POSITIVE STEADY STATES OF A DIFFUSIVE PREDATOR-PREY SYSTEM WITH PREDATOR CANNIBALISM

doi:10.1016/S0252-9602(17)30080-2

Abstract

Abstract:

The purpose of this paper is to investigate positive steady states of a diffusive predator-prey system with predator cannibalism under homogeneous Neumann boundary conditions. With the help of implicit function theorem and energy integral method, non-existence of non-constant positive steady states of the system is obtained, whereas coexistence of non-constant positive steady states is derived from topological degree theory. The results indicate that if dispersal rate of the predator or prey is sufficiently large, there is no non-constant positive steady states. However, under some appropriate hypotheses, if the dispersal rate of the predator is larger than some positive constant, for certain ranges of dispersal rates of the prey, there exists at least one non-constant positive steady state.

Key words: Predator-prey, existence and nonexistence, pattern formation

Biao WANG. POSITIVE STEADY STATES OF A DIFFUSIVE PREDATOR-PREY SYSTEM WITH PREDATOR CANNIBALISM[J].Acta mathematica scientia,Series B, 2017, 37(5): 1385-1398.

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References

[1] Sun G Q, Zhang G, Jin Z, Li L, et al. Predator cannibalism can give rise to regular spatial pattern in a predator-prey system. Nonlinear Dynam, 2009, 58(1):75-84
[2] Huang J H, Lu G, Ruan S G, et al. Existence of traveling wave solutions in a diffusive predator-prey model. J Math Biol, 2003, 46(2):132-152
[3] Yi F Q, Wei J J, Shi J P, et al. Bifurcation and spatiotemporal patterns in homogeneous diffusive predatorprey system. J Differential Equ, 2009, 246(5):1944-1977
[4] Garvie M R, Trenchea C. Spatiotemporal dynamics of two generic predator-prey models. J Biol Dynamics, 2010, 4(6):559-570
[5] Akhmeta M U, Beklioglub M, Ergenca T, Tkachenkoc V I, et al. An impulsive ratio-dependent predatorprey system with diffusion. Nonlinear Anal RWA, 2006, 7(5):1255-1267
[6] Dancer E N, Du Y H. Effects of certain degeneracies in the predator-prey model. SIAM J Math Anal, 2002, 34(2):292-314
[7] Du Y H, Lou Y. Some uniqueness and exact multiplicity results for a predator-prey model. Trans Amer Math Soc, 1997, 349(6):2443-2475
[8] Du Y H, Lou Y. S-shaped global bifurcation curve and Hopf bifurcation of positive solution to a predatorprey model. J Differential Equ, 1998, 144(2):390-440
[9] Du Y H, Wang M X. Asymptotic behaviour of positive steady states to a predator-prey model. Proc R Soc Edinb A, 2006, 136(4):759-778
[10] Hsu S B, Huang T W. Global stability for a class of predator-prey systems. SIAM J Appl Math, 1995, 55(3):763-783
[11] Jia Y F, Wu J H, Nie H, et al. The coexistence states of a predator-prey model with nonmonotonic functional response and diffusion. Acta Appl Math, 2009, 108(2):413-428
[12] Jia Y F, Xu H K, Agarwal R P, et al. Existence of positive solutions for a prey-predator model with refuge and diffusion. Appl Math Comput, 2011, 217(21):8264-8276
[13] Kadota T, Kuto K. Positive steady states for a prey-predator model with some nonlinear diffusion terms. J Math Anal. Appl, 2006, 323(2):1387-1401
[14] Kuang Y, Beretta B. Global qualitative analysis of a ratio-dependent predator-prey system. J Math Biol, 1998, 36(4):389-406
[15] Kuto K. Stability of steady-state solutions to a prey-predator system with cross-diffusion. J Differential Equ, 197(2):293-314
[16] Ma Z P, Li W T. Bifurcation analysis on a diffusive Holling-Tanner predator-prey model. Appl Math Modelling, 2013, 37(37):4371-4384
[17] Nakashima K, Yamada Y. Positive steady states for prey-predator models with cross-diffusion. Adv Differential Equ, 1996, 1(6):1099-1122
[18] Ruan S G, Xiao D M. Global analysis in a predator-prey system with nonmonotonic functional response. SIAM J Appl Math, 2001,61(4):1445-1472
[19] Shi H B, Li W T, Lin G, et al. Positive steady states of a diffusion predator-prey system with modified Holling-Tanner function response. Nonlinear Anal RWA, 2010,11(5):3711-3721
[20] Yamada Y. Stability of steady states for prey-predator diffusion equations with homogeneous Dirichlet conditions. SIAM J Math Anal, 1990, 21(2):327-345
[21] Zhou J. Positive steady state solutions of a Leslie-Gower predator-prey model with Holling type Ⅱ functional response and density-dependent diffusion. Nonlinear Anal TMA, 2013, 82(9):47-65
[22] Yang L, Zhang Y M. Positive steady states and dynamics for a diffusive predator-prey system with a degeneracy. Acta Math Sci, 2016, 36B(2):537-548
[23] Zhou W S, Zhao H X, Wei X D, Xu G K, et al. Existence of positive steady states for a predator-prey model with diffusion. Comm Pure Appl Anal, 2013, 12(5):2189-2201
[24] Lou Y, Wang B. Local dynamics of a diffusive predator-prey model in spatially heterogeneous environment. J Fixed Point Theory Appl, 2017, 19(1):755-772
[25] Du Y H, Hsu S B. A diffusive predator-prey model in heterogeneous environment. J. Differential Equ, 2004, 203(2):331-364
[26] Du Y H, Shi J P. Some recent results on diffusive predator-prey models in spatially heterogeneous envionment//Nonlinear Dynamics and Evolution Equations. Providence, RI:Amer Math Soc, 2006:95-135
[27] Du Y H, Shi J P. Allee effect and bistability in a spatially heterogenous predator-prey model. Trans Amer Math Soc, 2007, 359(9):4557-4593
[28] Wang B, Zhang Z C. Bifurcation analysis of a diffusive predator-prey model in spatially heterogeneous environment. Electron J Qual Theo, 2017, 2017(42):1-17
[29] Lou Y, Ni W M. Diffusion, self-diffusion and cross-diffusion. J Differential Equ, 1996, 131(1):79-131
[30] Lin C S, Ni W M, Takagi I, et al. Large amplitude stationary solutions to a chemotais systems. J Differential Equ, 1988, 72(1):1-27
[31] Gilbarg D, Trudinger N S. Elliptic Partial Differential Equation of Second Order. 2nd ed. Berlin:SpringerVerlag, 1983
[32] Nirenberg L. Topics in Nonlinear Functional Analysis. Providence, RI:American Mathematical Society, 2001
[33] Pang P Y H, Wang M X. Non-constant positive steady-states of a predator-prey system with non-monotonic functional response and diffusion. Proc Lond Math Soc, 2004, 88(1):135-157
[34] Peng R, Wang M X. Positive steady-states of the Holling-Tanner prey-predator model with diffusion. Proc R Soc Edinb. A, 2005, 135(1):149-164