Dynamics of Partial Differential Equations

Volume 5 (2008)

Number 3

A two-patch ecological system with nonlinear transfer rate and noise effect

Pages: 281 – 298

DOI: https://dx.doi.org/10.4310/DPDE.2008.v5.n3.a4


Zhaosheng Feng (Department of Mathematics, University of Texas–Pan American, Edinburg, Texas, USA)

Qishao Lu (Department of Mathematics, Beijing University of Aeronautics and Astronautics, Beijing, China)

Suqi Ma (Department of Mathematics, China Agricultural University, Beijing, China)


In this paper, we study the dynamical behavior of a species which inhabits two independent habitat patches. Due to the long range foraging behavior, frequent transfers happen between two patches with an exponentially decaying nonlinear transfer rate. Periodic oscillation is observed as a Hopf bifurcation occurs at some critical values of the delay τ. By applying the center manifold theorem, the Poincaré normal form and the approximate periodic solution near the critical delay values are obtained. The complete synchronization of variations of the population size of species in two patches is analyzed and numerical simulations under various parametric conditions are illustrated. The moment stability of the solution of the stochastic delay equation is also considered by applying the Itô integral.


delay differential equation, stochastic equation, Poincaré normal form, Hopf bifurcation, center manifold, noise effect, synchronization solution

2010 Mathematics Subject Classification

Primary 37M20, 92D40. Secondary 65P30.

Published 1 January 2008