# 一类带有时滞效应的修正的Leslie-Gower捕食者–食饵模型动力学分析Dynamics Analysis of a Modified Leslie-Gower Predator-Prey Model with Time Delay

DOI: 10.12677/AAM.2021.104131, PDF, HTML, XML, 下载: 12  浏览: 39

Abstract: This paper constructs a modified Leslie-Gower predator-prey model with Allee effect and prey refuge. The aim of this paper is to study the effect of time delay on the population dynamics. By using delay differential equation theory, we demonstrate the existence of Hopf bifurcation which is induced by time delay. According to center manifold theorem and normal form theory, the direction of Hopf bifurcation and stability of periodic solution are discussed. Additionally, numerical simulations reveal that the population density and the stability of positive equilibrium are sensitive for variations of time delay, Allee effect and prey refuge.

1. 引言

$\left\{\begin{array}{l}\frac{\text{d}u}{\text{d}t}=u\left({r}_{1}-{a}_{1}u-{c}_{1}v\right)\\ \frac{\text{d}v}{\text{d}t}=v\left({r}_{2}-\frac{{c}_{2}v}{u}\right)\end{array}$ (1.1)

$\left\{\begin{array}{l}\frac{\text{d}u}{\text{d}t}=u\left({r}_{1}-{a}_{1}u-\frac{h}{b+u}-{c}_{1}\left(1-m\right)v\right)\\ \frac{\text{d}v}{\text{d}t}=v\left({r}_{2}-\frac{{c}_{2}v\left(t-\tau \right)}{\left(1-m\right)u\left(t-\tau \right)+k}\right)\end{array}$ (1.2)

2. 动力学性质

2.1. 正平衡点的存在性与稳定性

$A=-{a}_{1}{c}_{2}-{c}_{1}{\left(1-m\right)}^{2}{r}_{2}$

$D=b{r}_{1}{c}_{2}-h{c}_{2}-b{c}_{1}k\left(1-m\right){r}_{2},$

$B={r}_{1}{c}_{2}-b{a}_{1}{c}_{2}-b{c}_{1}{\left(1-m\right)}^{2}{r}_{2}-{c}_{1}k\left(1-m\right){r}_{2}.$

${H}_{1}:m<\frac{h{c}_{2}+{c}_{1}k{r}_{2}-{r}_{1}{c}_{2}}{k{c}_{1}{r}_{2}},B>0,\Delta ={B}^{2}-4AD>0.$

${u}_{4}=\frac{-B+\sqrt{{B}^{2}-4AD}}{2A},{v}_{4}=\frac{{r}_{2}\left(\left(1-m\right){u}_{4}+k\right)}{{c}_{2}}$

${u}_{5}=\frac{-B-\sqrt{{B}^{2}-4AD}}{2A},{v}_{4}=\frac{{r}_{2}\left(\left(1-m\right){u}_{5}+k\right)}{{c}_{2}}$

${J}_{{E}_{i}}=\left(\begin{array}{cc}{u}_{i}\left(\frac{h}{{\left(b+{u}_{i}\right)}^{2}}-{a}_{1}\right)& -{c}_{1}\left(1-m\right){u}_{i}\\ \frac{{r}_{2}^{2}\left(1-m\right)}{{c}_{2}}& -{r}_{2}\end{array}\right),i=4,5$

$tr\left({J}_{{E}_{i}}\right)={u}_{i}\left(\frac{h}{{\left(b+{u}_{i}\right)}^{2}}-{a}_{1}\right)-{r}_{2},$ (2.1.1)

$det\left({J}_{{E}_{i}}\right)=\frac{{u}_{i}{r}_{2}}{{c}_{2}{\left(b+{u}_{i}\right)}^{2}}\left(\left({a}_{1}{c}_{2}+{c}_{1}{\left(1-m\right)}^{2}{r}_{2}\right){\left(b+{u}_{i}\right)}^{2}-h{c}_{2}\right).$ (2.1.2)

$G\left(u\right)=\left(\left({a}_{1}{c}_{2}+{c}_{1}{\left(1-m\right)}^{2}{r}_{2}\right){\left(b+{u}_{i}\right)}^{2}-h{c}_{2}\right),$

${P}_{1}=\frac{2bA+\sqrt{-4h{c}_{2}A}}{-2A}.$

$tr\left({J}_{{E}_{5}}\right)=\frac{-1}{8{\left(b+{u}_{5}\right)}^{2}{A}^{3}}\left[{L}_{1}-{L}_{2}\sqrt{{\left(E-bA\right)}^{2}+4h{c}_{2}A}\right],$

${L}_{1}=h\left[4{A}^{2}\left(\left(E+bA\right)+2{r}_{2}{c}_{2}+b{a}_{1}{c}_{2}\right)-12AE{a}_{1}{c}_{2}\right]+4{\left(E-bA\right)}^{2}\left(A{r}_{2}-{a}_{1}E\right),$

${L}_{2}=h\left[-4{A}^{2}+4A{a}_{1}{c}_{2}\right]+4\left(E-bA\right)\left({a}_{1}E-A{r}_{2}\right),$

$E={r}_{1}{c}_{2}-{c}_{1}k\left(1-m\right){r}_{2}.$

$A<0$ 可知 $tr\left({J}_{{E}_{5}}\right)$ 的符号与函数 ${L}_{1}-{L}_{2}\sqrt{{\left(E-bA\right)}^{2}+4h{c}_{2}A}$ 一致。

${h}_{1}=\frac{-4{\left(E-bA\right)}^{2}\left(A{r}_{2}-{a}_{1}E\right)}{4{A}^{2}\left(\left(E+bA\right)+2{r}_{2}{c}_{2}+b{a}_{1}{c}_{2}\right)-12AE{a}_{1}{a}_{2}},$

