Dynamics Analysis for a Prey–Predator Evolutionary Game System with Delays

Abstract

In this paper, we couple population dynamics and evolutionary game theory to establish a prey–predator system in which individuals in the predator population need to choose between group hunting strategies and isolated hunting strategies. This system includes two types of delay: fitness delay and hunting delay. In the absence of delays, we discuss the stability of boundary and interior equilibria. In addition, the condition that the non-delayed system undergoes transcritical bifurcation is obtained. For the delayed system, we explore the stability of the interior equilibrium and obtain the conditions for the existence of Hopf bifurcation. The conditions for determining the direction and stability of the Hopf bifurcation and the periodic variation in the periodic solution are introduced by using the normal form theory and center manifold theory. Finally, we simulate non-delayed and delayed systems. The results indicate that when the availability of prey is high, the isolated hunting strategy is the dominant strategy. When the availability of prey is low, mixed strategies appear and the proportion of the group hunting strategy increases as the availability of prey decreases. Furthermore, large delays lead to the disappearance of the mixed hunting strategy and its replacement by pure hunting strategies.

Appendix

We assume that $0< {\tau _2 ^*} < {\tau _{20}}$ and ${\tau _2 ^*} < \tau _{10}^*$ for the sake of generality. Let ${\tau _1}{{ = }}\tau _{10}^{{*}}{{ + }}\psi ,\,\psi \in \mathrm{R}$ so that $\psi =0$ is the Hopf bifurcation value of system (4.1). Let

$$\begin{aligned} {{{x}}_1}(t) = x(t) - x_ + ^*, \quad \,{m_1}(t) = m(t) - m_ + ^*. \end{aligned}$$

For convenience, we remove the subscripts of $x_1(t)$ and $m_1(t)$, which means that ${{{x}}_1}(t)$ and ${m_1}(t)$ are still denoted by x(t) and m(t), respectively. The delay is normalized by using the scale ${{t}} \mapsto \left( {\frac{t}{{{\tau _1}}}} \right)$. Then, we rewrite system (4.1) as a functional differential equation:

$$\begin{aligned} \dot{X}(t) = L_{\psi }\left( X_t\right) +s(\psi ,X_t), \end{aligned}$$

where

$$\begin{aligned} X\left( t \right) = \left( {\begin{array}{*{20}{c}} {x\left( t \right) }\\ {m\left( t \right) } \end{array}} \right) , \end{aligned}$$

$L_{\psi }:C\rightarrow \mathrm R$, and $s:\mathrm{R}\times C\rightarrow \mathrm R$ are given by

$$\begin{aligned} {L_\psi }\left( \varphi \right) = \left( {{\tau _1} + \psi } \right) \left( {{B_1}\varphi \left( 0 \right) + {B_2}\varphi \left( { - 1} \right) + {B_3}\varphi \left( { - \frac{{{\tau _2 ^*}}}{{{\tau _1}}}} \right) } \right) , \end{aligned}$$

where

$$\begin{aligned} {{B_1} = \left( {\begin{array}{*{20}{c}} 0&{}0\\ 0&{}a \end{array}} \right) ,{} {B_2} = \left( {\begin{array}{*{20}{c}} {{d_1}}&{}{{d_2}}\\ 0&{}0 \end{array}} \right) ,{} {B_3} = \left( {\begin{array}{*{20}{c}} 0&{}0\\ a&{}0 \end{array}} \right) ,\,s\left( {x,m} \right) = \left( {\begin{array}{*{20}{c}} {{s_1}}\\ {{s_2}} \end{array}} \right) }, \end{aligned}$$

and $\varphi = {\left( {{\varphi _1},{\varphi _2}} \right) ^T} \in C\left( {\left[ { - 1,0} \right] ,{\mathrm{R}^2}} \right)$. $C\left( {\left[ { - 1,0} \right] ,{\mathrm{R}^2}} \right)$ means the space consisting of the entire set of continuous functions whose definition domain is $[-1,0]$ and whose value domain is the set of two-dimensional real numbers. The nonlinear ${s_1}$ and ${s_2}$ are given by

