Hopf bifurcation induced by time delay and influence of Allee effect in a diffusive predator–prey system with herd behavior and prey chemotaxis

Abstract

We formulate and concern a diffusive delayed predator–prey system with herd behavior and fear effect; an Allee effect term and prey chemotaxis are both considered. We first investigate the stability of the system under the strong Allee effect and weak Allee effect without spatial diffusion or time delay. For the spatiotemporal system, the constant positive steady states and semi-trivial steady states are presented. Then, the dynamic behaviors of the diffusive system are demonstrated in detail, and the conditions of Turing instability caused by prey chemotaxis are explored. In addition, we regard the time delay as a bifurcation parameter to investigate the stability of reaction–diffusion system. The normal form theory and center manifold theorem are applied to derive the properties of Hopf bifurcation of the delayed diffusive system. Finally, series of computer simulations are given to verify the theoretical analysis and show how fear effect affects the stability of system.

Appendix: Proof for Theorem 4.3

Here, we calculate relevant conditions for some properties of Hopf bifurcations for system (4), and we will apply the normal formal theory and center manifold theorem of PDEs which can be studied in detail in ( [36, 37]). In a general way, let $j\in N_{0}$, $\tau _{*}=\tau _{n}^{j}$, $n\in \{0,1,2\cdot \cdot \cdot N_{2}\}$. Now, we present a new coefficient $\varsigma \in R$, let $\tau =\varsigma +\tau _{*}$ such that $\varsigma =0$ becomes the Hopf bifurcation value. Denote ${\tilde{u}}(\cdot , t)=u(\cdot , \tau t)-u_{*}$, ${\tilde{v}}(\cdot , t)=v(\cdot , \tau t)-v_{*}$, ${\tilde{U}}(t)=({\tilde{u}}(\cdot , \tau t), {\tilde{v}}(\cdot , \tau t))$. Based on the above, we know Eq. 23) has two simply purely imaginary roots which can be written as $\pm i\omega _{*}$.

For simplicity of the notation, then dropping the tildes, rewriting system (4) as:

$$\begin{aligned}&\left\{ \begin{array}{l} \frac{\partial u}{\partial t}=\tau \left[ d_{1}\triangle u+\frac{(u+u_{*})(1-u-u_{*})(u+u_{*}-m)}{1+k(v+v_{*})}\right. \\ \qquad \left. -b\sqrt{(u+u_{*})}(v+v_{*})\right] , \\ \frac{\partial v}{\partial t}=\tau \left[ d_{v}\triangle v+c\sqrt{u(t-\tau )+u_{*}}(v+v_{*})\right. \\ \qquad \left. -d(v+v_{*})\right] ,\\ \end{array} \right. \end{aligned}$$

(34)

$$\begin{aligned}&\frac{\text {d}U(t)}{dt}\nonumber \\&\quad =\tau D_{1}\Delta U(t)+L(\tau )U_{t}+F(U_{t},\varsigma ),\nonumber \\&\psi =(\psi _{1}, \psi _{2})^{T }, \end{aligned}$$

(35)

with

$$\begin{aligned} D_{1}\Delta U(t)= & {} \left( \begin{array}{c} d_{1}\Delta u \\ d_{2}\Delta v\\ \end{array} \right) , L(\tau )\psi \\= & {} \left( \begin{array}{cc} a_{11}\psi _{1}(0) &{} a_{12}\psi _{2}(0) \\ a_{21}\psi _{1}(-1) &{} 0\\ \end{array} \right) , \\ F(\psi ,\varsigma )\\= & {} \tau \left( \begin{array}{c} f^{(1)}(\tau ) \\ f^{(2)}(\tau )\\ \end{array} \right) \\= & {} \tau \left( \begin{array}{c} \Sigma _{i+j\ge 2}\frac{1}{i!j!}f^{(1)}_{ij}\psi _{1}^{i}(0)\psi _{2}^{j}(0) \\ \Sigma _{i+j+p\ge 2}\frac{1}{i!j!p!}f^{(2)}_{ij}\psi _{1}^{i}(0)\psi _{2}^{j}(0)\psi _{1}^{p}(-1)\\ \end{array} \right) , \end{aligned}$$