${h}_{2}=\frac{-4\left(E-bA\right)\left({a}_{1}E-A{r}_{2}\right)}{-4{A}^{2}+4A{a}_{1}{c}_{2}}.$

1) ${h}_{2}>{h}_{1}$

${L}_{1}^{2}-{L}_{2}^{2}\left({\left(E-bA\right)}^{2}+4h{c}_{2}A\right)={h}^{2}\left(-4A{c}_{2}{\left(4A{a}_{1}{c}_{2}-4{A}^{2}\right)}^{2}\right)+h{M}_{1}+{M}_{2},$

${M}_{1}=64{A}^{3}\left[\left({a}_{1}b-{r}_{2}\right)\left(E{a}_{1}-A{r}_{2}\right){c}_{2}^{2}+\left(b{A}^{2}\left({a}_{1}b+3{r}_{2}\right)-4EA\left({a}_{1}b+\frac{1}{4}{r}_{2}\right)+{E}^{2}{a}_{1}\right){c}_{2}+{A}^{2}Eb\right],$

${M}_{2}=64{A}^{6}{b}^{3}{r}_{2}-64{A}^{5}E{b}^{2}{r}_{2}-128{A}^{4}{E}^{2}b{r}_{2}+128{A}^{4}{E}^{2}b{r}_{2}-64{A}^{3}{E}^{3}{a}_{1}b.$

${\Delta }_{1}={M}_{1}^{2}+16{M}_{2}A{c}_{2}{\left(4A{a}_{1}{c}_{2}-4{A}^{2}\right)}^{2}.$

${h}_{3}=\frac{-{M}_{1}-\sqrt{{\Delta }_{1}}}{-8A{c}_{2}{\left(4A{a}_{1}{c}_{2}-4{A}^{2}\right)}^{2}},{h}_{4}=\frac{-{M}_{1}+\sqrt{{\Delta }_{1}}}{-8A{c}_{2}{\left(4A{a}_{1}{c}_{2}-4{A}^{2}\right)}^{2}}.$

2) ${h}_{2}<{h}_{1}$

i) ${M}_{1}>0$ ；ii) ${M}_{1}<0,{\Delta }_{1}<0$ ；iii) ${M}_{1}<0,{\Delta }_{1}>0,h\in \left(0,{h}_{3}\right)\cup \left({h}_{4},+\infty \right)$

a) ${h}_{2}>{h}_{1},0，(b) ${h}_{2}>{h}_{1},{h}_{1}0,{h}_{3}

c) ${h}_{2}<{h}_{1},0，(d) ${h}_{2}<{h}_{1},{M}_{1}>0,{h}_{2}

e) ${h}_{2}<{h}_{1},{M}_{1}<0,{h}_{2}0,h\in \left(0,{h}_{3}\right)\cup \left({h}_{4},+\infty \right)$

f) ${h}_{2}<{h}_{1},{M}_{1}<0,{h}_{2}

2.2. Hopf分岔

$\left\{\begin{array}{l}x\left(t\right)=u\left(t\right)-{u}_{5}\\ y\left(t\right)=v\left(t\right)-{v}_{5}\end{array}$

$\left\{\begin{array}{l}\frac{\text{d}x\left(t\right)}{\text{d}t}={a}_{11}x\left(t\right)+{a}_{12}y\left(t\right)\\ \frac{\text{d}y\left(t\right)}{\text{d}t}={a}_{21}x\left(t-\tau \right)+{a}_{22}y\left(t-\tau \right)+{a}_{23}y\left(t\right)\end{array}$ (2.2.1)

${a}_{11}={u}_{5}\left(\frac{h}{{\left(b+{u}_{5}\right)}^{2}}-{a}_{1}\right),{a}_{12}=-{c}_{1}\left(1-m\right){u}_{5},$

${a}_{21}=\frac{{r}_{2}^{2}\left(1-m\right)}{{c}_{2}},{a}_{22}=-{r}_{2},{a}_{23}=0.$

$\left[\begin{array}{cc}{a}_{11}& {a}_{12}\\ {a}_{21}{\text{e}}^{-\lambda \tau }& {a}_{22}{\text{e}}^{-\lambda \tau }\end{array}\right]$

${\lambda }^{2}-\left({a}_{11}+{a}_{22}{\text{e}}^{-\lambda \tau }\right)\lambda +\left({a}_{11}{a}_{22}-{a}_{12}{a}_{21}\right){\text{e}}^{-\lambda \tau }=0.$ (2.2.2)

$\left\{\begin{array}{l}-{\omega }^{2}\left(\tau \right)+{a}_{22}\omega \left(\tau \right)\mathrm{sin}\omega \tau +\left({a}_{11}{a}_{22}-{a}_{12}{a}_{21}\right)\mathrm{cos}\omega \tau =0\\ -{a}_{11}\omega \left(\tau \right)-{a}_{22}\omega \left(\tau \right)\mathrm{cos}\omega \tau +\left({a}_{11}{a}_{22}-{a}_{12}{a}_{21}\right)\mathrm{sin}\omega \tau =0\end{array}$ (2.2.3)

${\omega }^{4}\left(\tau \right)+\left({a}_{11}^{2}-{a}_{22}^{2}\right){\omega }^{2}\left(\tau \right)-{\left({a}_{11}{a}_{22}-{a}_{12}{a}_{21}\right)}^{2}=0.$ (2.2.4)

${\omega }_{0}=\sqrt{\frac{\left({a}_{22}^{2}-{a}_{11}^{2}\right)+\sqrt{{\left({a}_{22}^{2}-{a}_{11}^{2}\right)}^{2}+4{\left({a}_{11}{a}_{22}-{a}_{12}{a}_{21}\right)}^{2}}}{2}}$ .