$$\begin{aligned} s_1&=g_1x(t)x(t-1)+g_2x(t)m(t-1)+g_3x(t-1)m(t-1)+g_4x^2(t)x(t-1)\\&\quad +g_5x^2(t)m(t-1)+g_6x(t)x(t-1)m(t-1)+g_7x^2(t)x(t-1)m(t-1),\\ {s_2}&=h_1\left( m^2(t)+m(t)x\left( t-\frac{\tau _2^*}{\tau _1}\right) \right) , \end{aligned}$$

where

$$\begin{aligned} g_1&=(1-2x^*_+)\left( \frac{\beta _1b_1m^*_+-c_1}{n}+\beta _2b_2m^*_+\right) ,\\ g_2&=(1-2x^*_+)\left( \frac{\beta _1b_1}{n}x^*_+-\beta _2b_2(1-x^*_+)\right) ,\\ g_3&=x^*_+(1-x^*_+)\left( \frac{\beta _1b_1}{n}-\beta _2b_2\right) , \\ g_4&=c_2-\frac{\beta _1b_1m^*_+-c_1}{n}-\beta _2b_2m^*_+, \\ g_5&=\beta _2b_2(1-x^*_+)-\frac{\beta _1b_1x^*_+}{n}, \\ g_6&=(1-2x^*_+)(\frac{\beta _1b_1}{n}+\beta _2b_2),\\ g_7&=-\left( \frac{\beta _1b_1}{n}+\beta _2b_2\right) ,\\ h_1&=\varepsilon q(\beta _2-\beta _1). \end{aligned}$$

By the Riesz representation theorem, we know that a bounded variational matrix function $\gamma \left( {u ,\psi } \right)$ such that

$$\begin{aligned} {L_\psi }\left( \varphi \right) = \int _{ - 1}^0 \mathrm{d} \gamma \left( {u ,\psi } \right) \varphi \left( u \right) ,\,\varphi \in C\left( {\left[ { - 1,0} \right] ,{\mathrm{R}^2}} \right) . \end{aligned}$$

We select

$$\begin{aligned} \gamma \left( {u ,\psi } \right) = \left\{ \begin{aligned} \left( {\tau _{10}^* + \psi } \right) \left( {{B_1} + {B_2} + {B_3}} \right) ,\,u = 0, \\ \left( {\tau _{10}^* + \psi } \right) \left( {{B_2} + {B_3}} \right) ,\,-\frac{\tau _2^*}{\tau _1}\le u<0, \\ \left( {\tau _{10}^* + \psi } \right) {B_2},\,-1<u<-\frac{\tau _2^*}{\tau _1}, \\ 0,\,u = - 1. \\ \end{aligned} \right. \end{aligned}$$

(7.1)

Define

$$\begin{aligned} F\left( \psi \right) \varphi \left( u \right) = \left\{ \begin{aligned} \frac{{\mathrm{d}\varphi \left( u \right) }}{{\mathrm{d}u }},\,u \in \left[ { - 1,0} \right) , \\ \int _{ - 1}^0 {\left( {d\gamma \left( {\zeta ,\psi } \right) } \right) \varphi \left( \zeta \right) ,\,u = 0,} \\ \end{aligned} \right. \end{aligned}$$

and

$$\begin{aligned} R\left( \psi \right) \varphi \left( u \right) = \left\{ \begin{aligned} 0,\,u \in \left[ { - 1,0} \right) , \\ n\left( {\psi ,\varphi } \right) ,\,u = 0, \\ \end{aligned} \right. \end{aligned}$$

where

$$\begin{aligned}&n\left( {\psi ,\varphi } \right) = \left( {\tau _{10}^* + \psi } \right) {\left[ {{n'_1},{n'_2}} \right] ^T},\\&{n'_1}=g_1x(0)x(-1)+g_2x(0)m(-1)+g_3x(-1)m(-1)+g_4x^2(0)x(-1)\\&\qquad \ \ +g_5x^2(0)m(-1)+g_6x(0)x(-1)m(-1)+g_7x^2(0)x(-1)m(-1),\\&{n'_2}=h_1\left( m^2(0)+m(0)x\left( -\frac{\tau _2^*}{\tau _1}\right) \right) . \end{aligned}$$

