带有交错扩散的Leslie-Gower型三种群系统的稳态模式

摘要/Abstract

摘要：

讨论了带有Neumann边界条件的一类Leslie-Gower型三种群系统, 在一定的条件之下, 虽然系统对应的扩散(没有交错扩散)系统的唯一正平衡解
是稳定的, 系统中的交错扩散可导致Turing不稳定性的产生. 特别地, 建立了该系统非常数共存解的存在性. 结果表明, 交错扩散可引起系统中出现非常数正稳态解(稳态模式).

关键词: 交错扩散, 捕食系统, 先验估计, 非常数正稳态

Abstract:

This paper is concerned with a Leslie-Gower type three species model subject to the homogeneous Neumann boundary condition. We will show that under certain hypotheses, even though the unique positive equilibrium is asymptotically stable for the dynamics with diffusion, Turing instability can produce due to the presence of the cross-diffusion. In particular, we establish the existence of non-constant positive steady states of this system. The results indicate that cross-diffusion can create stationary patterns.

Key words: Cross-diffusion, Predator-prey system, Priori estimates, Non-constant positive steady state

中图分类号:

35J48

胡广平, 李小玲. 带有交错扩散的Leslie-Gower型三种群系统的稳态模式[J]. 数学物理学报, 2013, 33(1): 16-27.

HU Guang-Ping, LI Xiao-Ling. Stationary Patterns of a Leslie-Gower Type Three Species Model with Cross-Diffusions[J]. Acta mathematica scientia,Series A, 2013, 33(1): 16-27.

参考文献

[1] Aziz-Alaoui M A. Study of Leslie-Gower type tritrophic population model. Chaos Solitons & Fractals, 2002, 8: 1275--1293

[2] Chen L, J\"{u}ngel A. Analysis of a parabolic cross-diffusion population model without self-diffusion. J Differential Equations, 2006, 224: 39--59

[3] Chen W, Wang M. Qualitative analysis of predator-prey models with Beddington-DeAngelis functional response and diffusion. Math Comput Modelling, 2005, 42: 31--44

[4] Du Y, Hsu S B. Adiffusive predator-prey model in heterogeneous environment. J Differential Equations, 2004, 203: 331--364

[5] Gilbarg D, Trudinger N S. Elliptic Partial Differential Equations of Second Order. Berlin: Springer-Verlag, 2001

[6] Hei L J, Yu Y. Non-constant positive steady state of one resource and two consumers model with diffusion. J Math Anal Appl, 2008, 339: 566--581

[7] Henry D. Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics. Berlin, New York: Springer-Verlag, 1993

[8] Hu G P, Li X L. Stationary patterns for a Leslie-Gower type three species model with diffusion. Preprint

[9] Ko W, Ryu K. Non-constant positive steady-state of a diffusive predator-prey system in homogeneous environment. J Math Anal Appl, 2007, 327: 539--549

[10] Ko W, Ryu K. Qualitative analysis of a predator-prey model with Holling type II functinal response incorporating a prey refuge. J Differential Equations, 2006, 231: 534--550

[11] Leslie P H, Gower J C. A properties of a stochastic model for the predator-prey type of interaction between two species. Biometrica, 1960, 47: 219--234

[12] Lin L S, Ni W M, Takagi I. Large amplitude stationary solutions to a chemotaxis systems. J Differential Equations, 1988, 72: 1--27

[13] Lou Y, Ni W M. Diffusion, self-diffusion and cross-diffusion. J Differential Equations, 1996, 131: 79--131

[14] Lou Y, Mart\'inez S, Ni W. On 3 \times 3 Lotka-Volterra competition systems with cross-diffusion. Discrete Contin Dyn Syst, 2000, 6: 175--190

[15] Nindjin A F, Aziz-Alaoui M A. Persistence and global stability in a delayed Leslie-Gower type three species food chain. J Math Anal Appl, 2008, 340: 340--357

\REF{[16]}Ni W M, Tang M. Turing patterns in the Lengyel-Epstein system for the CIMA reaction. Trans Amer Math Soc, 2005, 357: 3953--3969

[17] Peng R, Sun F. Turing pattern of the oregonator model. Nonlinear Anal TMA, 2010, 72: 2337--2345

[18] Peng R, Wang M. Positive steady states of the Holling-Tanner prey-predator model with diffusion. Proc Roy Soc Edinburgh, 2005, 135A: 149--164

[19] Peng R, Wang M. Global stability of the equilibrium of a diffusive Holling-Tanner prey-predator model. Appl Math Lett, 2007, 20: 664--670

[20] Pang P Y H, Wang M. Non-constant positive steady states of a predator-prey system with non-monotonic functional response and diffusion. Proc Lond Math Soc, 2004, 88: 135--157

[21] Pang P Y H, Wang M. Strategy and stationary pattern in a three-species predator-prey. J Differential Equations, 2004, 200: 245--273

[22] Ruan S. Turing instability and travelling waves in diffusive plankton models with delayed nutrient recycling. IMA J Appl Math, 1998, 61: 15--32

[23] Shi H B, Li W T, Lin G. Positive steady states of a diffusive predator-prey system with modified Holling-Tanner functional sponse. Nonlinear Anal RWA, 2010, 11: 3711--3721

[24] 史红波, 李万同, 林果. 一类修正的Leslie-Gower型扩散捕食系统的定性分析. 数学物理学报, 2011, 31A: 1403--1415

[25] 陈滨, 王明新. 一类三种群捕食模型的正解. 数学物理学报, 2008, 28A: 1256--1266

[26] Wang M X. Stationary patterns for a prey-predator model with prey-dependent and ratio-dependent functional responses and diffusion.
Phys D, 2004, 196: 172--192

[27] Wang M X. Stationary patterns caused by cross-diffusion for a three-species prey-predator model. Comput Math Appl, 2006, 52: 707--720

[28] Zeng X. Non-constant positive steady states of a prey-predator system with cross-diffusions. J Math Anal Appl, 2007, 332: 989--1009