JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2015, Vol. 50 ›› Issue (03): 80-87.doi: 10.6040/j.issn.1671-9352.0.2014.339

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Dynamics research in a predator-prey system with a nonlinear growth rate

YANG Wen-bin, LI Yan-ling   

  1. College of Mathematics and Information Science, Shaanxi Normal University, Xi'an 710062, Shaanxi, China
  • Received:2014-07-21 Revised:2014-11-10 Online:2015-03-20 Published:2015-03-13

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Abstract: The paper is concerned with a predator-prey diffusive dynamics subject to homogeneous Dirichlet boundary conditions, where the predator population reproduces by the nonlinear function 1/(1+ev). Existence and uniqueness of coexistence states for the predator-prey system are investigated. Moreover, some asymptotic behaviors of time-dependent solutions are shown and some numerical simulations are done to complement the analytical results. The main tools used here include the implicit function theorem, the bifurcation theory and the perturbation technique.

Key words: reaction-diffusion equations, uniqueness, numerical simulation, stability, positive solution

CLC Number: 

  • O175.26
[1] BLAT J, BROWN K J. Global bifurcation of positive solutions in some systems of elliptic equations[J]. SIAM J Math Anal, 1986, 17(6):1339-1353.
[2] DE lA SEN M. The generalized Beverton-Holt equation and the control of populations[J]. Appl Math Model, 2008, 32(11):2312-2328.
[3] DE lA SEN M, ALONSO-QUESADA S. Control issues for the Beverton-Holt equation in ecology by locally monitoring the environment carrying capacity: Non-adaptive and adaptive cases[J]. Appl Math Comput, 2009, 215(7): 2616-2633.
[4] ERBACH A, LUTSCHER F, SEO G. Bistability and limit cycles in generalist predator-prey dynamics[J]. Ecol Comple, 2013, 14:48-55.
[5] HSU S B, HWANG T W, KUANG Y. Global analysis of the Michaelis-Menten-type ratio-dependent predator-prey system [J]. J Math Biol, 2001, 42(6):489-506.
[6] KUANG Y, BERETTA E. Global qualitative analysis of a ratio-dependent predator-prey system[J]. J Math Biol, 1998, 36(4):389-406.
[7] PAO C V. Nonlinear parabolic and elliptic equations[M]. New York: Plenum Press, 1992.
[8] PROTTER M H, WEINBERGER H F. Maximum principles in differential equations[M]. New York: Springer-Verlag, 1984.
[9] SCHAEFER H H. Topological vector spaces[M]. New York: Springer, 1971.
[10] SMOLLER J. Shock waves and reaction-diffusion equations[M]. New York: Springer-Verlag, 1994.
[11] TANG Sanyi, CHEKE R A, XIAO Yanni. Optimal implusive harvesting on non-autonomous Beverton-Holt difference equations[J]. Nonlinear Anal, 2006, 65(12): 2311-2341.
[12] YAMADA Y. Positive solutions for Lotka-Volterra systems with cross-diffusion [M]//Battelli F, Fecˇkan M. Handbook of differential equations: stationary partial differential equations: Vol VI. Amsterdam: Elsevier, 2008:411-501.
[13] YI Fengqi, WEI Junjie, SHI Junping. Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system [J]. J Differential Equations, 2009, 246(5):1944-1977.
[14] YOSHIDA T, JONES L E, ELLNER S P, et al. Rapid evolution drives ecological dynamics in a predator-prey system [J]. Nature, 2003, 424(17): 303-306.
[15] ZHANG Guohong, WANG Wendi, WANG Xiaoli. Coexistence states for a diffusive one-prey and two-predators model with B-D functional response [J]. J Math Anal Appl, 2012, 387(2): 931-948.
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