Then, system (4.1) becomes

$$\begin{aligned} {{\dot{X}}_t} = F\left( \psi \right) {X_t} + R\left( \psi \right) {X_t}, \end{aligned}$$

where ${X_t} = X\left( {t + u } \right)$ for $u \in \left( { - 1,0} \right)$. For $\phi \in {C^1}\left( {\left[ { - 1,0} \right] ,{{\left( {{\mathrm{R}^2}} \right) }^*}} \right)$, define

$$\begin{aligned} {F^*}\phi \left( v \right) = \left\{ \begin{aligned} - \frac{{\mathrm{d}\phi \left( v \right) }}{{\mathrm{d}v}},\,v \in \left( { - 1,0} \right] , \\ \int _{ - 1}^0 {\phi \left( { - \zeta } \right) d\gamma \left( {\zeta ,0} \right) ,\,v = 0,} \\ \end{aligned} \right. \end{aligned}$$

and a bilinear form

$$\begin{aligned} \left\langle {\phi \left( v \right) ,\varphi \left( u \right) } \right\rangle = \bar{\phi }\left( 0 \right) \varphi \left( 0 \right) - \int _{ - 1}^0 {\int _{\zeta = 0}^u {\bar{\phi }\left( {\zeta - u } \right) d\gamma \left( u \right) \varphi \left( \zeta \right) d\zeta } } , \end{aligned}$$

(7.2)

where $\gamma \left( u \right) = \gamma \left( {u ,0} \right)$, $\bar{\phi }(0)$ is the conjugate complex of $\phi (0)$ , $F=F(0)$ and $F^{*}$ are adjoint operators. Based on the above discussion, it is clear that $\pm i{\omega ^*}\tau _{10}^*$ are both eigenvalues of F(0) and eigenvalues of $F^{*}$. Suppose that the eigenvector of F(0) corresponding to $i{\omega ^*}\tau _{10}^*$ is $p \left( u \right) {\text { = }}{\left( {1,{\delta _2}} \right) ^T}{e^{i{\omega ^*}\tau _{10}^*u }}\left( {u \in [ - 1,0]} \right)$ and the eigenvector of $F^{*}$ corresponding to $-i{\omega ^*}\tau _{10}^*$ is ${p ^*}\left( v \right) {\text { = G}}{\left( {1,\delta _2^*} \right) }{e^{i{\omega ^*}\tau _{10}^*v}}\left( {v \in [0,1]} \right)$. From (7.1) and by the definition of F(0) , it follows that

$$\begin{aligned} \tau _{10}^*\left( {\begin{array}{*{20}{c}} {i{\omega ^*} - {d_1}{e^{ - i{\omega ^*}\tau _{10}^*}}}&{}{ - {d_2}{e^{ - i{\omega ^*}\tau _{10}^*}}}\\ { - a{e^{ - i{\omega ^*}\tau _2^*}}}&{}{i{\omega ^*} - a} \end{array}} \right) p\left( 0 \right) = \left( {\begin{array}{*{20}{c}} 0\\ 0 \end{array}} \right) , \end{aligned}$$

which implies

$$\begin{aligned} p \left( 0 \right) = {\left( {1,{\delta _2}} \right) ^T} = {\left( {1,\frac{{a{e^{ - i{\omega ^*}{\tau ^*}}}}}{{i{\omega ^*} - a}}} \right) ^T}. \end{aligned}$$

