_1(\eta-\tau)d\tau\Big)^\frac{1}{2}\Big[\int_0^t\int_{B(r)}e^{-\omega\eta} |\nabla u|^2 dsd\eta+\int_0^t\int_{B(r)}e^{-\omega\eta}|u_\eta|^2dsd\eta\Big]\\ &\leq\frac{T}{2\omega}\sqrt{h_1(0)}\Big[\frac{\partial}{\partial r}G(r, t)\Big]. \end{align}$

(5.15)

Similar to (5.15), we have

$\begin{align} \Big|J_2\Big|\leq\frac{T}{2\omega}\sqrt{h_2(0)}\Big[\frac{\partial}{\partial r}G(r, t)\Big]. \end{align}$

(5.16)

Inserting (5.14)–(5.16) into (5.5) and (5.6), we have

$\begin{align} \Big|G(r, t)\Big|\leq c_9\frac{1}{\omega}\Big[\frac{\partial}{\partial r}G(r, t)\Big], \end{align}$

(5.17)

where $c_9 = \frac{T}{2}(\sqrt{h_1(0)}+\sqrt{h_2(0)})+2$ .

We can follow the similar arguments given in the previous sections to obtain the following theorem.

Theorem 5.1. Let $(u, v)$ be the solution of (5.1), (5.2), (2.1) and (2.2) in $\Omega$ , and (5.3) and (5.4) hold. For fixed $t$ ,

(1) If $\exists\ R_0\geq r_0$ , $G(R_0, t)\geq0$ , then

$G(r, t)\geq G(R_0, t)e^{\frac{\omega}{b_9}(r-R_0)}.$

(2) If $\forall\ r\geq r_0$ , $G(r, t) < 0$ , then

$-G(r, t)\leq \Big[-G(r_0, t)\Big]e^{-\frac{\omega}{b_9}(r-r_0)}.$

Again, the rate of growth or decay obtained in this case is arbitrarily large

Remark 5.1. It is clear that the above analysis can be adapted without difficulties to the equation (see ^[28,29])

$u_{tt}-k_0\triangle u+\int_0^tdiv[a(x)h(t-s)\nabla u(s)]ds+b(x)g(u_t)+f(u) = 0$

and the equation (see ^[30])

$|u_{t}|^\sigma u_{tt}-k_0\triangle u-\Delta u_{tt}+\int_0^th(t-s)\Delta u(s)]ds-\gamma\Delta u_t = 0$

with some suitable $g$ and $a(x)+b(x)\geq b_{10} > 0$ and $k_0, \sigma, \gamma > 0.$

6. Conclusions

In this paper, we have considered several situations where the solutions of Eqs (1.1) and (1.2) either grow or decay exponentially or polynomially. We emphasize that the Poincaré inequality on the cross sections is not used in this paper. Thus, our results also hold for the two-dimensional case. On the other hand, there are some deeper problems to be studied in this paper. We can continue to study the continuous dependence of coefficients in the equation as that in ^[31]. These are the issues we will continue to study in the future.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors express their heartfelt thanks to the editors and referees who have provided some important suggestions. This work was supported by the Guangzhou Huashang College Tutorial Scientific Research Project (2022HSDS09), National Natural Science Foundation of China (11371175) and the Research Team Project of Guangzhou Huashang College (2021HSKT01).

Conflict of interest

The authors declare that there is no conflict of interest.