$\mathrm{cos}{\omega }_{0}\tau =\frac{-{\omega }_{0}^{2}{a}_{21}{a}_{12}}{{a}_{22}^{2}{\omega }_{0}^{2}+{\left({a}_{11}{a}_{22}-{a}_{12}{a}_{21}\right)}^{2}},$

${\tau }_{j}=\frac{1}{{\omega }_{0}}\mathrm{arccos}\frac{-{\omega }_{0}^{2}{a}_{21}{a}_{12}}{{a}_{22}^{2}{\omega }_{0}^{2}+{\left({a}_{11}{a}_{22}-{a}_{12}{a}_{21}\right)}^{2}}+\frac{2j\pi }{{\omega }_{0}},j=0,1,2,\cdots .$

${\left(\frac{\text{d}\lambda }{\text{d}\tau }\right)}^{-1}=\left(\frac{2\lambda -{a}_{11}}{\lambda \left({a}_{11}\lambda -{\lambda }^{2}\right)}-\frac{\tau }{\lambda }-\frac{{a}_{22}}{\lambda \left({a}_{11}{a}_{22}-{a}_{12}{a}_{21}\right)-{a}_{22}{\lambda }^{2}}\right).$

$\tau ={\tau }_{0}$ 代入上述方程 ${\left(\frac{\text{d}\lambda }{\text{d}\tau }\right)}^{-1}$ 中，通过分离实部与虚部可得

${Re{\left(\frac{\text{d}\lambda }{\text{d}\tau }\right)}^{-1}|}_{\tau ={\tau }_{0}}=\frac{{a}_{11}^{2}{\left({a}_{11}{a}_{22}-{a}_{12}{a}_{21}\right)}^{2}+2{\omega }_{0}^{2}{\left({a}_{11}{a}_{22}-{a}_{12}{a}_{21}\right)}^{2}+{a}_{22}^{2}{\omega }_{0}^{4}}{\left({a}_{11}^{2}{\omega }_{0}^{2}+{\omega }_{0}^{4}\right)\left({\left({a}_{11}{a}_{22}-{a}_{12}{a}_{21}\right)}^{2}+{a}_{22}^{2}{\omega }_{0}^{2}\right)}>0.$

2.3. Hopf分岔方向

$x\left(t\right)=u\left(t\tau \right)-{u}_{5},y\left(t\right)=v\left(t\tau \right)-{v}_{5},$

$\tau ={\tau }_{0}+\delta ,\varphi ={\left(x,y\right)}^{\text{T}},{\varphi }_{t}\left(\theta \right)=\varphi \left(t+\theta \right),\theta \in \left[-1,0\right].$

$\left(\begin{array}{c}\frac{\text{d}x}{\text{d}t}\\ \frac{\text{d}y}{\text{d}t}\end{array}\right)=\tau \left({B}_{1}\left(\begin{array}{c}x\left(t\right)\\ y\left(t\right)\end{array}\right)+{B}_{2}\left(\begin{array}{c}x\left(t-1\right)\\ y\left(t-1\right)\end{array}\right)+f\right)$ (2.3.1)

${a}_{11}=-{a}_{1}{u}_{5}+\frac{h{u}_{5}}{{\left(b+{u}_{5}\right)}^{2}},{a}_{12}=-{c}_{1}\left(1-m\right){u}_{5},$

${a}_{21}=\frac{{r}_{2}^{2}\left(1-m\right)}{{c}_{2}},{a}_{22}=-{r}_{2}.$

${f}_{1}=\left(\frac{hb}{{\left(b+{u}_{5}\right)}^{3}}-{a}_{1}\right){x}^{2}\left(t\right)-{c}_{1}\left(1-m\right)x\left(t\right)y\left(t\right),$

$\begin{array}{c}{f}_{2}=-\frac{{\left(1-m\right)}^{2}{r}_{2}^{2}}{{c}_{2}\left(\left(1-m\right){u}_{5}+k\right)}{x}^{2}\left(t-1\right)+\frac{{r}_{2}\left(1-m\right)}{\left(1-m\right){u}_{5}+k}x\left(t-1\right)y\left(t\right)\\ \text{\hspace{0.17em}}+\frac{{r}_{2}\left(1-m\right)}{\left(1-m\right){u}_{5}+k}x\left(t-1\right)y\left(t-1\right)-\frac{{c}_{2}}{\left(1-m\right){u}_{5}+k}y\left(t\right)y\left(t-1\right).\end{array}$

${L}_{\delta }\left(\varphi \right)={\int }_{-1}^{0}\text{d}\eta \left(\theta ,\delta \right)\varphi \left(\theta \right),\varphi \in C\left(\left[-1,0\right],{R}^{2}\right).$

$\eta \left(\theta ,\delta \right)={\tau }_{0}+\delta \left[{B}_{1}\kappa \left(\theta \right)-{B}_{2}\kappa \left(\theta +1\right)\right].$

$\kappa \left(\theta \right)=\left\{\begin{array}{l}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }\theta \ne 0\\ 1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\theta =0\end{array}$

$P\left(\delta \right)\varphi \left(\theta \right)=\left\{\begin{array}{l}\frac{\text{d}\varphi \left(\theta \right)}{\text{d}\theta },\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-1\le \theta <0\\ {\int }_{-1}^{0}\text{d}\eta \left(s,\delta \right)\varphi \left(s\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\theta =0\end{array}$

$Q\left(\delta \right)\varphi \left(\theta \right)=\left\{\begin{array}{l}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-1\le \theta <0\\ f\left(\delta \right)\varphi \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\theta =0\end{array}$

$\frac{\text{d}U}{\text{d}t}=P\left(\delta \right){U}_{t}+Q\left(\delta \right){U}_{t}$ (2.3.2)

$U={\left(x,y\right)}^{\text{T}},U\left(t\right)=U\left(t+\theta \right),\theta \in \left[-1,0\right]$

${P}^{*}\psi \left(s\right)=\left\{\begin{array}{l}-\frac{\text{d}\psi \left(s\right)}{\text{d}s},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0

