Non-constant positive solutions of a general Gause-type predator-prey system with self- and cross-diffusions

Pan Xue¹ ; Yunfeng Jia² ; Cuiping Ren¹ ; Xingjun Li¹

¹ School of General Education, Xi’an Eurasia University, Xi’an, Shaanxi 710065, China.
² School of Mathematics and Statistics, Shaanxi Normal University, Xi’an, Shaanxi 710062, China.

Mathematical modelling of natural phenomena, Tome 16 (2021), article no. 25.

Résumé

In this paper, we investigate the non-constant stationary solutions of a general Gause-type predator-prey system with self- and cross-diffusions subject to the homogeneous Neumann boundary condition. In the system, the cross-diffusions are introduced in such a way that the prey runs away from the predator, while the predator moves away from a large group of preys. Firstly, we establish a priori estimate for the positive solutions. Secondly, we study the non-existence results of non-constant positive solutions. Finally, we consider the existence of non-constant positive solutions and discuss the Turing instability of the positive constant solution.

Détail

DOI : 10.1051/mmnp/2021017

Bibliographie

[1] J.H.P. Dawes, M.O. Souza A derivation of Hollingís type I, II and III functional responses in predator-prey systems J. Theoret. Biol 2013 11 22

[2] D.T. Dimitrov, H.V. Kojouharov Complete mathematical analysis of predator-prey models with linear prey growth and Beddington-DeAngelis Appl. Math. Comput 2005 523 538

[3] D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equation of Second Order. Springer-Verlag, Berlin (2001).

[4] N.S. Goel, S.C. Maitra On the Volterra and other nonlinear models of interacting populations Rev. Modern Phys 1971 231 276

[5] Y. Jia A sufficient and necessary condition for the existence of positive solutions for a prey-predator system with Ivlev-type functional response Appl. Math. Lett 2011 1084 1088

[6] Y. Jia, J. Wu, H.-K. Xu Positive solutions for a predator-prey interaction model with Hollings-type functional response and diffusion Taiwan. J. Math 2011 2013 2034

[7] Y. Jia, J. Wu, H.-K. Xu Positive solutions of a Lotka-Volterra competition model with cross-diffusion Comput. Math. Appl 2014 1220 1228

[8] Y. Jia, P. Xue Effects of the self- and cross-diffusion on positive steady states for a generalized predator-prey system Nonlinear Anal. Real World Appl 2016 229 241

[9] W. Ko, K. Ryu A qualitative study on general Gause-type predator-prey models with constant diffusion rates J. Math. Anal. Appl 2008 217 230

[10] J. Leray, J. Schauder Topologie et équations fonctionnelles Ann. Sci. École Norm. Sup. 1934 45 78

[11] C.-S. Lin, W.-M. Ni, I. Takagi Large amplitude stationary solutions to a chemotaxis system J. Differ. Equ 1988 1 27

[12] Y. Lou, W.-M. Ni, S. Yotsutani On a limiting system in the Lotka-Volterra competition with cross-diffusion Discret. Contin. Dyn. Syst 2004 435 458

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

[14] A. Madzvamuse, H.S. Ndakwo, R. Barreira Cross-diffusion-driven instability for reaction-diffusion systems: analysis and simulations J. Math. Biol 2014 262 292

[15] Q. Meng, L. Yang Steady state in a cross-diffusion predator-prey model with the Beddington-DeAngelis functional response Nonlinear Anal. Real World Appl 2019 401 413

[16] W.-M. Ni Diffusion, cross-diffusion and their spike-layer steady states Notices Am. Math. Soc 1998 9 18

[17] K. Oeda Effect of cross-diffusion on the stationary problem of a prey-predator model with a protection zone J. Differ. Equ 2011 3988 4009

[18] A. Okubo, Diffusion and Ecological Problems: Mathematical Models. Springer-Verlag, Berlin (1980).

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

[20] R. Peng, J. Shi Non-existence of non-constant positive steady states of two Holling-type II predator-prey systems: strong interaction case J. Differ. Equ 2009 866 886

[21] G. Seo, D.L. Deangelis A predator-prey model with a Holling type I functional response including a predator mutual interference J. Nonlinear Sci 2011 811 833

[22] A.A. Shaikh, H. Das, N. Ali, C. Sonck Study of LG-Holling type III predator-prey model with disease in predator J. Appl. Math. Comput 2018 235 255

[23] J. Sugie Two-parameter bifurcation in a predator-prey system of Ivlev type J. Math. Anal. Appl 1998 349 371

[24] J. Shi, Z. Xie, K. Little Cross-diffusion induced instability and stability in reaction-diffusion systems J. Appl. Anal. Comput 2011 95 119

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