References

[1]	F. Courchamp, L. Berec, J. Gascoigne, Allee Effects in Ecology and Conservation, Oxford University Press, 2008.
[2]	E. M. Takyi, K. Cooper, A. Dreher, C. McCrorey, The (de) stabilizing effect of juvenile prey cannibalism in a stage-structured model, Math. Biosci. Eng., 20 (2022), 3355–3378. https://doi.org/10.3934/mbe.2023158 doi: 10.3934/mbe.2023158
[3]	E. M. Takyi, K. Cooper, A. Dreher, C. McCrorey, Dynamics of a predator–prey system with wind effect and prey refuge, J. Appl. Nonlinear Dyn., 12 (2023), 427–440. https://doi.org/10.5890/JAND.2023.09.001 doi: 10.5890/JAND.2023.09.001
[4]	R. D. Parshad, S. Wickramasooriya, K. Antwi-Fordjour, A. Banerjee, Additional food causes predators to explode—unless the predators compete, Int. J. Bifurcation Chaos, 33 (2023), 2350034.
[5]	R. K. Upadhyay, R. D. Parshad, K. Antwi-Fordjour, E. Quansah, S. Kumari, Global dynamics of stochastic predator–prey model with mutual interference and prey defense, J. Appl. Math. Comput., 60 (2019), 169–190. https://doi.org/10.1007/s12190-018-1207-7 doi: 10.1007/s12190-018-1207-7
[6]	R. D. Alexander, The evolution of social behavior, Ann. Rev. Ecol. Syst., 5 (1974), 325–383. https://doi.org/10.1146/annurev.es.05.110174.001545 doi: 10.1146/annurev.es.05.110174.001545
[7]	S. Périquet, L. Todd-Jones, M. Valeix, B. Stapelkamp, N. Elliot, M. Wijers, et al., Influence of immediate predation risk by lions on the vigilance of prey of different body size, Behavioral Ecol., 23 (2012), 970–976.
[8]	A. Treves, Theory and method in studies of vigilance and aggression, Anim. Behav., 60 (2001), 711–722. https://doi.org/10.1006/anbe.2000.1528 doi: 10.1006/anbe.2000.1528
[9]	S. Liley, S. Creel, What best explains vigilance in elk: characteristics of prey, predators, or the environment?, Behav. Ecol., 19 (2007), 245–254. https://doi.org/10.1093/beheco/arm116 doi: 10.1093/beheco/arm116
[10]	R. A. Martin, N. Hammerschlag, Marine predator–prey contests: Ambush and speed versus vigilance and agility, Mar. Biol. Res., 8 (2012), 90–94. https://doi.org/10.1080/17451000.2011.614255 doi: 10.1080/17451000.2011.614255
[11]	T. M. Caro, Cheetah mothers' vigilance: Looking out for prey or for predators?, Behav. Ecol. Sociobiol., 20 (1987), 351–361. https://doi.org/10.1007/BF00300681 doi: 10.1007/BF00300681
[12]	M. M. Dehn, Vigilance for predators: Detection and dilution effects, Behav. Ecol. Sociobiol., 26 (1990), 337–342.
[13]	D. Fortin, M. S. Boyce, E. H. Merrill, J. M. Fryxell, Foraging costs of vigilance in large mammalian herbivores, Oikos, 107 (2004), 172–180. https://doi.org/10.1111/j.0030-1299.2004.12976.x doi: 10.1111/j.0030-1299.2004.12976.x
[14]	A. W. Illius, C. Fitzgibbon, Costs of vigilance in foraging ungulates, Anim. Behav., 47 (1994), 481–484. https://doi.org/10.1006/anbe.1994.1067 doi: 10.1006/anbe.1994.1067
[15]	C. D. FitzGibbon, A cost to individuals with reduced vigilance in groups of Thomson's gazelles hunted by cheetahs, Anim. Behav., 37 (1989), 508–510. https://psycnet.apa.org/doi/10.1016/0003-3472(89)90098-5 doi: 10.1016/0003-3472(89)90098-5
[16]	S. M. Durant, Living with the enemy: avoidance of hyenas and lions by cheetahs in the serengeti, Behav. Ecol., 11 (2000), 624–632. https://doi.org/10.1093/beheco/11.6.624 doi: 10.1093/beheco/11.6.624
[17]	T. Kimbrell, R. D. Holt, P. Lundberg, The influence of vigilance on intraguild predation, J. Theor. Biol., 249 (2007), 218–234. https://doi.org/10.1016/j.jtbi.2007.07.031 doi: 10.1016/j.jtbi.2007.07.031
[18]	M. Hossain, R. Kumbhakar, N. Pal, Dynamics in the biparametric spaces of a three-species food chain model with vigilance, Chaos Solitons Fractals, 162 (2022), 112438. https://doi.org/10.1016/j.chaos.2022.112438 doi: 10.1016/j.chaos.2022.112438
[19]	M. Watson, N. J. Aebischer, W. Cresswell, Vigilance and fitness in grey partridges perdix perdix: the effects of group size and foraging-vigilance trade-offs on predation mortality, J. Anim. Ecol., 76 (2007), 211–221.
[20]	M. L. Lührs, M. Dammhahn, An unusual case of cooperative hunting in a solitary carnivore, J. Ethol., 28 (2010), 379–383. https://doi.org/10.1007/s10164-009-0190-8 doi: 10.1007/s10164-009-0190-8
[21]	M. T. Alves, F. M. Hilker, Hunting cooperation and Allee effects in predators, J. Theor. Biol., 419 (2017), 13–22. https://doi.org/10.1016/j.jtbi.2017.02.002 doi: 10.1016/j.jtbi.2017.02.002
[22]	D. Scheel, C. Packer, Group hunting behaviour of lions: a search for cooperation, Anim. Behav., 41 (1991), 697–709.