$〈\psi \left(s\right),\varphi \left(\theta \right)〉=\stackrel{¯}{\psi }\left(0\right)\varphi \left(0\right)-{\int }_{-1}^{0}{\int }_{\varsigma =0}^{\theta }\stackrel{¯}{\psi }\left(\varsigma -\theta \right)\text{d}\eta \left(\theta \right)\varphi \left(\varsigma \right).$

$P\left(0\right)$ 对应特征值 $i{\tau }_{0}{\omega }_{0}$ 的特征向量为 $q\left(\theta \right)$${P}^{*}$ 对应特征值 $-i{\tau }_{0}{\omega }_{0}$ 的特征向量为 ${q}^{*}\left( s \right)$

$q\left(\theta \right)={\left(1,\alpha \right)}^{\text{T}}{\text{e}}^{i{\omega }_{0}{\tau }_{0}\theta },{q}^{*}\left(s\right)=D{\left(1,\beta \right)}^{\text{T}}{\text{e}}^{i{\omega }_{0}{\tau }_{0}s}.$

$P\left(0\right)q\left(\theta \right)=q\left(\theta \right)i{\tau }_{0}{\omega }_{0}.$

$\left[i{\omega }_{0}I-\left({K}_{1}+{K}_{2}\right){\text{e}}^{-i{\omega }_{0}{\tau }_{0}}\right]q\left(0\right)=0.$

$\left[\begin{array}{cc}i{\omega }_{0}-{a}_{11}& -{a}_{12}\\ -{a}_{21}{\text{e}}^{-i{\omega }_{0}{\tau }_{0}}& i{\omega }_{0}-{a}_{22}{\text{e}}^{-i{\omega }_{0}{\tau }_{0}}\end{array}\right]\left(\begin{array}{c}1\\ \alpha \end{array}\right)=\left(\begin{array}{c}0\\ 0\end{array}\right),$

$\left[\begin{array}{cc}i{\omega }_{0}+{a}_{11}& {a}_{21}{\text{e}}^{-i{\omega }_{0}{\tau }_{0}}\\ {a}_{12}& i{\omega }_{0}+{a}_{22}{\text{e}}^{-i{\omega }_{0}{\tau }_{0}}\end{array}\right]\left(\begin{array}{c}1\\ \beta \end{array}\right)=\left(\begin{array}{c}0\\ 0\end{array}\right).$

$\alpha =\frac{i{\omega }_{0}-{a}_{11}}{{a}_{12}},\beta =-\frac{\left(i{\omega }_{0}+{a}_{11}\right){\text{e}}^{i{\tau }_{0}{\omega }_{0}}}{{a}_{21}}.$

$\begin{array}{c}{q}^{*}\left(s\right),q\left(\theta \right)={\stackrel{¯}{q}}^{*}\left(0\right)q\left(0\right)-{\int }_{-1}^{0}{\int }_{0}^{\theta }{\stackrel{¯}{q}}^{*}\left(\varsigma -\theta \right)\text{d}\eta \left(\theta \right)q\left(\varsigma \right)\text{d}\varsigma \\ =\stackrel{¯}{D}\left(\begin{array}{cc}1& \stackrel{¯}{\beta }\end{array}\right)\left(\begin{array}{c}1\\ \alpha \end{array}\right)-{\int }_{-1}^{0}{\int }_{0}^{\theta }\stackrel{¯}{D}\left(\begin{array}{cc}1& \stackrel{¯}{\beta }\end{array}\right){\text{e}}^{-i{\omega }_{0}{\tau }_{0}\left(\varsigma -\theta \right)}\text{d}\eta \left(\theta \right)\left(\begin{array}{c}1\\ \alpha \end{array}\right){\text{e}}^{-i{\omega }_{0}{\tau }_{0}\varsigma }\text{d}\varsigma \\ =\stackrel{¯}{D}\left(1+\alpha \stackrel{¯}{\beta }\right)+{\tau }_{0}\stackrel{¯}{D}{\text{e}}^{-i{\tau }_{0}{\omega }_{0}}\left(\frac{\stackrel{¯}{\beta }{r}_{2}^{2}\left(1-m\right)}{{c}_{2}}-\stackrel{¯}{\beta }{r}_{2}\alpha \right).\end{array}$

$D={\left(1+\stackrel{¯}{\alpha }\beta +{\tau }_{\text{0}}{\text{e}}^{i{\tau }_{0}{\omega }_{0}}\left(\frac{\beta {r}_{2}^{2}\left(1-m\right)}{{c}_{2}}-\beta {r}_{2}\stackrel{¯}{\alpha }\right)\right)}^{-1},$

$〈{q}^{*}\left(s\right),q\left(\theta \right)〉=1,〈{q}^{*}\left(s\right),\stackrel{¯}{q}\left(\theta \right)〉=0.$

$z\left(t\right)=〈{q}^{*},{U}_{t}〉,W\left(t,\theta \right)={U}_{t}\left(\theta \right)-2Re\left\{z\left(t\right)q\left(\theta \right)\right\},$ (2.3.3)

$W\left(t,\theta \right)=W\left(z,\stackrel{¯}{z},\theta \right)={W}_{20}\left(\theta \right)\frac{{z}^{2}}{2}+{W}_{11}\left(\theta \right)z\stackrel{¯}{z}+{W}_{02}\left(\theta \right)\frac{{\stackrel{¯}{z}}^{2}}{2}+{W}_{30}\left(\theta \right)\frac{{z}^{3}}{6}+\cdots$ (2.3.4)

