Spatiotemporal patterns induced by Turing and Turing-Hopf bifurcations in a predator-prey system

Chen, Mengxin; Wu, Ranchao; Chen, Liping

doi:10.1016/j.amc.2020.125300

“…There have been several reports on the dynamic behavior of this ratio-dependent predator-prey system (2). The Turing and Turing-Hopf bifurcations of system (2) were investigated and the spatiotemporal patterns of system (2) in two-dimensional space were discovered by Chen and Wu [3]. In [16], under homogeneous Neumann boundary conditions, the existence conditions of Hopf bifurcation, Turing instability of spatial uniformity and Turing-Hopf bifurcation in onedimensional space were shown.…”

Section: Introductionmentioning

confidence: 99%

Dynamic analysis of a Leslie–Gower-type predator–prey system with the fear effect and ratio-dependent Holling III functional response

Chen

¹

,

Zhang

²

2022

NAMC

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In this paper, we extend a Leslie–Gower-type predator–prey system with ratio-dependent Holling III functional response considering the cost of antipredator defence due to fear. We study the impact of the fear effect on the model, and we find that many interesting dynamical properties of the model can occur when the fear effect is present. Firstly, the relationship between the fear coefficient K and the positive equilibrium point is introduced. Meanwhile, the existence of the Turing instability, the Hopf bifurcation, and the Turing–Hopf bifurcation are analyzed by some key bifurcation parameters. Next, a normal form for the Turing–Hopf bifurcation is calculated. Finally, numerical simulations are carried out to corroborate our theoretical results.

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“…In this work, we shall research the dynamics of the general activator-substrate model (1) and networked model (2), respectively. More precisely, the boundedness of the positive non-constant steady state will be studied for the elliptic system of the continuous general activator-substrate model (1). With the help of the maximum principle and the Harnack's inequality one shows that it admits the upper bounds and the lower bounds for any positive solution (u(x), v(x)) under certain conditions.…”

Section: Introductionmentioning

confidence: 99%

Spatiotemporal Patterns in a General Networked Activator-Substrate Model

Chen

¹

,

Zheng

²

,

Wu

³

et al. 2021

Preprint

Self Cite

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For the purpose of understanding the spatiotemporal pattern formation in the random networked system, a general activator-substrate model with network structure is introduced. Firstly, we investigate the boundedness of the non-constant steady state of the elliptic system of the continuous media system. It is found that the non-constant steady state admits their upper and lower bounds with certain conditions. Then, one investigates some properties and non-existence of the non-constant steady state with the no-flux boundary conditions. The main results show that the diffusion rate of activator should greater than the diffusion rate of substrate. Otherwise, there might be no pattern formation of the system. Afterwards, a general random networked activator-substrate model is made public. The conditions of the stability, the Hopf bifurcation, the Turing instability and a co-dimensional-two Turing-Hopf bifurcation are yield by the method of stability analysis and bifurcation theorem. Finally, we choose a suitable sub-system of the general activator-substrate model to verify the theoretical results, and full numerical simulations are well verified these results. Especially, an interesting finding is that the stability of the positive equilibrium will switch from unstable to stable one with the change of the connection probability of the nodes, this is different from the pattern formation in the continuous media systems.

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“…Hence, many coupled reaction-diffusion equations are used to understand this process in a continuous space. An interesting and challenging research area is understanding the Turing patterns by a coupled reaction-diffusion equation [1][2][3][4]. Along this way, there has been a great deal of accumulated achievements to enrich pattern formations within coupled reaction-diffusion systems.…”

Section: Introductionmentioning

confidence: 99%

“…Also, the asymptotic stability of constant steady states, the steady state bifurcations from constant steady states are analyzed both in one-dimensional kernel and in two-dimensional kernel cases have been investigated by Wang et al in [24]. For more results on activator-substrate model (1) one can refer to Refs. [25,28].…”

Section: Introductionmentioning

confidence: 99%

“…In this work, we shall research the dynamics of the general activator-substrate model (1) and networked model (2), respectively. More precisely, the boundedness of the positive non-constant steady state will be studied for the elliptic system of the continuous general activator-substrate model (1). With the help of the maximum principle and the Harnack's inequality one shows that it admits the upper bounds and the lower bounds for any positive solution (u(x), v(x)) under certain conditions.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Spatiotemporal patterns in a general networked activator–substrate model

Chen

¹

,

Zheng

²

,

Wu

³

et al. 2021

Self Cite

View full text Add to dashboard Cite

For the purpose of understanding the spatiotemporal pattern formation in the random networked system, a general activator-substrate model with network structure is introduced. Firstly, we investigate the boundedness of the non-constant steady state of the elliptic system of the continuous media system. It is found that the non-constant steady state admits their upper and lower bounds with certain conditions. Then, one investigates some properties and non-existence of the non-constant steady state with the no-flux boundary conditions. The main results show that the diffusion rate of activator should greater than the diffusion rate of substrate. Otherwise, there might be no pattern formation of the system. Afterwards, a general random networked activator-substrate model is made public. The conditions of the stability, the Hopf bifurcation, the Turing instability and a co-dimensional-two Turing-Hopf bifurcation are yield by the method of stability analysis and bifurcation theorem. Finally, we choose a suitable sub-system of the general activator-substrate model to verify the theoretical results, and full numerical simulations are well verified these results. Especially, an interesting finding is that the stability of the positive equilibrium will switch from unstable to stable one

show abstract

Spatiotemporal patterns induced by Turing and Turing-Hopf bifurcations in a predator-prey system

Cited by 19 publications

References 56 publications

Dynamic analysis of a Leslie–Gower-type predator–prey system with the fear effect and ratio-dependent Holling III functional response

Dynamic analysis of a Leslie–Gower-type predator–prey system with the fear effect and ratio-dependent Holling III functional response

Spatiotemporal Patterns in a General Networked Activator-Substrate Model

Spatiotemporal patterns in a general networked activator–substrate model

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