[23]	S. Creel, N. M. Creel, Communal hunting and pack size in African wild dogs, lycaon pictus, Anim. Behav., 50 (1995), 1325–1339. https://doi.org/10.1016/0003-3472(95)80048-4 doi: 10.1016/0003-3472(95)80048-4
[24]	C. Boesch, Cooperative hunting in wild chimpanzees, Anim. Behav., 48 (1994), 653–667. https://doi.org/10.1006/anbe.1994.1285 doi: 10.1006/anbe.1994.1285
[25]	P. A. Schmidt, L. D. Mech, Wolf pack size and food acquisition, Am. Nat., 150 (1997), 513–517. https://doi.org/10.1086/286079 doi: 10.1086/286079
[26]	P. E. Stander, Cooperative hunting in lions: The role of the individual, Behav. Ecol. Sociobiol., 29 (1992), 445–454. https://doi.org/10.1007/BF00170175 doi: 10.1007/BF00170175
[27]	R. D. Estes, J. Goddard, Prey selection and hunting behavior of the African wild dog, J. Wildl. Manage., 31 (1967), 52–70. https://doi.org/10.2307/1249030 doi: 10.2307/1249030
[28]	J. C. Bednarz, Cooperative hunting Harris' Hawks (parabuteo unicinctus), Science, 239 (1988), 1525–1527. https://doi.org/10.1126/science.239.4847.1525 doi: 10.1126/science.239.4847.1525
[29]	T. J. Pitcher, A. E. Magurran, I. J. Winfield, Fish in larger shoals find food faster, Behav. Ecol. Sociobiol., 10 (1982), 149–151. https://doi.org/10.1177/019262338201000227 doi: 10.1177/019262338201000227
[30]	J. A. Vucetich, R. O. Peterson, T. A. Waite, Raven scavenging favours group foraging in wolves, Anim. Behav., 67 (2004), 1117–1126. https://doi.org/10.1016/j.anbehav.2003.06.018 doi: 10.1016/j.anbehav.2003.06.018
[31]	L. Berec, Impacts of foraging facilitation among predators on predator-prey dynamics, Bull. Math. Biol., 72 (2010), 94–121. https://doi.org/10.1007/s11538-009-9439-1 doi: 10.1007/s11538-009-9439-1
[32]	S. Pal, N. Pal, S. Samanta, J. Chattopadhyay, Fear effect in prey and hunting cooperation among predators in a Leslie-Gower model, Math. Biosci. Eng., 16 (2019), 5146. https://doi.org/10.3934/mbe.2019258 doi: 10.3934/mbe.2019258
[33]	T. Singh, R. Dubey, V. N. Mishra, Spatial dynamics of predator-prey system with hunting cooperation in predators and type I functional response, AIMS Math., 5 (2020), 673–684. https://doi.org/10.3934/math.2020045 doi: 10.3934/math.2020045
[34]	D. Wu, M. Zhao, Qualitative analysis for a diffusive predator–prey model with hunting cooperative, Phys. A Stat. Mech. Appl., 515 (2019), 299–309. https://doi.org/10.1016/j.physa.2018.09.176 doi: 10.1016/j.physa.2018.09.176
[35]	S. Pal, N. Pal, J. Chattopadhyay, Hunting cooperation in a discrete-time predator–prey system, Int. J. Bifurcation Chaos, 28 (2018), 1850083. https://doi.org/10.1142/S0218127418500839 doi: 10.1142/S0218127418500839
[36]	B. Mondal, S. Sarkar, U. Ghosh, Complex dynamics of a generalist predator–prey model with hunting cooperation in predator, Eur. Phys. J. Plus, 137 (2021), 43.
[37]	M. Hossain, S. Garai, S. Jafari, N. Pal, Bifurcation, chaos, multistability, and organized structures in a predator–prey model with vigilance, Chaos Interdiscip. J. Nonlinear Sci., 32 (2022).
[38]	M. Pierre, Global existence in reaction-diffusion systems with control of mass: a survey, Milan J. Math., 78 (2010), 417–455. https://doi.org/10.1007/s00032-010-0133-4 doi: 10.1007/s00032-010-0133-4
[39]	M. Pierre, D. Schmitt, Blowup in reaction-diffusion systems with dissipation of mass, SIAM Rev., 42 (2000), 93–106.
[40]	E. A. Barbashin, Introduction to the theory of stability, translated from the Russian by Transcripta Service, London, T. Lukes Wolters-Noordhoff Publishing, Groningen, 1970.
[41]	J. L. Salle, S. Lefschetz, Stability by Liapunov's direct method, Academic Press, New York and London, 1961.
[42]	J. E. Marsden, M. McCracken, The Hopf bifurcation and its applications, Springer Science & Business Media, 2012.
[43]	L. Perko, Differential equations and dynamical systems, Springer Science & Business Media, Springer-Verlag, New York, 2013.
[44]	A. Dhooge, W. Govaerts, Y. A. Kuznetsov, H. G. E. Meijer, B. Sautois, New features of the software matcont for bifurcation analysis of dynamical systems, Math. Comput. Modell. Dyn. Syst., 14 (2008), 147–175. https://doi.org/10.1515/nf-2008-0101 doi: 10.1515/nf-2008-0101
[45]	L. M. Hassan, D. Arends, S. A. Rahmatalla, M. Reissmann, H. Reyer, K. Wimmers, et al., Genetic diversity of Nubian ibex in comparison to other ibex and domesticated goat species, Eur. J. Wildl. Res., 64 (2018), 1–10. https://doi.org/10.1007/s15033-018-1109-2 doi: 10.1007/s15033-018-1109-2
[46]	C. Iribarren, B. P. Kotler, Foraging patterns of habitat use reveal landscape of fear of Nubian ibex Capra nubiana, Wildl. Biol., 18 (2012), 194–201. https://doi.org/10.2981/11-041 doi: 10.2981/11-041