$\delta =0$ 时，对于 ${U}_{t}\in {C}_{0}$，有下式成立：

$\begin{array}{l}\frac{\text{d}z\left(t\right)}{\text{d}t}=〈{q}^{*},\stackrel{˙}{U}\left(t\right)〉=i{\omega }_{0}{\tau }_{0}z\left(t\right)+〈{\stackrel{¯}{q}}^{*}\left(\theta \right),Q\left(0\right)\left(W\left(z,\stackrel{¯}{z},\theta \right)+2Re\left\{z\left(t\right)q\left(\theta \right)\right\}\right)〉\\ ={\stackrel{¯}{q}}^{*}\left(0\right)f\left(0\right)\left(W\left(z,\stackrel{¯}{z},0\right)+2Re\left\{z\left(t\right)q\left(0\right)\right\}\right)\\ \triangleq i{\omega }_{0}{\tau }_{0}z\left(t\right)+g\left(z,\stackrel{¯}{z}\right).\end{array}$

$\begin{array}{c}g\left(z,\stackrel{¯}{z}\right)={\stackrel{¯}{q}}^{*}\left(0\right)f\left(0\right)\left(W\left(z,\stackrel{¯}{z},0\right)+2Re\left\{z\left(t\right)q\left(0\right)\right\}\right)=\stackrel{¯}{D}{\tau }_{0}\left({f}_{1}+\stackrel{¯}{\beta }{f}_{2}\right)\\ ={g}_{20}\frac{{z}^{2}}{2}+{g}_{11}z\stackrel{¯}{z}+{g}_{02}\frac{{\stackrel{¯}{z}}^{2}}{2}+{g}_{21}\frac{{z}^{2}\stackrel{¯}{z}}{2}+\cdots \end{array}$ (2.3.5)

$q\left(0\right)={\left(1,\alpha \right)}^{\text{T}},$

$\begin{array}{c}{U}_{t}\left(\theta \right)={\left({x}_{t}\left(\theta \right),{y}_{t}\left(\theta \right)\right)}^{\text{T}}=2Re\left\{z\left(t\right)q\left(\theta \right)\right\}+W\left(t,\theta \right)\\ =zq\left(\theta \right)+\stackrel{¯}{z}\text{ }\stackrel{¯}{q}\left(\theta \right)+W\left(t,\theta \right).\end{array}$

$x\left(t\right)=z+\stackrel{¯}{z}+{W}_{20}^{\left(1\right)}\left(0\right)\frac{{z}^{2}}{2}+{W}_{11}^{\left(1\right)}\left(0\right)z\stackrel{¯}{z}+{W}_{02}^{\left(1\right)}\left(0\right)\frac{{\stackrel{¯}{z}}^{2}}{2},$

$y\left(t\right)=\alpha z+\stackrel{¯}{\alpha }\stackrel{¯}{z}+{W}_{20}^{\left(2\right)}\left(0\right)\frac{{z}^{2}}{2}+{W}_{11}^{\left(2\right)}\left(0\right)z\stackrel{¯}{z}+{W}_{02}^{\left(2\right)}\left(0\right)\frac{{\stackrel{¯}{z}}^{2}}{2},$

$x\left(t-1\right)=z{\text{e}}^{-i{\omega }_{0}{\tau }_{0}}+\stackrel{¯}{z}{\text{e}}^{i{\omega }_{0}{\tau }_{0}}+{W}_{20}^{\left(1\right)}\left(-1\right)\frac{{z}^{2}}{2}+{W}_{11}^{\left(1\right)}\left(-1\right)z\stackrel{¯}{z}+{W}_{02}^{\left(1\right)}\left(-1\right)\frac{{\stackrel{¯}{z}}^{2}}{2},$

$y\left(t-1\right)=\alpha z{\text{e}}^{-i{\omega }_{0}{\tau }_{0}}+\stackrel{¯}{\alpha }\stackrel{¯}{z}{\text{e}}^{i{\omega }_{0}{\tau }_{0}}+{W}_{20}^{\left(2\right)}\left(-1\right)\frac{{z}^{2}}{2}+{W}_{11}^{\left(2\right)}\left(-1\right)z\stackrel{¯}{z}+{W}_{02}^{\left(2\right)}\left(-1\right)\frac{{\stackrel{¯}{z}}^{2}}{2}.$

${g}_{20}=2\stackrel{¯}{D}{\tau }_{0}\left[{f}_{11}+{f}_{12}\alpha +{f}_{21}\stackrel{¯}{\beta }{\text{e}}^{-2i{\omega }_{0}{\tau }_{0}}+{f}_{22}\stackrel{¯}{\beta }\alpha {\text{e}}^{-i{\omega }_{0}{\tau }_{0}}+{f}_{23}\stackrel{¯}{\beta }\alpha {\text{e}}^{-2i{\omega }_{0}{\tau }_{0}}+{f}_{24}{\alpha }^{2}\stackrel{¯}{\beta }{\text{e}}^{-i{\omega }_{0}{\tau }_{0}}\right],$

$\begin{array}{c}{g}_{11}=\stackrel{¯}{D}{\tau }_{0}\left[2{f}_{11}+{f}_{12}\left(\stackrel{¯}{\alpha }+\alpha \right)+2{f}_{21}\stackrel{¯}{\beta }+{f}_{22}\stackrel{¯}{\beta }\left(\stackrel{¯}{\alpha }{\text{e}}^{-i{\omega }_{0}{\tau }_{0}}+\alpha {\text{e}}^{i{\omega }_{0}{\tau }_{0}}\right)\\ \text{\hspace{0.17em}}+{f}_{23}\stackrel{¯}{\beta }\left(\stackrel{¯}{\alpha }+\alpha \right)+{f}_{24}\stackrel{¯}{\beta }\left(\alpha \stackrel{¯}{\alpha }{\text{e}}^{i{\omega }_{0}{\tau }_{0}}+\alpha \stackrel{¯}{\alpha }{\text{e}}^{-i{\omega }_{0}{\tau }_{0}}\right)\right],\end{array}$