It is similarly easy to obtain

$$\begin{aligned} {p ^*}\left( v \right) = G\left( {1,\delta _2^*} \right) {e^{i{\omega ^*}\tau _{10}^*v}} = G\left( {1,\frac{{ - {d_2}{e^{i{\omega ^*}\tau _{10}^*}}}}{{i{\omega ^*} + a}}} \right) {e^{i{\omega ^*}\tau _{10}^*v}}. \end{aligned}$$

Next, we need to determine G so that $\left\langle {{p ^*}\left( v \right) ,p \left( u \right) } \right\rangle {\text { = }}1$ and $\left\langle {{p ^*}\left( v \right) ,\bar{p} \left( u \right) } \right\rangle {\text { = 0}}$. According to (7.2), we have

$$\begin{aligned} \left\langle {{p ^*}\left( v \right) ,p \left( u \right) } \right\rangle= & {} \bar{G}\left( {1,\bar{\delta }_2^*} \right) {\left( {1,{\delta _2}} \right) ^T} - \int _{ - 1}^0 {\int _{\zeta = 0}^u {\bar{G}\left( {1,\bar{\delta }_2^*} \right) {e^{ - i{\omega ^*}\tau _{10}^*\left( {\zeta - u } \right) }}d\gamma \left( u \right) {{\left( {1,{\delta _2}} \right) }^T}{e^{i{\omega ^*}\tau _{10}^*\zeta }}d\zeta } }\\= & {} \bar{G}\left\{ {1 + {\delta _2}\bar{\delta }_2^* - \int _{ - 1}^0 {\left( {1,\bar{\delta }_2^*} \right) u {e^{i{\omega ^*}\tau _{10}^*u }}d\gamma \left( u\right) {{\left( {1,{\delta _2}} \right) }^T}} } \right\} \\= & {} \bar{G}\left\{ {1+\delta _2\bar{\delta }^*_2+(d_1+d_2\delta _2)\tau _{10}^*e^{-i\omega ^*\tau _{10}^*}+a\bar{\delta }_2^*e^{-i\omega ^*\tau _{10}^*}} \right\} . \end{aligned}$$

So we choose G as

$$\begin{aligned} G = {\left[ {1+\bar{\delta }_2\delta _2^*+(d_1+d_2\bar{\delta }_2^*)\tau _{10}^*e^{i\omega ^*\tau _{10}^*}+a\delta ^*_2\tau _2^*e^{i\omega ^*\tau _{10}^*}} \right] ^{ - 1}}. \end{aligned}$$

According to the algorithm in [16, 37], we can obtain:

$$\begin{aligned} g_{20}&=2\bar{G}\tau _{10}^*\left[ \left( g_1+g_2\delta _2\right) e^{-i\omega ^*\tau _{10}^*}+g_3\delta _2e^{-2i\omega ^*\tau _{10}^*}+\bar{\delta }_2^*h_1\left( \delta _2^2+\delta _2e^{-i\omega ^*\tau _{2}^*}\right) \right] ,\\ g_{11}&=\bar{G}\tau _{10}^*\left[ g_1\left( e^{i\omega ^*\tau _{10}^*}+e^{-i\omega ^*\tau _{10}^*}\right) +g_2\left( \bar{\delta }_2e^{i\omega ^*\tau _{10}^*}+\delta _2e^{-i\omega ^*\tau _{10}^*}\right) \right. \\&\quad \left. +g_3(\bar{\delta }_2+\delta _2)+h_1\bar{\delta }_2^*\left( 2\delta _2\bar{\delta }_2+\delta _2e^{i\omega ^*\tau _{2}^*}+\bar{\delta }_2e^{-i\omega ^*\tau _{2}^*}\right) \right] ,\\ g_{02}&=2\bar{G}\tau _{10}^*\left[ \left( g_1+g_2\bar{\delta }_2\right) e^{i\omega ^*\tau _{10}^*}+\bar{\delta }_2^*h_1\left( \bar{\delta }_2^2+\bar{\delta }_2e^{i\omega ^*\tau _{2}^*}\right) +g_3\bar{\delta }_2e^{2i\omega ^*\tau _{10}^*}\right] ,\\ g_{21}&=2\bar{G}\tau _{10}^*\left[ g_1\left( \frac{W_{20}^{(1)}(-1)}{2}+W_{11}^{(1)}(0)e^{-i\omega ^*\tau _{10}^*}+\frac{W_{20}^{(1)}(0)}{2}e^{i\omega ^*\tau _{10}^*}\right) \right. \\&\quad +g_2\left( \frac{W_{20}^{(2)}(-1)}{2}+\frac{W_{20}^{(1)}(0)}{2}\bar{\delta }_2e^{i\omega ^*\tau _{10}^*}+W_{11}^{(2)}(-1)+W_{11}^{(1)}(0)\delta _2e^{-i\omega ^*\tau _{10}^*}\right) \\&\quad +g_3\left( \frac{W_{20}^{(2)}(-1)}{2}e^{i\omega ^*\tau _{10}^*}+\frac{W_{20}^{(2)}(-2)}{2}\bar{\delta }_2e^{i\omega ^*\tau _{10}^*}\right. \\&\quad \left. +\frac{W_{11}^{(2)}(-1)}{2}e^{-i\omega ^*\tau _{10}^*}+W_{11}^{(1)}(-1)\delta _2e^{-i\omega ^*\tau _{10}^*}\right) \\&\quad +g_4\left( 2e^{-i\omega ^*\tau _{10}^*}+e^{i\omega ^*\tau _{10}^*}\right) +g_5\left( 2\delta _2e^{-i\omega ^*\tau _{10}^*}+\bar{\delta }_2e^{i\omega ^*\tau _{10}^*}\right) \\&\quad +g_6\left( \delta _2e^{-2i\omega ^*\tau _{10}^*}+\delta _2+\bar{\delta }_2\right) \\&\quad \left. +h_1\bar{\delta }_2^*\left( \bar{\delta }_2{W_{20}^{(2)}}(0)+2\delta _2W_{11}^{(2)}(0)+\frac{\bar{\delta }_2}{2}W_{20}^{(1)}\left( -\frac{\tau _2^*}{\tau _{10}^*}\right) \right. \right. \\&\quad \left. \left. +\frac{W_{20}^{(2)}(0)}{2}e^{i\omega ^*\tau _{2}^*}+\delta _2W_{11}^{(1)}\left( -\frac{\tau _2^*}{\tau _{10}^*}\right) +W_{11}^{(2)}(0)e^{-i\omega ^*\tau _{2}^*}\right) \right] , \end{aligned}$$

where

$$\begin{aligned} {W_{20}^{(1)}}\left( \vartheta \right)= & {} \frac{{i{g_{20}}{e^{i{\omega ^*}\tau _{10}^*\vartheta }}}}{{{\omega ^*}\tau _{10}^*}} + \frac{{i{{\bar{g}}_{02}} {e^{ - i{\omega ^*}\tau _{10}^*\vartheta }}}}{{3{\omega ^*}\tau _{10}^*}} + {A_1}{e^{2i{\omega ^*}\tau _{10}^*\vartheta }}, \\ {W_{20}^{(2)}}\left( \vartheta \right)= & {} \frac{{i{g_{20}}\delta _2{e^{i{\omega ^*}\tau _{10}^*\vartheta }}}}{{{\omega ^*}\tau _{10}^*}} + \frac{{i{{\bar{g}}_{02}}\bar{\delta }_2{e^{ - i{\omega ^*}\tau _{10}^*\vartheta }}}}{{3{\omega ^*}\tau _{10}^*}} + {A_1}{e^{2i{\omega ^*}\tau _{10}^*\vartheta }}, \\ {W_{11}^{(1)}}\left( \vartheta \right)= & {} - \frac{{i{g_{11}}{e^{i{\omega ^*}\tau _{10}^*\vartheta }}}}{{{\omega ^*}\tau _{10}^*}} + \frac{{i{{\bar{g}}_{11}}{e^{ - i{\omega ^*}\tau _{10}^*\vartheta }}}}{{{\omega ^*}\tau _{10}^*}} + {A_2}, \\ {W_{11}^{(2)}}\left( \vartheta \right)= & {} - \frac{{i{g_{11}}\delta _2{e^{i{\omega ^*}\tau _{10}^*\vartheta }}}}{{{\omega ^*}\tau _{10}^*}} + \frac{{i{{\bar{g}}_{11}}\bar{\delta }_2{e^{ - i{\omega ^*}\tau _{10}^*\vartheta }}}}{{{\omega ^*}\tau _{10}^*}} + {A_2}. \end{aligned}$$

