Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

Spatial dynamics of a diffusive predator-prey model with stage structure

Pages: 1831 - 1853, Volume 20, Issue 6, August 2015      doi:10.3934/dcdsb.2015.20.1831

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Liang Zhang - School of Mathematics and Statistics, Lanzhou University , and Key Laboratory of Applied Mathematics and Complex Systems of Gansu province, Lanzhou, Gansu 730000, China (email)
Zhi-Cheng Wang - School of Mathematics and Statistics, Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou, Gansu 730000, China (email)

Abstract: In this paper, we propose a nonlocal and time-delayed reaction-diffusion predator-prey model with stage structure. It is assumed that prey individuals undergo two stages, immature and mature, and the conversion of consumed prey biomass into predator biomass has a retardation. In terms of the principal eigenvalue of nonlocal elliptic eigenvalue problems, we establish the uniform persistence and global extinction for the model. In particular, the uniform persistence implies the existence of positive steady states. Finally, we investigate a specially spatially homogeneous predator-prey system and show the complicated dynamics of the system due to the non-local delay in the prey equation.

Keywords:  Predator-prey model, non-local delays, persistence and extinction, principal eigenvalues, fluctuation methods, global attractivity.
Mathematics Subject Classification:  35K57, 35B35, 35B40, 92D25, 93B60.

Received: October 2013;      Revised: April 2014;      Available Online: June 2015.