${g}_{02}=2\stackrel{¯}{D}{\tau }_{0}\left[{f}_{11}+{f}_{12}\stackrel{¯}{\alpha }+{f}_{21}\stackrel{¯}{\beta }{\text{e}}^{2i{\omega }_{0}{\tau }_{0}}+{f}_{22}\stackrel{¯}{\beta }\stackrel{¯}{\alpha }{\text{e}}^{i{\omega }_{0}{\tau }_{0}}+{f}_{23}\stackrel{¯}{\beta }\stackrel{¯}{\alpha }{\text{e}}^{2i{\omega }_{0}{\tau }_{0}}+{f}_{24}{\stackrel{¯}{\alpha }}^{2}\stackrel{¯}{\beta }{\text{e}}^{i{\omega }_{0}{\tau }_{0}}\right],$

$\begin{array}{c}{g}_{21}=2\stackrel{¯}{D}{\tau }_{0}\left[{f}_{11}\left({W}_{20}^{1}\left(0\right)+2{W}_{11}^{1}\left(0\right)\right)+{f}_{12}\left({W}_{11}^{2}\left(0\right)+\frac{1}{2}{W}_{20}^{2}\left(0\right)+\frac{1}{2}{W}_{20}^{1}\left(0\right)\stackrel{¯}{\alpha }+{W}_{11}^{1}\left(0\right)\alpha \right)\\ \text{\hspace{0.17em}}+\stackrel{¯}{\beta }\left({f}_{21}\left({W}_{20}^{1}\left(-1\right){\text{e}}^{i{\omega }_{0}{\tau }_{0}}+2{W}_{11}^{1}\left(0\right){\text{e}}^{-i{\omega }_{0}{\tau }_{0}}\right)\right)\\ \text{\hspace{0.17em}}+{f}_{22}\left(\frac{1}{2}{W}_{20}^{2}\left(0\right){\text{e}}^{i{\omega }_{0}{\tau }_{0}}+{W}_{11}^{2}\left(0\right){\text{e}}^{-i{\omega }_{0}{\tau }_{0}}+\frac{1}{2}{W}_{20}^{1}\left(-1\right)\stackrel{¯}{\alpha }+\alpha {W}_{11}^{1}\left(-1\right)\right)\\ \text{\hspace{0.17em}}+{f}_{23}\left(\frac{1}{2}{W}_{20}^{2}\left(-1\right){\text{e}}^{i{\omega }_{0}{\tau }_{0}}+{W}_{11}^{2}\left(-1\right){\text{e}}^{-i{\omega }_{0}{\tau }_{0}}+\frac{1}{2}{W}_{20}^{1}\left(-1\right)\stackrel{¯}{\alpha }{\text{e}}^{i{\omega }_{0}{\tau }_{0}}+{W}_{11}^{1}\left(-1\right)\alpha {\text{e}}^{-i{\omega }_{0}{\tau }_{0}}\right)\\ \text{\hspace{0.17em}}+{f}_{24}\left(\frac{1}{2}{W}_{20}^{2}\left(0\right)\stackrel{¯}{\alpha }{\text{e}}^{i{\omega }_{0}{\tau }_{0}}+{W}_{11}^{2}\left(0\right)\alpha {\text{e}}^{-i{\omega }_{0}{\tau }_{0}}+\alpha {W}_{11}^{2}\left(-1\right)+\frac{1}{2}\stackrel{¯}{\alpha }{W}_{20}^{2}\left(-1\right)\right)\right],\end{array}$

${f}_{11}=\left(\frac{hb}{{\left(b+{u}_{5}\right)}^{3}}-{a}_{1}\right),{f}_{12}=-{c}_{1}\left(1-m\right),{f}_{21}=\frac{{\left(1-m\right)}^{2}{r}_{2}^{2}}{{c}_{2}\left({u}_{5}\left(1-m\right)+k\right)},$

${f}_{22}=\frac{{r}_{2}\left(1-m\right)}{{u}_{5}\left(1-m\right)+k},{f}_{23}=\frac{{r}_{2}\left(1-m\right)}{{u}_{5}\left(1-m\right)+k},{f}_{24}=-\frac{{c}_{2}}{{u}_{5}\left(1-m\right)+k}.$

$\begin{array}{c}\stackrel{˙}{W}={\stackrel{˙}{U}}_{t}-2Re\left\{\stackrel{˙}{z}q\right\}={\stackrel{˙}{U}}_{t}-\stackrel{˙}{z}q-\stackrel{˙}{\stackrel{¯}{z}}\text{ }\stackrel{¯}{q}=P{U}_{t}+Q{U}_{t}-\stackrel{˙}{z}q-\stackrel{˙}{\stackrel{¯}{z}}\text{ }\stackrel{¯}{q}\\ =\left\{\begin{array}{l}PW-2Re\left\{{\stackrel{¯}{q}}^{*}\left(\theta \right)f\left(0\right)q\left(\theta \right)\right\},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0<\theta \le 1\\ PW-2Re\left\{{\stackrel{¯}{q}}^{*}\left(0\right)f\left(0\right)q\left(0\right)\right\}+f\left(0\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\theta =0\end{array}\\ =PW+H\left(z,\stackrel{¯}{z},\theta \right)\end{array}$

$H\left(z,\stackrel{¯}{z},\theta \right)=\frac{{H}_{20}\left(\theta \right){z}^{2}}{2}+{H}_{11}\left(\theta \right)z\stackrel{¯}{z}+\frac{{H}_{02}\left(\theta \right){\stackrel{¯}{z}}^{2}}{2}+\cdots .$

