Rich dynamics in a non-local population model over three patches

Abstract

We consider a system of nonlinear delay differential equations that describes the growth of the mature population of a species with age-structure living over three patches. We analyze existence of nonnegative homogeneous equilibria and their stability and discuss possible Hopf bifurcation from these equilibria. More precisely, by employing both the standard Hopf bifurcation theory and the symmetric bifurcation theory for functional differential equations, we obtain very rich dynamics for the system, includResearch of P. Weng was supported by ing bistable equilibria, transient oscillations, synchronous periodic solutions, phase-locked periodic solutions, mirror-reflecting waves and standing waves.

Dynamics of a structured population on two patches

“…We have considered three types of birth function but there are many others that could be considered. Finally, the two-patch model has recently been extended to three patches [12]. We wonder if an n-patch model can be derived and studied systematically?…”

“…A linear birth function is of the form b(w) = λw where λ is a positive constant; a Ricker birth function is of the form b(w) = λ 1 we −λ 2 w where λ 1 , λ 2 are positive constants; and an Allee birth function is of the form b(w) = α 1 w 2 e −α 2 w where α 1 , α 2 are positive constants. These three types of birth function have all appeared regularly in the literature [3][4][5][10][11][12]. The system (3) has been studied with Ricker birth functions in [3,5] and with Allee birth functions in [2,4].…”

Global stability of an age-structured population model on several temporally variable patches

“…( 2001 ), Takeuchi ( 1986a ), Takeuchi ( 1986b ), Terry ( 2011 ) and Weng et al. ( 2010 ). Age-structured population growth model in continuous environment typically represent movement as diffusion (Webb 2008 ).…”

Persistence analysis of the age-structured population model on several patches

“…The system of delay differential equations developed by So, Wu and Zou in [16] deals with population divided into two age classes (immature and adult) and assumes that population inhabits two identical patches and that the vital rates are timeindependent. In [21], Weng, Xiao and Zou extended the model of So, Wu and Zou in [16] and considered dynamics of population on three patches. Terry in [20] investigates population of two stages and on two patches and discusses persistence of population for different birth functions and dispersion patterns.…”