This article has been cited by:

1.	S.A. Pedro, J.M. Tchuenche, HIV/AIDS dynamics: Impact of economic classes with transmission from poor clinical settings, 2010, 267, 00225193, 471, 10.1016/j.jtbi.2010.09.019
2.	C. P. Bhunu, S. Mushayabasa, H. Kojouharov, J. M. Tchuenche, Mathematical Analysis of an HIV/AIDS Model: Impact of Educational Programs and Abstinence in Sub-Saharan Africa, 2011, 10, 1570-1166, 31, 10.1007/s10852-010-9134-0
3.	Sarada S. Panchanathan, Diana B. Petitti, Douglas B. Fridsma, The development and validation of a simulation tool for health policy decision making, 2010, 43, 15320464, 602, 10.1016/j.jbi.2010.03.013
4.	Cristiana J. Silva, Delfim F.M. Torres, A SICA compartmental model in epidemiology with application to HIV/AIDS in Cape Verde, 2017, 30, 1476945X, 70, 10.1016/j.ecocom.2016.12.001
5.	S. D. Hove-Musekwa, F. Nyabadza, H. Mambili-Mamboundou, Modelling Hospitalization, Home-Based Care, and Individual Withdrawal for People Living with HIV/AIDS in High Prevalence Settings, 2011, 73, 0092-8240, 2888, 10.1007/s11538-011-9651-7
6.	Emmanuelina L. Kateme, Jean M. Tchuenche, Senelani D. Hove-Musekwa, HIV/AIDS Dynamics with Three Control Strategies: The Role of Incidence Function, 2012, 2012, 2090-5572, 1, 10.5402/2012/864795
7.	F. Nyabadza, Z. Mukandavire, S.D. Hove-Musekwa, Modelling the HIV/AIDS epidemic trends in South Africa: Insights from a simple mathematical model, 2011, 12, 14681218, 2091, 10.1016/j.nonrwa.2010.12.024
8.	Xiaodong Wang, Chunxia Wang, Kai Wang, Extinction and persistence of a stochastic SICA epidemic model with standard incidence rate for HIV transmission, 2021, 2021, 1687-1847, 10.1186/s13662-021-03392-y

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)

通讯作者: 陈斌, bchen63@163.com

1.
沈阳化工大学材料科学与工程学院沈阳 110142

Mathematical Biosciences and Engineering

4.4

Metrics

Article views(2124) PDF downloads(157) Cited by(3)

Preview PDF

Download XML

Export Citation

Article outline

Show full outline

Figures and Tables

Figures(5) / Tables(1)

Mathematical Biosciences and Engineering

Dynamical analysis of a predator-prey system with prey vigilance and hunting cooperation in predators

Related Papers:

Abstract

1. Introduction

2. The Phragmén-Lindelöf type alternative result under A1

3. The Phragmén-Lindelöf type alternative result under A2

4. The Phragmén-Lindelöf type alternative result when $\rho(q^2)=b_1+b_2q^{2\beta}$

5. A nonlinear system of viscoelastic type

6. Conclusions

Use of AI tools declaration

Acknowledgments

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Figures and Tables

Other Articles By Authors

Catalog

Mathematical Biosciences and Engineering

Dynamical analysis of a predator-prey system with prey vigilance and hunting cooperation in predators

Related Papers:

Abstract

1. Introduction

2. The Phragmén-Lindelöf type alternative result under A1

3. The Phragmén-Lindelöf type alternative result under A2

4. The Phragmén-Lindelöf type alternative result when ρ(q2)=b1+b2q2β \rho(q^2)=b_1+b_2q^{2\beta}

5. A nonlinear system of viscoelastic type

6. Conclusions

Use of AI tools declaration

Acknowledgments

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Figures and Tables

Other Articles By Authors

Related pages

Tools

Export File

Citation

Format

Content

Catalog

4. The Phragmén-Lindelöf type alternative result when $\rho(q^2)=b_1+b_2q^{2\beta}$