$\begin{array}{c}\stackrel{˙}{W}\left(z,\stackrel{¯}{z}\right)={W}_{z}\stackrel{˙}{z}+{W}_{\stackrel{\stackrel{¯}{z}}{z}}\stackrel{˙}{\stackrel{¯}{z}}={W}_{20}\left(\theta \right)z\stackrel{˙}{z}+{W}_{11}\left(\theta \right)\stackrel{¯}{z}\stackrel{˙}{z}+{W}_{02}\stackrel{¯}{z}\text{ }\stackrel{˙}{\stackrel{¯}{z}}+{W}_{11}\left(\theta \right)z\stackrel{˙}{\stackrel{¯}{z}}+O\left({|\left(z,\stackrel{¯}{z}\right)|}^{3}\right)\\ ={W}_{20}\left(\theta \right)z\left(i{\omega }_{0}{\tau }_{0}z+g\left(z,\stackrel{¯}{z}\right)\right)+{W}_{11}\left(\theta \right)\stackrel{¯}{z}\left(i{\omega }_{0}{\tau }_{0}\stackrel{¯}{z}+\stackrel{¯}{g}\left(z,\stackrel{¯}{z}\right)\right)\\ \text{\hspace{0.17em}}+{W}_{11}z\left(-i{\omega }_{0}{\tau }_{0}\stackrel{¯}{z}+\stackrel{¯}{g}\left(z,\stackrel{¯}{z}\right)\right)+{W}_{02}\left(\theta \right)\stackrel{¯}{z}\left(-i{\omega }_{0}{\tau }_{0}\stackrel{¯}{z}+\stackrel{¯}{g}\left(z,\stackrel{¯}{z}\right)\right)+O\left({|\left(z,\stackrel{¯}{z}\right)|}^{3}\right)\\ =i{\omega }_{0}{\tau }_{0}{W}_{20}\left(\theta \right){z}^{2}-i{\omega }_{0}{\tau }_{0}{W}_{02}\left(\theta \right){\stackrel{¯}{z}}^{2}+O\left({|\left(z,\stackrel{¯}{z}\right)|}^{3}\right)\end{array}$

$\left(2i{\omega }_{0}{\tau }_{0}I-P\left(0\right)\right){W}_{20}\left(\theta \right)={H}_{20}\left(\theta \right),$

$P\left(0\right){W}_{11}\left(\theta \right)=-{H}_{11}\left(\theta \right),$ (2.3.6)

$\left(2i{\omega }_{0}{\tau }_{0}I+P\left(0\right)\right){W}_{02}\left(\theta \right)=-{H}_{02}\left( \theta \right)$

$\begin{array}{c}H\left(z,\stackrel{¯}{z},\theta \right)=-2Re\left\{{\stackrel{¯}{q}}^{*}\left(\theta \right)Q\left(0\right)q\left(\theta \right)\right\}=-g\left(z,\stackrel{¯}{z}\right)q\left(\theta \right)-\stackrel{¯}{g}\left(z,\stackrel{¯}{z}\right)\stackrel{¯}{q}\left(\theta \right)\\ =-\left({g}_{20}\frac{{z}^{2}}{2}+{g}_{11}z\stackrel{¯}{z}+{g}_{02}\frac{{\stackrel{¯}{z}}^{2}}{2}+\cdots \right)q\left(\theta \right)-\left({\stackrel{¯}{g}}_{20}\frac{{\stackrel{¯}{z}}^{2}}{2}+{\stackrel{¯}{g}}_{11}z\stackrel{¯}{z}+{\stackrel{¯}{g}}_{02}\frac{{\stackrel{¯}{z}}^{2}}{2}+\cdots \right)\stackrel{¯}{q}\left(\theta \right).\end{array}$

${H}_{20}\left(\theta \right)=-{g}_{20}q\left(\theta \right)-{\stackrel{¯}{g}}_{02}\stackrel{¯}{q}\left(\theta \right),$

${H}_{11}\left(\theta \right)=-{g}_{11}q\left(\theta \right)-{\stackrel{¯}{g}}_{11}\stackrel{¯}{q}\left(\theta \right).$ (2.3.7)

$\stackrel{̇}{{W}_{20}}\left(\theta \right)=2i{\omega }_{0}{\tau }_{0}{W}_{20}\left(\theta \right)+{g}_{20}q\left(\theta \right)+{\stackrel{¯}{g}}_{02}\stackrel{¯}{q}\left(\theta \right),$

${W}_{20}\left(\theta \right)=\frac{i{g}_{20}q\left(0\right)}{{\omega }_{0}{\tau }_{0}}{\text{e}}^{i{\omega }_{0}{\tau }_{0}\theta }+\frac{i{\stackrel{¯}{g}}_{02}\stackrel{¯}{q}\left(0\right)}{3{\omega }_{0}{\tau }_{0}}{\text{e}}^{-i{\omega }_{0}{\tau }_{0}\theta }+{G}_{1}{\text{e}}^{2i{\omega }_{0}{\tau }_{0}\theta },$

${W}_{11}\left(\theta \right)=\frac{-i{g}_{11}q\left(0\right)}{{\omega }_{0}{\tau }_{0}}{\text{e}}^{i{\omega }_{0}{\tau }_{0}\theta }+\frac{i{\stackrel{¯}{g}}_{11}\stackrel{¯}{q}\left(0\right)}{{\omega }_{0}{\tau }_{0}}{\text{e}}^{-i{\omega }_{0}{\tau }_{0}\theta }+{G}_{2}.$

$\theta =0$ 时，有

${H}_{20}\left(0\right)=-{g}_{20}q\left(0\right)-{\stackrel{¯}{g}}_{02}\stackrel{¯}{q}\left(0\right)+{\tau }_{0}{\left({z}_{1},{z}_{2},0\right)}^{\text{T}},$

${H}_{11}\left(0\right)=-{g}_{11}q\left(0\right)-{\stackrel{¯}{g}}_{11}\stackrel{¯}{q}\left(0\right)+{\tau }_{0}{\left({z}_{3},{z}_{4},0\right)}^{\text{T}}.$