where

$$\begin{aligned}&f^{(1)}(u,v,w)=\frac{(u+u_{*})(1-u-u_{*})(u+u_{*}-m)}{1+k(v+v_{*})}\nonumber \\&\quad -b\sqrt{u+u_{*}}(v+v_{*}),\nonumber \\&f^{(2)}(u,v,w)=c\sqrt{w+u_{*}}(v+v_{*})-d(v+v_{*}). \end{aligned}$$

(36)

Applying the Riesz representation theorem [38], $\eta (\theta ,\tau _{*})$ $(\theta \in [-1,0])$ is a $2\times 2$ matrix function, such that

$$\begin{aligned}&-\tau _{*}D(\frac{n}{l})^{2}\phi (0)\\&\quad +L(\tau _{*})(\phi )=\int _{-1}^{0}\text {d}[\eta (\theta ,\tau _{*})]\phi (\theta )\\&\qquad for \phi \in ([-1,0],R^{2}), \end{aligned}$$

$\eta (\theta ,\tau _{*})$

$$\begin{aligned} \eta (\theta ,\tau _{*})=A\delta (\theta )+B\delta (\theta +1), \end{aligned}$$

with

$$\begin{aligned} A= \tau _{*}\left( \begin{array}{cc} a_{11}-d_{1}(\frac{n}{l})^{2} &{} a_{12}\\ 0 &{} -d_{2}(\frac{n}{l})^{2}\\ \end{array} \right) , ~~ B=\tau _{*}\left( \begin{array}{cc} 0&{}0 \\ -a_{21} &{} 0\\ \end{array} \right) . \end{aligned}$$

Denote $\Upsilon (\theta )\in C^{1}([-1,0],R^{2})$, define

$$\begin{aligned} {\mathcal {C}}(\Upsilon (\theta ))=\left\{ \begin{array}{l} \frac{\text {d}\Upsilon (\theta )}{\text {d}\theta },~~~~~~~~~~~\theta \in [-1,0),\\ \int _{-1}^{0}\text {d}\eta (\theta )\Upsilon (\theta ),\theta =0. \end{array} \right. \end{aligned}$$

And For $\Psi (\vartheta )\in C^{1}([-1,0],R^{2})$, define

$$\begin{aligned} \mathcal {C^{*}}(\Psi (\vartheta ))=\left\{ \begin{array}{l} \frac{-\text {d}\Psi (\vartheta )}{\text {d}\vartheta },~~~~~~~~~~~~\vartheta \in [0,1),\\ \int _{-1}^{0}\text {d}\eta (\vartheta )\Upsilon (-\vartheta ),\vartheta =0. \end{array} \right. \end{aligned}$$

$\mathcal {C^{*}}$ is the formal adjoint of ${\mathcal {C}}$ under the bilinear pairing

$$\begin{aligned}&\langle \Psi (\vartheta ),\Upsilon (\theta )\rangle ={\overline{\Psi }}(0),\Upsilon (0)\\&\quad -\int _{-1}^{0}\int _{0}^{\theta }{\overline{\Psi }}(s-\theta )\text {d}\eta (\theta )\Upsilon (s)\text {d}s. \end{aligned}$$

As mentioned above $\pm i\omega _{*}$ are the eigenvalues of the operator ${\mathcal {C}}$ , thus, it is easy to get the $\pm i\omega _{*}$ are also the eigenvalues of $\mathcal {C^{*}}$.

