A predator-prey diffusion model in age-dependent population dynamics
type: Item
instance of: scholarly article
title: A predator-prey diffusion model in age-dependent population dynamics
zbMATH Open document ID: 0703.92020
author: S. Takigawa
published in: Hiroshima Mathematical Journal
publication date: 1989-01-01T00:00:00Z
review text: An age-structured predator-prey model in which the predation rate is a function of prey age and in which both predator and prey are allowed to diffuse in a one-dimensional habitat is considered. The prey dynamics are governed by the McKendrick-von Foerster equation with an added Laplacian to account for diffusion. The predator's dynamics are taken to be age independent. The functional response is of Holling II type. The prey birth rate is assumed linear. Zero flux conditions are imposed on the ends of the finite habitat interval. The author proves two results. First, if the prey's inherent net reproductive rate N is less than one then both predator and prey populations tend to zero. Secondly, if \(N>1\) then there exists a unique stationary solution in which the predator is absent. This stationary solution is stable if the growth rate of the predator S (at this stationary solution) is negative and is unstable if \(S>0\). Unfortunately the author does not address the possibility of coexistence. For example, a branch of coexistence states presumably bifurcates from this stationary solution at \(S=0\).
Mathematics Subject Classification ID: 92D25
zbMATH DE Number: 4154277
zbMATH Keywords: Holling type II response
zbMATH Keywords: age-structured predator-prey model
zbMATH Keywords: McKendrick- von Foerster equation
zbMATH Keywords: Laplacian
zbMATH Keywords: diffusion
zbMATH Keywords: Zero flux conditions
zbMATH Keywords: finite habitat interval
zbMATH Keywords: net reproductive rate
zbMATH Keywords: stationary solution
reviewed by: J. M. Cushing
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4154277