${z}_{1}={f}_{11}+{f}_{12}\alpha ,$

${z}_{2}={f}_{21}{\text{e}}^{-2i{\omega }_{0}{\tau }_{0}}+{f}_{22}{\text{e}}^{-i{\omega }_{0}{\tau }_{0}}\alpha +{f}_{23}\alpha {\text{e}}^{-2i{\omega }_{0}{\tau }_{0}}+{f}_{24}{\alpha }^{2}{\text{e}}^{-i{\omega }_{0}{\tau }_{0}},$

${z}_{3}=2{f}_{11}+{f}_{12}\left(\alpha +\stackrel{¯}{\alpha }\right),$

${z}_{4}=2{f}_{21}+{f}_{22}\left({\text{e}}^{-i{\omega }_{0}{\tau }_{0}}\stackrel{¯}{\alpha }+{\text{e}}^{i{\omega }_{0}{\tau }_{0}}\alpha \right)+{f}_{23}\left(\alpha +\stackrel{¯}{\alpha }\right)+{f}_{24}\left(\alpha \stackrel{¯}{\alpha }{\text{e}}^{-i{\omega }_{0}{\tau }_{0}}+\alpha \stackrel{¯}{\alpha }{\text{e}}^{i{\omega }_{0}{\tau }_{0}}\right).$

${\int }_{-1}^{0}\text{d}\eta \left(\theta \right){W}_{20}\left(\theta \right)=2i{\omega }_{0}{\tau }_{0}{W}_{20}\left(0\right)-{H}_{20}\left(0\right),{\int }_{-1}^{0}\text{d}\eta \left(\theta \right){W}_{11}\left(\theta \right)=-{H}_{11}\left(0\right).$

${H}_{20}\left(0\right)=2i{\omega }_{0}{\tau }_{0}{W}_{20}\left(0\right)-{\tau }_{0}{B}_{1}{W}_{20}\left(0\right)-{\tau }_{0}{B}_{2}{W}_{20}\left(-1\right),$

${H}_{11}\left(0\right)=-{\tau }_{0}{B}_{1}{W}_{11}\left(0\right)-{\tau }_{0}{B}_{2}{W}_{11}\left(-1\right).$

$\begin{array}{c}{G}_{1}=\frac{1}{2i{\omega }_{0}{\tau }_{0}I-{\tau }_{0}{B}_{1}-{\tau }_{0}{B}_{2}{\text{e}}^{-2i{\omega }_{0}{\tau }_{0}}}\left({H}_{20}\left(0\right)+2{g}_{20}q\left(0\right)+\frac{2}{3}{\stackrel{¯}{g}}_{02}\stackrel{¯}{q}\left(0\right)+\frac{{B}_{1}i{g}_{20}q\left(0\right)}{{\omega }_{0}}\\ \text{\hspace{0.17em}}+\frac{{B}_{1}i{\stackrel{¯}{g}}_{02}\stackrel{¯}{q}\left(0\right)}{3{\omega }_{0}}+{B}_{2}\frac{i{g}_{20}q\left(0\right)}{{\omega }_{0}}{\text{e}}^{-i{\omega }_{0}{\tau }_{0}}+{B}_{2}\frac{i{\stackrel{¯}{g}}_{02}\stackrel{¯}{q}\left(0\right)}{3{\omega }_{0}}{\text{e}}^{i{\omega }_{0}{\tau }_{0}}\right),\end{array}$

${G}_{2}=\frac{1}{{\tau }_{0}\left({B}_{1}+{B}_{2}\right)}\left(-{H}_{11}\left(0\right)+\frac{{B}_{1}i{g}_{11}q\left(0\right)}{{\omega }_{0}}-\frac{{B}_{1}i{\stackrel{¯}{g}}_{11}\stackrel{¯}{q}\left(0\right)}{{\omega }_{0}}+\frac{{B}_{2}i{g}_{11}q\left(0\right)}{{\omega }_{0}}{\text{e}}^{-i{\omega }_{0}{\tau }_{0}}-\frac{{B}_{2}i{\stackrel{¯}{g}}_{11}\stackrel{¯}{q}\left(0\right)}{{\omega }_{0}}{\text{e}}^{i{\omega }_{0}{\tau }_{0}}\right).$

${c}_{1}\left(0\right)=\frac{i}{2{\omega }_{0}{\tau }_{0}}\left({g}_{20}{g}_{11}-2{|{g}_{11}|}^{2}-\frac{{|{g}_{02}|}^{2}}{3}\right)+\frac{{g}_{21}}{2},$

${\beta }_{2}=Re\left({c}_{1}\left(0\right)\right),{\mu }_{2}=-\frac{Re\left({c}_{1}\left(0\right)\right)}{Re\left({\lambda }^{\prime }\left({\tau }_{0}\right)\right)},$

${T}_{2}=-\frac{Im\left\{{c}_{1}\left(0\right)\right\}+{\mu }_{2}Im\left\{{\lambda }^{\prime }\left({\tau }_{0}\right)\right\}}{{\omega }_{0}{\tau }_{0}}.$

3. 数值模拟

${r}_{1}=0.6,{r}_{2}=0.3,{c}_{1}=0.1,{c}_{2}=0.1,k=0.1,{a}_{1}=0.01,b=0.7.$

Figure 1. $m=0.5,h=0.5,\tau =1.5$. (a) Time series; (b) Phase portrait

Figure 2. $m=0.5,h=0.5,\tau =1.9$. (a) Time series; (b) Phase portrait

Figure 3. Analysis of sensitivity with $m=0.5,h=0.5$. The solid blue line denotes the stable equilibrium; the solid red line represents the maximal and minimal density of predator population. The point “Hb” is the Hopf bifurcation point

Figure 4. (a) Analysis of sensitivity with $m=0.5,\tau =1.8$ ; (b) Analysis of sensitivity with $h=0.5,\tau =1.8$. The solid green line denotes the stable equilibrium; the dashed red line indicates the positive equilibrium which is unstable. The solid blue lines represent the maximal and minimal density of predator populations. The point “Hb” is the Hopf bifurcation point

4. 结论

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