Pattern formation in a diffusive predator–prey system with cross-diffusion effects

We consider the effect of time delay and cross diffusion on the dynamics of a modified Leslie-Gower predator-prey model incorporating a prey refuge. Based on the stability analysis, we demonstrate that delayed feedback may generate Hopf and Turing instability under some conditions, resulting in spatial patterns. One of the most interesting findings is that the model exhibits complex pattern replication: the model dynamics exhibits a delay and diffusion controlled formation growth not only to spots, stripes, and holes, but also to spiral pattern self-replication. The results indicate that time delay and cross diffusion play important roles in pattern formation.

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On global bifurcation for a cross-diffusion predator–prey system with prey-taxis

Computers & Mathematics with Applications ◽

10.1016/j.camwa.2018.05.037 ◽

2018 ◽

Vol 76 (5) ◽

pp. 1014-1025 ◽

Cited By ~ 1

Author(s):

Chenglin Li

Keyword(s):

Global Bifurcation ◽

Predator Prey ◽

Cross Diffusion ◽

Prey System

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Spatiotemporal pattern formation of a ratio-dependent invasion-diffusion predator-prey system with disease in the predator

Journal of Interdisciplinary Mathematics ◽

10.1080/09720502.2017.1367526 ◽

2019 ◽

Vol 22 (2) ◽

pp. 101-120

Author(s):

Chenglin Li

Keyword(s):

Pattern Formation ◽

Spatiotemporal Pattern ◽

Predator Prey ◽

Spatiotemporal Pattern Formation ◽

Prey System ◽

Ratio Dependent

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Pattern Formation, Long-Term Transients, and the Turing–Hopf Bifurcation in a Space- and Time-Discrete Predator–Prey System

Bulletin of Mathematical Biology ◽

10.1007/s11538-010-9593-5 ◽

2010 ◽

Vol 73 (8) ◽

pp. 1812-1840 ◽

Cited By ~ 37

Author(s):

Luiz Alberto Díaz Rodrigues ◽

Diomar Cristina Mistro ◽

Sergei Petrovskii

Keyword(s):

Hopf Bifurcation ◽

Pattern Formation ◽

Predator Prey ◽

Space And Time ◽

Prey System

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Neimark-Sacker-Turing Instability and Pattern Formation in a Spatiotemporal Discrete Predator-Prey System with Allee Effect

Discrete Dynamics in Nature and Society ◽

10.1155/2018/8713651 ◽

2018 ◽

Vol 2018 ◽

pp. 1-17

Author(s):

Huayong Zhang ◽

Xuebing Cong ◽

Tousheng Huang ◽

Shengnan Ma ◽

Ge Pan

Keyword(s):

Pattern Formation ◽

Allee Effect ◽

Initial Conditions ◽

Turing Instability ◽

Spatiotemporal Chaos ◽

Closed Curves ◽

Predator Prey ◽

Prey System ◽

Route To Chaos ◽

The Rich

A spatiotemporal discrete predator-prey system with Allee effect is investigated to learn its Neimark-Sacker-Turing instability and pattern formation. Based on the occurrence of stable homogeneous stationary states, conditions for Neimark-Sacker bifurcation and Turing instability are determined. Numerical simulations reveal that Neimark-Sacker bifurcation triggers a route to chaos, with the emergence of invariant closed curves, periodic orbits, and chaotic attractors. The occurrence of Turing instability on these three typical dynamical behaviors leads to the formation of heterogeneous patterns. Under the effects of Neimark-Sacker-Turing instability, pattern evolution process is sensitive to tiny changes of initial conditions, suggesting the occurrence of spatiotemporal chaos. With application of deterministic initial conditions, transient symmetrical patterns are observed, demonstrating that ordered structures can exist in chaotic processes. Moreover, when local kinetics of the system goes further on the route to chaos, the speed of symmetry breaking becomes faster, leading to more fragmented and more disordered patterns at the same evolution time. The rich spatiotemporal complexity provides new comprehension on predator-prey coexistence in the ways of spatiotemporal chaos.

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