Suppose that $P(\theta )=(P_{1},P_{2})^\mathrm {T}e^{i\omega _{*}\tau _{*}\theta }=(1,P_{2})^\mathrm {T}e^{i\omega _{*}\tau _{*}\theta }$, $\theta \in [-1,0]$ is the eigenvector of ${\mathcal {C}}$, $i\omega _{*}\tau _{*}$ is the root of ${\mathcal {C}}$, and $Q(\vartheta )=R(Q_{1},Q_{2})^\mathrm {T}e^{i\omega _{*}\tau _{*}\vartheta }=R(1,Q_{2})^\mathrm {T}e^{i\omega _{*}\tau _{*}\vartheta }$ is the eigenvector of $\mathcal {C^{*}}$, $-i\omega _{*}\tau _{*}$ is the root of $\mathcal {C^{*}}$. Refer to [39] for detailed methods and definitions. According to the equation $\langle P^{*}(\theta ),Q(\vartheta )\rangle =1$, then we obtain

$$\begin{aligned} P_{2}{=}\frac{i\omega -a_{11}+d_{1}\frac{n^{2}}{l^{2}}}{a_{12}}, R=(1+P_{2}{\bar{Q}}_{2}+a_{21}{\bar{Q}}_{2}e^{i\omega \tau }), \end{aligned}$$

$$\begin{aligned} Q_{2}=\frac{-i\omega \tau -a_{11}+d_{1}\frac{n^{2}}{l^{2}}}{a_{21}}e^{-i\omega }. \end{aligned}$$

Thus, we can compute $f^{(2)}_{100}=0,$ then we have

$$\begin{aligned}&f^{(1)}_{10}=\frac{(1-2u_{*})(u_{*}-m)+u_{*}(1-u_{*})}{1+kv_{*}}\\&-\frac{bv_{*}\sqrt{u_{*}}}{2u_{*}},~f^{(1)}_{02}\\&=\frac{2k^{2}u_{*}(1-u_{*})(u_{*}-m)}{(1+kv_{*})^{3}}, f^{(1)}_{20}=\frac{-6u_{*}+2m+2}{1+kv_{*}}\\&+\frac{bv_{*}u_{*}^{-\frac{3}{2}}}{4},~\\&f^{(1)}_{21}=\frac{k(6u_{*}-2m-2)}{(1+kv_{*})^{2}}\\&+\frac{bu_{*}^{-\frac{3}{2}}}{4},\\&f^{(1)}_{11}=\frac{k(3u_{*}^{2}-2u_{*}m-2{u_{*}}+m)}{(1+kv_{*})^{2}}\\&-\frac{b\sqrt{u_{*}}}{2u_{*}},~f^{(1)}_{22}\\&=\frac{-2k^{2}(6u_{*}-2m-2)}{(1+kv_{*})^{3}}, f^{(2)}_{001}=\frac{cv_{*}\sqrt{u_{*}}}{2u_{*}},~f^{(2)}_{002}\\&=-\frac{cu_{*}^{-\frac{1}{2}}}{4}v_{*},~f^{(2)}_{003}\\&=\frac{3c}{8}u_{*}^{-\frac{5}{2}}, f^{(1)}_{12}=\frac{2k^{2}(3u_{*}-2u_{*}m-2_{u_{*}}+m)}{(1+kv_{*})^{3}},~f^{(2)}_{010}\\&=c\sqrt{u_{*}}-d,~f^{(2)}_{012}=-\frac{cu_{*}^{-\frac{1}{2}}}{4}. \end{aligned}$$

Through the calculation above, we have

$$\begin{aligned} F=(\tau +\varsigma )\left( \begin{array}{cc} \frac{1}{2}f^{(1)}_{20}\psi _{1}^{2}(0)+f^{(1)}_{11}\psi _{1}(0)\psi _{2}(0)+\frac{1}{2}f^{(1)}_{02}\psi _{2}^{2}(0)+\frac{1}{6}f^{(1)}_{30}\psi _{1}^{3}(0)\psi _{2}(0)\\ +\frac{1}{2}f^{(1)}_{21}\psi _{1}^{2}(0)\psi _{2}(0)\\ +\frac{1}{2}f^{(1)}_{12}\psi _{1}(0)\psi _{2}^{2}(0)+\frac{1}{6}\psi _{2}^{3}(0)+\cdot \cdot \cdot \\ +\frac{1}{2}f^{(2)}_{002}\psi _{1}^{2}(-1)+f^{(2)}_{011}\psi _{1}(-1)\psi _{2}(-1)+\\ +\frac{1}{2}f^{(2)}_{020}\psi _{2}^{2}(-1)\\ +\frac{1}{6}f^{(2)}_{003}\psi _{1}^{3}(-1)+\frac{1}{2}f^{(2)}_{012}\psi _{1}^{2}(-1)\psi _{2}(0). \end{array} \right) . \end{aligned}$$

