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Turing–Hopf bifurcation analysis of a predator–prey model with herd behavior and cross-diffusion

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Abstract

In this paper, we consider a predator–prey model with herd behavior and cross-diffusion subject to homogeneous Neumann boundary condition. Firstly, the existence and priori bound of a solution for the model without cross-diffusion are shown. Then, by computing and analyzing the normal form on the center manifold associated with the Turing–Hopf bifurcation, we find a wealth of spatiotemporal dynamics near the Turing–Hopf bifurcation point under suitable conditions. Furthermore, some numerical simulations to illustrate the theoretical analysis are carried out.

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Acknowledgments

The authors highly appreciate the anonymous reviewers and editor for providing valuable suggestions which helped us to improve the manuscript. The work is supported by the National Natural Science Foundation of China (No. 11571257) and the Science and Technology Project of Department of Education of Jiangxi Province (No. GJJ150771).

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Authors and Affiliations

  1. Department of Mathematics, Tongji University, Shanghai, 200092, People’s Republic of China

    Xiaosong Tang & Yongli Song

  2. College of Mathematics and Physics, Jinggangshan University, Ji’an, 343009, People’s Republic of China

    Xiaosong Tang

  3. Department of Mathematics, Swinburne University of Technology, Hawthorn, VIC, 3122, Australia

    Tonghua Zhang

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  1. Xiaosong Tang
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Correspondence to Yongli Song.

Appendix: calculation of \(B_{210}, B_{102}, B_{111}\) and \(B_{003}\)

Appendix: calculation of \(B_{210}, B_{102}, B_{111}\) and \(B_{003}\)

$$\begin{aligned} B_{210}= & {} C_{210}+\frac{3}{2}\left( D_{210}+E_{210}\right) ,\\ B_{102}= & {} C_{102}+\frac{3}{2}\left( D_{102}+E_{102}\right) ,\\ B_{111}= & {} C_{111}+\frac{3}{2}\left( D_{111}+E_{111}\right) ,\\ B_{003}= & {} C_{003}+\frac{3}{2}\left( D_{003}+E_{003}\right) , \end{aligned}$$

where

$$\begin{aligned} C_{210}= & {} \frac{1}{\pi }q_{0}^\mathrm{T}F_{210},\quad C_{102}=\frac{1}{\pi }q_{0}^\mathrm{T}F_{102},\\ C_{111}= & {} \frac{1}{\pi }q_{k_{*}}^\mathrm{T}F_{111},\quad C_{003}=\frac{3}{2\pi }q_{k_{*}}^\mathrm{T}F_{003}, \end{aligned}$$