$A_{1}$ and $A_{2}$ are given by

$$\begin{aligned} \begin{array}{l} {A_1} = 2\left( {\begin{array}{*{20}{c}} {2i{\omega ^*} - {d_1}{e^{ - 2i{\omega ^*}\tau _{10}^*}}}&{}{ - {d_2}{e^{ - 2i{\omega ^*}\tau _{10}^*}}}\\ { - a{e^{ - 2i{\omega ^*}\tau _2^*}}}&{}{2i{\omega ^*} - a} \end{array}} \right) \left( {\begin{array}{*{20}{c}} {\left( {{g_1} + {g_2}{\delta _2}} \right) {e^{ - i{\omega ^*}\tau _{10}^*}} + {g_3}{\delta _2}{e^{ - 2i{\omega ^*}\tau _{10}^*}}}\\ {{h_1}{\delta _2}\left( {{\delta _2} + {e^{ - i{\omega ^*}\tau _2^*}}} \right) } \end{array}} \right) ,\\ {A_2} = 2{\left( {\begin{array}{*{20}{c}} { - {d_1}}&{}{ - {d_2}}\\ { - a}&{}{ - a} \end{array}} \right) ^{ - 1}}\\ \qquad \qquad \quad \left( {\begin{array}{*{20}{c}} {{g_1}\left( {{e^{i{\omega ^*}\tau _{10}^*}} + {e^{ - i{\omega ^*}\tau _{10}^*}}} \right) + {g_2}\left( {{{\bar{\delta }}_2}{e^{i{\omega ^*}\tau _{10}^*}} + {\delta _2}{e^{ - i{\omega ^*}\tau _{10}^*}}} \right) + {g_3}\left( {{{\bar{\delta }}_2} + {\delta _2}} \right) }\\ {{h_1}\left( {2{\delta _2}{{\bar{\delta }}_2} + {\delta _2}{e^{i{\omega ^*}\tau _2^*}} + {{\bar{\delta }}_2}{e^{ - i{\omega ^*}\tau _2^*}}} \right) } \end{array}} \right) . \end{array} \end{aligned}$$

Finally, the following values can be obtained:

$$\begin{aligned} {\rho _1}\left( 0 \right)&= \frac{i\left[ {{g_{20}}{g_{11}} - 2{{\left| {{g_{11}}} \right| }^2} - \frac{{{{\left| {{g_{02}}} \right| }^2}}}{3}} \right] }{{2{\omega ^*}\tau _{10}^*}} + \frac{{{g_{21}}}}{2},\\ {\eta _2}&= - \frac{{{\text {Re}} \left[ {{\rho _1}\left( 0 \right) } \right] }}{{{\text {Re}} \left[ {\lambda '\left( {\tau _{10}^*} \right) } \right] }},\\ \kappa&= 2{\text {Re}} \left[ {{\rho _1}\left( 0 \right) } \right] , \\ {T_2}&= - \frac{{{\text {Im}} \left[ {{\rho _1}\left( 0 \right) } \right] + {u_2}{\text {Im}} \left( {\lambda '\left( {\tau _{10}^*} \right) } \right) }}{{{\omega ^*}\tau _{10}^*}}. \end{aligned}$$