Therefore, we can compute the following quantities $g_{20}=\tau _{*}{\bar{R}}\left[ f_{20}^{(1)}+2f_{11}^{(1)}P_{2}+f_{02}^{(1)}P_{2}^{2}\right.$ $\qquad \,\,\left. +\bar{Q_{2}}\left( f_{002}^{(2)}e^{-2i\omega \tau _{*}}+2f_{011}\right) e^{-i\omega \tau _{*}}P_{2}\right] ,$ $g_{11}=\tau _{*}{\bar{R}}\left[ f_{20}^{(1)}+f_{02}^{(1)}P_{2}{\bar{P}}_{2}+2f_{11}^{(1)}Re\{P_{2}\}+{\bar{Q}}_{2}\right.$ $\qquad \,\,\left. \left( f_{002}+2f_{011}Re\{P_{2}\}\right) e^{-i\omega \tau _{*}}\right] ,$ $g_{02}={\bar{g}}_{20},$

$$\begin{aligned} g_{21}= & {} \frac{\tau _{*}{\bar{R}}}{l\pi }\left\{ \int _{0}^{l\pi }\left[ f_{20}^{(1)}\left( 2w_{11}^{(1)}(0)+w_{20}^{(1)}(0)\right) \right. \right. \\&+\left. \left. f_{02}^{(1)}\left( 2P_{2}w_{11}^{(2)}(0)+{\bar{P}}_{2}w_{20}^{(2)}(0)\right) \right] \cos ^{2}\frac{nx}{l}\mathrm {d}x\right\} \\&\quad +\frac{\tau _{*}{\bar{R}}}{l\pi }\left\{ \int _{0}^{l\pi }\left[ f_{11}^{(1)}\left( 2w_{11}^{(2)}(0)+w_{20}^{(2)}(0)\right) \right. \right. \\&+\left. \left. {\bar{P}}_{2}w_{20}^{(1)}(0)+2P_{2}w_{11}^{(1)}(0)\right] \cos ^{2}\frac{nx}{l}\mathrm {d}x\right\} \\&\quad +\frac{\tau _{*}{\bar{R}}}{l\pi }\left\{ \left[ f_{30}^{(1)}+f_{03}^{(1)}P_{2}{\bar{P}}_{2}+f_{12}^{(1)}\left( P_{2}^{2}+2|P_{2}|^{2}\right) \right. \right. \\&+\left. \left. f_{21}^{(1)}(2P_{2}+2{\bar{P}}_{2})\right] \int _{0}^{l\pi }\cos ^{4}\frac{nx}{l}\mathrm {d}x\right\} \\&\quad +\frac{1}{l\pi }\left\{ \left[ f_{003}^{(2)}e^{-i\omega \tau }+f_{012}^{(2)}\left( 2P_{2}\right. \right. \right. \\&\left. \left. \left. +{\bar{P}}e^{-2i\omega \tau }\right) \right] \int _{0}^{l\pi }\cos ^{4}\frac{nx}{l}\mathrm {d}x\right\} \\&\quad +\frac{1}{l\pi }\int _{0}^{l\pi }\left[ f_{011}^{(2)}\left( 2w_{11}(-1)^{(2)}e^{-i\omega \tau _{*}}+w_{20}^{2}(0)e^{i\omega \tau _{*}}\right. \right. \\&+\left. \left. {\bar{P}}_{2}w_{20}^{(1)}(-1)+2P_{2}w_{11}^{(1)}(-1)\right) \right. \\&\quad \left. +f_{002}^{(2)}\left( 2w_{11}^{(1)}(-1)e^{-i\omega _{*}\tau _{*}}+w_{20}^{(1)}(-1)e^{i\omega _{*}\tau _{*}}\right) \right. \\&+\left. 2f_{011}^{(2)}w_{11}^{(1)}(-1)\right] \int _{0}^{l\pi }\cos ^{2}\frac{nx}{l}\mathrm {d}x, \end{aligned}$$

