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Equivariant Hopf bifurcation arising in circular-distributed predator-prey interaction with taxis

Yaqi Chen, Xianyi Zeng, Ben Niu

Abstract

In this paper, we study the Rosenzweig-MacArthur predator-prey model with predator-taxis and time delay defined on a disk. Theoretically, we studied the equivariant Hopf bifurcation around the positive constant steady-state solution. Standing and rotating waves have been investigated through the theory of isotropic subgroups and Lyapunov-Schmidt reduction. The existence conditions, the formula for the periodic direction and the periodic variation of bifurcation periodic solutions are obtained. Numerically, we select appropriate parameters and conduct numerical simulations to illustrate the theoretical results and reveal quite complicated dynamics on the disk. Different types of rotating and standing waves, as well as more complex spatiotemporal patterns with random initial values, are new dynamic phenomena that do not occur in one-dimensional intervals.

Equivariant Hopf bifurcation arising in circular-distributed predator-prey interaction with taxis

Abstract

In this paper, we study the Rosenzweig-MacArthur predator-prey model with predator-taxis and time delay defined on a disk. Theoretically, we studied the equivariant Hopf bifurcation around the positive constant steady-state solution. Standing and rotating waves have been investigated through the theory of isotropic subgroups and Lyapunov-Schmidt reduction. The existence conditions, the formula for the periodic direction and the periodic variation of bifurcation periodic solutions are obtained. Numerically, we select appropriate parameters and conduct numerical simulations to illustrate the theoretical results and reveal quite complicated dynamics on the disk. Different types of rotating and standing waves, as well as more complex spatiotemporal patterns with random initial values, are new dynamic phenomena that do not occur in one-dimensional intervals.
Paper Structure (4 sections, 3 theorems, 29 equations, 6 figures, 1 table)

This paper contains 4 sections, 3 theorems, 29 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Equivariant Hopf bifurcation
  3. Numerical simulations
  4. Conclusion

Key Result

Lemma 2.1

Let For $\vartheta \in [-\tau,0]$, define and Then $\mathrm{(i)}$${\rm{Ker}}\mathscr{L}_{\tau_{{nm}}^k}$ is spanned by $\left\{\varphi_1,\varphi_2,\varphi_3,\varphi_4\right\}$ and ${\rm{Ker}}\mathscr{L}_{\tau_{{nm}}^k}^*$ is spanned by $\left\{\varphi_1^*,\varphi_2^*,\varphi_3^*,\varphi_4^*\right\}$. $\mathrm{(ii)}$$(\varphi_j^*,\varphi_j)=1$,

Figures (6)

  • Figure 1: Partial bifurcation curves on the $\chi-\tau$ plane. Parameters are $d_1=0.1,~d_2=0.2,~\alpha=1,~K=6,~d=0.1,~R=10$.
  • Figure 2: System (\ref{['system R-M']}) generates standing wave that has a fixed axis with $(\chi,\tau)=(0.38,9.88)$. Initial values are $u(t,r,\theta)=u^*(1+0.1\cdot\cos{t}\cdot\cos{\frac{2\pi r}{R}}\cdot\cos{\theta}),~v(t,r,\theta)=v^*(1+0.1\cdot\cos{t}\cdot\cos{\frac{2\pi r}{R}}\cdot\cos{\theta}),~t\in[-\tau,0)$. (a): u; (b): v.
  • Figure 3: System (\ref{['system R-M']}) generates standing wave that has two fixed axes with $(\chi,\tau)=(0.46,9.6)$. Initial values are $u(t,r,\theta)=u^*(1+0.1\cdot\cos{t}\cdot\cos{\frac{2\pi r}{R}}\cdot\cos{2\theta}),~v(t,r,\theta)=v^*(1+0.1\cdot\cos{t}\cdot\cos{\frac{2\pi r}{R}}\cdot\cos{2\theta}),~t\in[-\tau,0)$. (a): u; (b): v.
  • Figure 4: System (\ref{['system R-M']}) generates counterclockwise rotating wave with $(\chi,\tau)=(0.38,9.88)$. The patterns at four different time within a period are selected to show the periodic changes in population distribution. Initial values are $u(t,r,\theta)=u^*(1+0.1\cdot\cos{t}\cdot\cos{\frac{2\pi r}{R}}\cdot\sin{\theta})$, $v(t,r,\theta)=v^*(1+0.1\cdot\cos{t}\cdot\cos{\frac{2\pi r}{R}}\cdot\cos{\theta}),~t\in[-\tau,0)$. (a): u; (b): v.
  • Figure 5: System (\ref{['system R-M']}) generates clockwise rotating wave with $(\chi,\tau)=(0.46,9.6)$. The patterns at four different time within a period are selected to show the periodic changes in population distribution. Initial values are $u(t,r,\theta)=u^*(1+0.1\cdot\cos{t}\cdot\cos{\frac{2\pi r}{R}}\cdot\cos{2\theta}),~v(t,r,\theta)=v^*(1+0.1\cdot\cos{t}\cdot\cos{\frac{2\pi r}{R}}\cdot\sin{2\theta}),~t\in[-\tau,0)$. (a): u; (b): v.
  • ...and 1 more figures

Theorems & Definitions (4)

  • Lemma 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Remark 3.1