Dynamics of a delayed diffusive predator–prey model with hyperbolic mortality
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This paper is devoted to consider a time-delayed diffusive prey–predator model with hyperbolic mortality. We focus on the impact of time delay on the stability of positive constant solution of delayed differential equations and positive constant equilibrium of delayed diffusive differential equations, respectively, and we investigate the similarities and differences between them. Our conclusions show that when time delay continues to increase and crosses through some critical values, a family of homogenous and inhomogeneous periodic solutions emerge. Particularly, we find the minimum value of time delay, which is often hard to be found. We also consider the nonexistence and existence of steady state solutions to the reaction–diffusion model without time delay.
KeywordsDelayed predator–prey model Reaction–diffusion equation Hopf bifurcation Steady state solutions Stationary pattern
Mathematics Subject Classification34C23 35B32 35J25 92D40
This work was supported by the National Natural Science Foundation of China (No. 11501572) and by the Fundamental Research Funds for the Central Universities of China (No. 15CX02076A).
- 5.Moussaoui, A., Bassaid, S., Dads, E.H.A.: The impact of water level fluctuations on a delayed prey–predator model. Nonlinear Anal. Real World Appl. 21, 170–184 (2015)Google Scholar
- 18.Chen S. S., Shi J. P., Wei J. J.: Global stability and Hopf bifurcation in a delayed diffusive Leslie–Gower predator–prey system. Int. J. Bifurc. Chaos 22 (3) (2012)Google Scholar
- 28.Li, Y.: Steady-state solution for a general Schnakenberg model. Nonlinear Anal. Real World Appl. 12, 1985–1990 (2011)Google Scholar
- 29.Li, Y., Wang, M.X.: Stationary pattern of a diffusive prey–predator model with trophic intersections of three levels. Nonlinear Anal. Real World Appl. 14, 1806–1816 (2013)Google Scholar
- 31.Ruan S. G., Wei J. J.: On the zeros of transcendental functions with applications to stability of delay differential equations with two delays. Dynamics of continuous, Discrete and Impulsive Systems Series A: Math. Anal. 10: 863–874 (2003)Google Scholar
- 35.Nirenberg, L.: Topics in Nonlinear Functional Analysis. American Mathematical Society, Providence (2001)Google Scholar