with

$$\begin{aligned} F_{210}= & {} \frac{1}{2}\left( f_{3000}\left| p_{01}\right| ^2p_{01} +f_{0300}\left| p_{02}\right| ^2p_{02}\right. \\&+\,f_{2100}\left( p_{01}^{2} \overline{p}_{02} +2 \left| p_{01}\right| ^2p_{02}\right) \\&\left. +\,f_{1200}\left( p_{02}^2 \overline{p}_{01}+2 \left| p_{02}\right| ^2p_{01}\right) \right) ,\\ F_{102}= & {} \frac{1}{2}\left( f_{3000} p_{01}p_{k_{*}1}^{2}+f_{0300} p_{02} p_{k_{*}2}^{2}\right. \\&+\,f_{2100}\left( p_{02} p_{k_{*}1}^{2}+2 p_{01}p_{k_{*}1} p_{k_{*}2}\right) \\&\left. +\,f_{1200}\left( p_{01} p_{k_{*}2}^{2}+2 p_{02}p_{k_{*}1} p_{k_{*}2}\right) \right) ,\\ F_{111}= & {} \left( f_{3000} \left| p_{01}\right| ^{2}p_{k_{*}1}+f_{0300} \left| p_{02}\right| ^{2}p_{k_{*}2}\right. \\&+\,f_{2100}\left( \left| p_{01}\right| ^{2}p_{k_{*}2}+2p_{k_{*}1}\text{ Re }\left\{ p_{01}\overline{p}_{02}\right\} \right) \\&\left. +\,f_{1200}\left( \left| p_{02}\right| ^{2}p_{k_{*}1} +2p_{k_{*}2}\text{ Re }\left\{ p_{02}\overline{p}_{01}\right\} \right) \right) ,\\ F_{003}= & {} \frac{1}{3!}\left( f_{3000} p_{k_{*}1}^{3}+f_{0300} p_{k_{*}2}^{3}\right) \\&+\,\frac{1}{2}\left( f_{2100}p_{k_{*}1}^{2}p_{k_{*}2}+ f_{1200} p_{k_{*}1} p_{k_{*}2}^{2}\right) ,\\ D_{210}= & {} \frac{1}{3\pi \omega _{c} i }\left( -\left( q_{0}^\mathrm{T}F_{200}\right) \left( q_{0}^\mathrm{T}F_{110}\right) \right. \nonumber \\&\left. +\,\frac{1}{3}\left| q_{0}^\mathrm{T}F_{020}\right| ^{2}+2\left| q_{0}^\mathrm{T}F_{110}\right| ^{2}\right) , \\ D_{102}= & {} \frac{1}{3\pi \omega _{c} i}\left( -\left( q_{0}^\mathrm{T}F_{200}\right) \left( q_{0}^\mathrm{T}F_{002}\right) +\left( q_{0}^\mathrm{T}F_{110}\right) \right. \\&\left. \left( \overline{q}_{0}^\mathrm{T}F_{002}\right) +2\left( q_{0}^\mathrm{T}F_{002}\right) \left( q_{k_{*}}^\mathrm{T}F_{101}\right) \right) , \\ D_{111}= & {} -\frac{4}{3\pi \omega _{c}} \text{ Im }\left\{ \left( q_{0}^\mathrm{T}F_{110}\right) \left( q_{k_{*}}^\mathrm{T}F_{101} \right) \right\} ,\\ D_{003}= & {} -\frac{2}{3\pi \omega _{c} } \text{ Im }\left\{ \left( q_{0}^\mathrm{T}F_{002}\right) \left( q_{k_{*}}^\mathrm{T}F_{101}\right) \right\} , \end{aligned}$$

and

$$\begin{aligned} E_{210}= & {} \frac{1}{3\sqrt{\pi }}q_{0}^\mathrm{T}\left( \left( p_{01}f_{2000}+p_{02}f_{1100} \right) h^{(1)}_{0110}\right. \\&+\left( p_{02}f_{0200}+p_{01}f_{1100} \right) h^{(2)}_{0110} \\&+\left( \overline{p}_{01}f_{2000}+\overline{p}_{02}f_{1100} \right) h^{(1)}_{0200}\\&\left. +\left( \overline{p}_{02}f_{0200}+\overline{p}_{01}f_{1100} \right) h^{(2)}_{0200} \right) ,\\ E_{102}= & {} \frac{1}{3\sqrt{\pi }}q_{0}^\mathrm{T}\left( \left( p_{01}f_{2000}+p_{02}f_{1100} \right) h^{(1)}_{0002}\right. \\&+\left( p_{02}f_{0200}+p_{01}f_{1100} \right) h^{(2)}_{0002}\\&+\left( p_{k_{*}1}f_{2000}+p_{k_{*}2}f_{1100} \right) h^{(1)}_{k_{*}101}\nonumber \\&\left. +\left( p_{k_{*}2}f_{0200}+p_{k_{*}1}f_{1100} \right) h^{(2)}_{k_{*}101} \right) ,\\ E_{111}= & {} \frac{1}{3\sqrt{\pi }}q_{k_{*}}^\mathrm{T}\left( \left( p_{01}f_{2000}+p_{02}f_{1100}\right) h^{(1)}_{k_{*}011}\right. \\&+\left( p_{02}f_{0200}+p_{01}f_{1100}\right) h^{(2)}_{k_{*}011} \\&\left( \overline{p}_{01}f_{2000}+\overline{p}_{02}f_{1100} \right) h^{(1)}_{k_{*}101}\\&\left. +\left( \overline{p}_{02}f_{0200}+\overline{p}_{01}f_{1100} \right) h^{(2)}_{k_{*}101}\right) \\&+q_{k_{*}}^\mathrm{T}\left( \left( p_{k_{*}1}f_{2000}+p_{k_{*}2}f_{1100}\right) \right. \\&\left( \frac{1}{3\sqrt{\pi }}h_{0110}^{(1)}+ \frac{1}{3\sqrt{2\pi }}h_{(2k_{*})110}^{(1)}\right) \\&+\left( p_{k_{*}2}f_{0200}+p_{k_{*}1}f_{1100}\right) \\&\left. \left( \frac{1}{3\sqrt{\pi }}h_{0110}^{(2)}+ \frac{1}{3\sqrt{2\pi }}h_{(2k_{*})110}^{(2)}\right) \right) ,\\ E_{003}= & {} q_{k_{*}}^\mathrm{T}\left( \left( p_{k_{*}1}f_{2000}+p_{k_{*}2}f_{1100}\right) \right. \\&\left( \frac{1}{3\sqrt{\pi }}h_{0002}^{(1)}+\frac{1}{3\sqrt{2\pi }}h_{(2k_{*})002}^{(1)}\right) \\&+\left( p_{k_{*}2}f_{0200}+p_{k_{*}1}f_{1100}\right) \\&\left. \left( \frac{1}{3\sqrt{\pi }}h_{0002}^{(2)}+\frac{1}{3\sqrt{2\pi }}h_{(2k_{*})002}^{(2)}\right) \right) . \end{aligned}$$

