Acta mathematica scientia, Series B >
POSITIVE STEADY STATES AND DYNAMICS FOR A DIFFUSIVE PREDATOR-PREY SYSTEM WITH A DEGENERACY
Received date: 2014-11-11
Revised date: 2015-10-12
Online published: 2016-04-25
Supported by
The research was supported by the National Natural Science Foundation of China (11361053, 11201204, 11471148, 11471330, 145RJZA112).
In this article, we consider positive steady state solutions and dynamics for a spatially heterogeneous predator-prey system with modified Leslie-Gower and Holling-Type II schemes. The heterogeneity here is created by the degeneracy of the intra-specific pressures for the prey. By the bifurcation method, the degree theory, and a priori estimates, we discuss the existence and multiplicity of positive steady states. Moreover, by the comparison argument, we also discuss the dynamical behavior for the diffusive predator-prey system.
Key words： Predator-prey system; steady state solution; dynamical behavior
Lu YANG , Yimin ZHANG . POSITIVE STEADY STATES AND DYNAMICS FOR A DIFFUSIVE PREDATOR-PREY SYSTEM WITH A DEGENERACY[J]. Acta mathematica scientia, Series B, 2016 , 36(2) : 537 -548 . DOI: 10.1016/S0252-9602(16)30019-4
[1] Aziz-Alaoui M A, Okiye M D. Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-Type II schemes. Appl Math Lett, 2003, 16:1069-1075
[2] Blat J, Brown K J. Global bifurcation of positive solutions in some systems of elliptic equations. SIAM J Math Anal, 1986, 17:1339-1353
[3] Chen B, Wang M X. Qualitative analysis for a diffusive predator-prey model. Computers and Mathematics with Applications, 2008, 55:339-355
[4] Crandall M G, Rabinowitz P H. Bifurcation from simple eigenvalues. J Funct Anal, 1971, 8:321-340
[5] Dancer E N, Du Y H. Effects of certain degeneracies in the predator-prey model. SIAM J Math Anal, 2002, 34:292-314
[6] Dockery J, Hutson V, Mischaikow K, Pernarowski M. The evolution of slow dispersal rates:a reaction-diffusion model. J Math Biol, 1998, 37:61-83
[7] Du Y H. Order Structure and Topological Methods in Nonlinear Partial Differential Equations. Vol 1. Maximum Principles and Applications. Singapore:World Scientific, 2005
[8] Du Y H, Hsu S B. A diffusive predator-prey model in heterogeneous environment. J Differential Equ, 2004, 203:331-364
[9] Du Y H, Huang Q. Blow-up solutions for a class of semilinear elliptic and parabolic equations. SIAM J Math Anal, 1999, 31:1-18
[10] Du Y H, Peng R, Wang M X. Effect of a protection zone in the diffusive Leslie predator-prey model. J Differential Equ, 2009, 246:3932-3956
[11] Du Y H, Shi J P. A diffusive predator-prey model with a protection zone. J Differential Equ, 2006, 229:63-91
[12] Du Y H, Shi J P. Some recent results on diffusive predator-prey models in spatially heterogeneous environ-ment//Brummer H, Zhao X, Zou X. Nonlinear Dynamics and Evolution Equations. Fields Inst Commun. Vol 48. Providence, RI:Amer Math Soc, 2006:95-135
[13] Du Y H, Shi J P. Allee effect and bistability in a spatially heterogeneous predator-prey model. Trans Amer Math Soc, 2007, 359(9):4557-4593
[14] Du Y H, Wang M X. Asymptotic behaviour of positive steady-states to a predator-prey model. Proc Roy Soc Edinburgh A, 2006, 136:759-779
[15] Hutson V, Lou Y, Mischaikow K. Spatial heterogeneity of resources versus Lotka-Volterra dynamics. J Differential Equ, 2002, 185:97-136
[16] Hutson V, Lou Y, Mischaikow K. Convergence in competition models with small diffusion coefficients. J Differential Equ, 2005, 211:135-161
[17] Hutson V, Lou Y, Mischaikow K, Polá?ik P. Competing species near a degenerate limit. SIAM J Math Anal, 2003, 35:453-491
[18] Hutson V, Mischaikow K, Polá?ik P. The evolution of dispersal rates in a heterogeneous time-periodic environment. J Math Biol, 2001, 43:501-533
[19] Kadota T, Kuto K. Positive steady states for a prey-predator model with some nonlinear diffusion terms. J Math Anal Appl, 2006, 323:1387-1401
[20] Kuto K. Bifurcation branch of stationary solutions for a Lotka-Volterra cross-diffusion system in a spatially heterogeneous environment. Nonlinear Anal RWA, 2009, 10:943-965
[21] Ko W, Ryu K. Coexistence states of a nonlinear Lotka-Volterra type predator-prey model with cross-diffusion. Nonlinear Anal, 2009, 71:e1109-e1115
[22] Lou Y, Ni W M. Diffusion vs cross-diffusion. J Differential Equ, 1996, 131:79-131
[23] Lou Y, Ni W M. Diffusion, self-diffusion and cross-diffusion:an elliptic approach. J Differential Equ, 1999, 154:157-190
[24] Oeda K. Effect of cross-diffusion on the stationary problem of a prey-predator model with a protection zone. J Differential Equ, 2011, 250:3988-4009
[25] Peng R, Wang M X. Positive steady states of the Holling-Tanner prey-predator model with diffusion. Proc Roy Soc Edinburgh, 2005, 135A:149-164
[26] Shi J P. Persistence and bifurcation of degenerate solutions. J Funct Anal, 1999, 169(2):494-531
[27] Wang Y X, Li W T. Effects of cross-diffusion and heterogeneous environment on positive steady states of a prey-predator system. Nonlinear Anal RWA, 2013, 14:1235-1246
[28] Wang Y X, Li W T. Fish-Hook shaped global bifurcation branch of a spatially heterogeneous cooperative system with cross-diffusion. J Differential Equ, 2011, 251:1670-1695
/
| 〈 |
|
〉 |