where $n=0,1,2,\cdot \cdot \cdot ,$ then we have

$$\begin{aligned} w_{20}(\theta ){=}\left( \frac{ig_{20}}{\omega _{*}\tau _{*}}P(\theta )+\frac{ig_{02}}{3\omega _{*}\tau _{*}}{\overline{P}}(\theta )\right) \cdot f_{n}{+}E_{1}e^{2i\omega _{*}\tau _{*}\theta }, \end{aligned}$$

$$\begin{aligned} w_{11}(\theta )=\left( -\frac{ig_{11}}{\omega _{*}\tau _{*}}P(\theta )+\frac{i{\overline{g}}_{11}}{\omega _{*}\tau _{*}}{\overline{P}}(\theta )\right) \cdot f_{n}+E_{2}, \end{aligned}$$

the definition of $f_{n}$ has the same meaning as in reference [39].

According to the [39], we get

$$\begin{aligned} E_{1}= \left( \begin{array}{cc} 2i\omega _{*}+d_{1}(\frac{n}{l})^{2}-a_{11} &{} -a_{12} \\ -a_{21}e^{-2i\omega _{*}\tau _{*}} &{} 2i\omega _{*}+d_{2}(\frac{n}{l})^{2} \end{array} \right) ^{-1}\\ \times \left( \begin{array}{cc} f^{(1)}_{20}+2f^{(1)}_{11}P_{2}+f^{(1)}_{02}P^{2}_{2}\\ 2f^{(2)}_{011}P_{2}e^{-i\omega _{*}\tau _{*}} \end{array} \right) \cos ^{2}\frac{n}{l}x,\\ E_{2}=\left( \begin{array}{cc} d_{1}(\frac{n}{l})^{2}-a_{11} &{} -a_{12} \\ -a_{21} &{} d_{2}(\frac{n}{l})^{2} \end{array} \right) ^{-1}\times H, \end{aligned}$$

where

$$\begin{aligned} H=\left( \begin{array}{cc} f^{(1)}_{20}+f^{(1)}_{11}(P_{2}+{\overline{P}}_{2})+f^{(1)}_{02}P_{2}{\overline{P}}_{2}\\ f^{(2)}_{011}(P_{2}e^{-i\omega _{*}\tau _{*}}+{\overline{P}}_{2}e^{-i\omega _{*}\tau _{*}}) \end{array} \right) \cos ^{2}\frac{n}{l}x. \end{aligned}$$

Furthermore, we can determine $g_{21}$ by determining the $w_{20}$ and $w_{11}$.

Based on the above analysis, we have [40]

$$\begin{aligned} \left\{ \begin{array}{l} c_{1}(\tau _{*})=\frac{i\left( g_{11}g_{20}-2\mid g_{11}\mid ^{2}-\frac{\mid g_{02}\mid ^{2}}{3}\right) }{2\omega \tau _{*}}+\frac{g_{21}}{2}, \\ \mu _{2}=-\frac{\texttt {Re}\{c_{1}(\tau _{*})\}}{\texttt {Re}(\lambda ^{'}(\tau _{*}))},\\ \gamma _{2}=2\texttt {Re}\{c_{1}(\tau _{*})\},\\ T_{2}=-\frac{\texttt {Im}\{c_{1}(\tau _{*})\}+\mu _{2}{} \texttt {Im}\{\lambda ^{'}(\tau _{*})\}}{\omega \tau _{*}}. \end{array} \right. \end{aligned}$$

From the above calculation, correlational properties of the Hopf bifurcation according to the sign of $\mu _{2}$, $\gamma _{2}$, $T_{2}$ and $c_{1}(\tau _{*})$ can be obtained.