where \(h^{(i)}_{jm_{1}m_{2}m_{3}}, (i=1,2, j=0, 2k_{*}, m_{l}=1, 2, l=1, 2,3)\) are given by

$$\begin{aligned}&h_{0200}=\frac{1}{\sqrt{\pi }}\left( 2i\omega _{c}I-\mathscr {M}_0\right) ^{-1}\\&\left( F_{200}-\left( q_{0}^\mathrm{T}F_{200}p_{0}+\bar{q}_{0}^\mathrm{T}F_{200}\bar{p}_{0}\right) \right) ,\\&h_{0020}=\frac{1}{\sqrt{\pi }}\left( -2i\omega _{c}I-\mathscr {M}_0\right) ^{-1}\\&\left( F_{020}-\left( q_{0}^\mathrm{T}F_{020}p_{0}+\bar{q}_{0}^\mathrm{T}F_{020}\bar{p}_{0}\right) \right) ,\\&h_{0002}=-\frac{1}{\sqrt{\pi }}\mathscr {M}_{0}^{-1}\\&\left( F_{002}-\left( q_0^\mathrm{T}F_{002}p_0+\bar{q}_0^\mathrm{T}F_{002}\bar{p}_{0}\right) \right) ,\\&h_{0110}=-\frac{2}{\sqrt{\pi }}\mathscr {M}_{0}^{-1}\\&\left( F_{110}-\left( q_0^\mathrm{T}F_{110}p_0+\bar{q}_0^\mathrm{T}F_{110}\bar{p}_{0}\right) \right) ,\\&h_{k_{*}101}=\frac{2}{\sqrt{\pi }}\\&\left( i\omega _{c}I-\mathscr {M}_{k_{*}}\right) ^{-1}\left( F_{101}-q_{k_{*}}^\mathrm{T}F_{101}p_{k_{*}}\right) ,\\&h_{k_{*}011}=\frac{2}{\sqrt{\pi }}\left( -i\omega _{c}I-\mathscr {M}_{k_{*}}\right) ^{-1}\\&\left( F_{011}-q_{k_{*}}^\mathrm{T}F_{011}p_{k_{*}}\right) ,\\&h_{(2k_{*})002}=-\frac{1}{\sqrt{2\pi }}\mathscr {M}_{2k_{*}}^{-1}F_{002},\,h_{(2k_{*})110}=(0, 0)^\mathrm{T}. \end{aligned}$$

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Tang, X., Song, Y. & Zhang, T. Turing–Hopf bifurcation analysis of a predator–prey model with herd behavior and cross-diffusion. Nonlinear Dyn 86, 73–89 (2016). https://doi.org/10.1007/s11071-016-2873-3

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