Stable periodic solutions for two species, density dependent coevolution
type: Item
instance of: scholarly article
title: Stable periodic solutions for two species, density dependent coevolution
zbMATH Open document ID: 0558.92011
DOI: 10.1007/BF00276547
published in: Journal of Mathematical Biology
publication date: 1985-01-01T00:00:00Z
review text: This paper studies a four dimensional system of time-autonomous ordinary differential equations which models the interaction of two diploid, diallelic populations with overlapping generations. The variables are two population densities and an allele frequency in each of the populations. For single species models, the existence of periodic solutions requires that the genotype fitness functions be both frequency and density dependent. But, for two species exhibiting a predator-prey interaction, two examples are presented where there exists asymptotically stable cycles with fitness functions only density dependent. In the first example, the Hopf bifurcation theorem is used on a two parameter, polynomial vector field. The second example has a Michaelis- Menten or Holling term for the interaction between predator and prey; and, for this example, the existence and uniqueness of limit cycles for a wide range of parameter values has been established in the literature.
Mathematics Subject Classification ID: 92D25
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 34C30
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 34D05
Mathematics Subject Classification ID: 37-XX
zbMATH DE Number: 3891153
zbMATH Keywords: density dependent coevolution
zbMATH Keywords: four dimensional system of time-autonomous ordinary differential equations
zbMATH Keywords: interaction of two diploid, diallelic populations
zbMATH Keywords: overlapping generations
zbMATH Keywords: genotype fitness functions
zbMATH Keywords: predator-prey interaction
zbMATH Keywords: asymptotically stable cycles
zbMATH Keywords: Hopf bifurcation
zbMATH Keywords: polynomial vector field
zbMATH Keywords: Michaelis-Menten
zbMATH Keywords: Holling term
author: James F. Selgrade
author: Gene Namkoong
MaRDI profile type: Publication
Wikidata QID: Q110791683
full work available at URL: https://doi.org/10.1007/bf00276547
OpenAlex ID: W2049592110
description: scientific article; zbMATH DE number 3891153