Asymptotic behavior and multiplicity for a diffusive Leslie-Gower predator-prey system with Crowley-Martin functional response

Abstract A diffusive predator–prey system with modified Leslie–Gower and Crowley–Martin functional response is considered. The extinction and permanence of the time-dependent system are determined by virtue of the comparison principle. Then, the sufficient and necessary conditions for the existence of coexistence states are obtained. Furthermore, the stability, uniqueness and exact multiplicity of coexistence states are investigated by means of the combination of the perturbation theory, bifurcation theory and degree theory. Our results indicate that c 1 have an effect on the stability and exact multiplicity of coexistence states. Finally, some numerical simulations are presented to verify and complement the theoretical results.

[1]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[2]  Chris Cosner,et al.  On the Dynamics of Predator–Prey Models with the Beddington–DeAngelis Functional Response☆ , 2001 .

[3]  Philip H. Crowley,et al.  Functional Responses and Interference within and between Year Classes of a Dragonfly Population , 1989, Journal of the North American Benthological Society.

[4]  Global bifurcation of co-existence states for a predator-prey-mutualist model with diffusion , 2007 .

[5]  K. Balachandran,et al.  Stability and Hopf bifurcation of a diffusive predator–prey model with predator saturation and competition , 2013 .

[6]  Sze-Bi Hsu,et al.  A diffusive predator–prey model in heterogeneous environment , 2004 .

[7]  Lige Li,et al.  Coexistence theorems of steady states for predator-prey interacting systems , 1988 .

[8]  Wonlyul Ko,et al.  Analysis of diffusive two-competing-prey and one-predator systems with Beddington–Deangelis functional response , 2009 .

[9]  Jun Zhou.  Positive solutions of a diffusive Leslie–Gower predator–prey model with Bazykin functional response , 2014 .

[10]  Gaihui Guo,et al.  Multiplicity and uniqueness of positive solutions for a predator–prey model with B–D functional response☆ , 2010 .

[11]  Mingxin Wang,et al.  Positive solutions of a prey–predator model with predator saturation and competition☆ , 2008 .

[12]  Jianhua Wu,et al.  Global bifurcation of coexistence state for the competition model in the chemostat , 2000 .

[13]  Junping Shi,et al.  The existence, bifurcation and stability of positive stationary solutions of a diffusive Leslie–Gower predator–prey model with Holling-type II functional responses , 2013 .

[14]  M. Crandall,et al.  Bifurcation from simple eigenvalues , 1971 .

[15]  E. N. Dancer.  On the indices of fixed points of mappings in cones and applications , 1983 .

[16]  Yuan Lou,et al.  Some uniqueness and exact multiplicity results for a predator-prey model , 1997 .

[17]  Rui Peng,et al.  Non-existence of non-constant positive steady states of two Holling type-II predator–prey systems: Strong interaction case , 2009 .

[18]  Junjie Wei,et al.  Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system ✩ , 2009 .

[19]  Thilo Gross,et al.  Instabilities in spatially extended predator-prey systems: spatio-temporal patterns in the neighborhood of Turing-Hopf bifurcations. , 2007, Journal of theoretical biology.

[20]  Mingxin Wang,et al.  On multiplicity and stability of positive solutions of a diffusive prey-predator model , 2006 .

[21]  Jun Zhou,et al.  Positive solutions of a diffusive predator-prey model with modified Leslie-Gower and Holling-type II schemes ✩ , 2012 .

[22]  Chunlai Mu,et al.  Positive solutions for a three-trophic food chain model with diffusion and Beddington–Deangelis functional response☆ , 2011 .

[23]  Mingxin Wang,et al.  Stationary patterns for a prey-predator model with prey-dependent and ratio-dependent functional responses and diffusion , 2004 .

[24]  Jianhua Wu,et al.  Qualitative analysis for a ratio-dependent predator–prey model with stage structure and diffusion☆ , 2008 .

[25]  Gaihui Guo,et al.  The effect of predator competition on positive solutions for a predator–prey model with diffusion ☆ , 2012 .

[26]  Bin Chen,et al.  Qualitative analysis for a diffusive predator-prey model , 2008, Comput. Math. Appl..

[27]  C. V. Pao,et al.  Quasisolutions and global attractor of reaction-diffusion systems , 1996 .

[28]  Rui Peng,et al.  Qualitative analysis on a diffusive prey-predator model with ratio-dependent functional response , 2008 .

[29]  Yihong Du,et al.  Effects of Certain Degeneracies in the Predator-Prey Model , 2002, SIAM J. Math. Anal..

[30]  Mingxin Wang,et al.  Global asymptotic stability of positive steady states of a diffusive ratio-dependent prey-predator model , 2008, Appl. Math. Lett..

[31]  Sze-Bi Hsu,et al.  Global Stability for a Class of Predator-Prey Systems , 1995, SIAM J. Appl. Math..

[32]  S. Meyn,et al.  THE EXISTENCE OF AN “I” , 2020, The Nature of Order, Book 4: The Luminous Ground.

[33]  Mingxin Wang,et al.  Asymptotic behaviour of positive steady states to a predator—prey model , 2006, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[34]  Chris Cosner,et al.  EFFECTS OF DOMAIN SIZE ON THE PERSISTENCE OF POPULATIONS IN A DIFFUSIVE FOOD‐CHAIN MODEL WITH BEDDINGTON‐DeANGELIS FUNCTIONAL RESPONSE , 2001 .

[35]  M. Jazar,et al.  Global dynamics of a modified Leslie-Gower predator-prey model with Crowley-Martin functional responses , 2013 .

[36]  Rui Peng,et al.  Global stability of the equilibrium of a diffusive Holling-Tanner prey-predator model , 2007, Appl. Math. Lett..

[37]  Wonlyul Ko,et al.  Qualitative analysis of a predator-prey model with Holling type II functional response incorporating a prey refuge , 2006 .

[38]  Rui Peng,et al.  Positive steady states of the Holling–Tanner prey–predator model with diffusion , 2005, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.