Pattern selection in a predation model with self and cross diffusion

doi:10.1088/1674-1056/20/3/034702

中国物理B ›› 2011, Vol. 20 ›› Issue (3): 34702-034702.doi: 10.1088/1674-1056/20/3/034702

Pattern selection in a predation model with self and cross diffusion

王玮明¹, 林晔智², 王文娟³, 谭永基³

(1)College of Mathematics and Information Science, Wenzhou University, Wenzhou 325035, China;School of Mathematical Sciences, Fudan University, Shanghai 200433, China; (2)Computer Sci-Tech Department, East China Normal University, Shanghai 200062, China; (3)School of Mathematical Sciences, Fudan University, Shanghai 200433, China

收稿日期:2010-07-29 修回日期:2010-11-19 出版日期:2011-03-15 发布日期:2011-03-15
基金资助:
Project supported by the Natural Science Foundation of Zhejiang Province, China (Grant No. Y7080041), and the Shanghai Postdoctoral Scientific Program, China (Grant No. 09R21410700).

Pattern selection in a predation model with self and cross diffusion

Wang Wei-Ming(王玮明)^a)b)†, Wang Wen-Juan(王文娟)^b), Lin Ye-Zhi(林晔智)^c), and Tan Yong-Ji(谭永基)^b)

^a College of Mathematics and Information Science, Wenzhou University, Wenzhou 325035, China; ^b School of Mathematical Sciences, Fudan University, Shanghai 200433, China; ^c Computer Sci-Tech Department, East China Normal University, Shanghai 200062, China

Received:2010-07-29 Revised:2010-11-19 Online:2011-03-15 Published:2011-03-15
Supported by:
Project supported by the Natural Science Foundation of Zhejiang Province, China (Grant No. Y7080041), and the Shanghai Postdoctoral Scientific Program, China (Grant No. 09R21410700).

摘要/Abstract

摘要： In this paper, we present the amplitude equations for the excited modes in a cross-diffusive predator--prey model with zero-flux boundary conditions. From these equations, the stability of patterns towards uniform and inhomogenous perturbations is determined. Furthermore, we present novel numerical evidence of six typical turing patterns, and find that the model dynamics exhibits complex pattern replications: for μ₁<μ≤μ₂, the steady state is the only stable solution of the model; for μ₂<μ≤μ₄, by increasing the control parameter μ, the sequence H_π-hexagons → H₀-hexagon-stripe mixtures rightarrow stripes → H_π-hexagon-stripe mixtures → H₀-hexagons is observed; for μ>μ₄, the stripe pattern emerges. This may enrich the pattern formation in the cross-diffusive predator--prey model.

关键词: cross-diffusion, turing instability, pattern selection, amplitude equations

Abstract: In this paper, we present the amplitude equations for the excited modes in a cross-diffusive predator--prey model with zero-flux boundary conditions. From these equations, the stability of patterns towards uniform and inhomogenous perturbations is determined. Furthermore, we present novel numerical evidence of six typical turing patterns, and find that the model dynamics exhibits complex pattern replications: for $\mu_1<\mu\leq\mu_2$, the steady state is the only stable solution of the model; for $\mu_2<\mu\leq\mu_4$, by increasing the control parameter $\mu$, the sequence $H_{\pi}$-hexagons $\rightarrow$ $H_{\pi}$-hexagon-stripe mixtures $\rightarrow$ stripes $\rightarrow$ $H_{0}$-hexagon-stripe mixtures $\rightarrow$ $H_{0}$-hexagons is observed; for $\mu>\mu_4$, the stripe pattern emerges. This may enrich the pattern formation in the cross-diffusive predator--prey model.

Key words: cross-diffusion, turing instability, pattern selection, amplitude equations

中图分类号: (Pattern selection; pattern formation)

47.54.-r

87.23.Cc (Population dynamics and ecological pattern formation) 89.75.Kd (Patterns)

王玮明, 王文娟, 林晔智, 谭永基. Pattern selection in a predation model with self and cross diffusion[J]. 中国物理B, 2011, 20(3): 34702-034702.

Wang Wei-Ming(王玮明), Wang Wen-Juan(王文娟), Lin Ye-Zhi(林晔智), and Tan Yong-Ji(谭永基). Pattern selection in a predation model with self and cross diffusion[J]. Chin. Phys. B, 2011, 20(3): 34702-034702.

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Pattern selection in a predation model with self and cross diffusion

Pattern selection in a predation model with self and cross diffusion

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