Discrete or distributed delay? Effects on stability of population growth

• Published in: Mathematical biosciences and engineering : MBE Vol. 13; no. 1; pp. 19 - 41
• Main Authors: Beretta, Edoardo, Breda, Dimitri
• Format: Journal Article
• Language: English
• Published: United States AIMS Press 01.02.2016
• Subjects: Animals; Birth Rate; Computer Simulation; delta$-dirac distribution; gamma distribution; Humans; local asymptotic stability analysis; maturation delay; Models, Statistical; Mortality; Population Growth; stability switches; Survival Rate; Time Factors
• ISSN: 1551-0018
• DOI: 10.3934/mbe.2016.13.19

The growth of a population subject to maturation delay is modeled by using either a discrete delay or a delay continuously distributed over the population. The occurrence of stability switches (stable-unstable-stable) of the positive equilibrium as the delay increases is investigated in both cases. Necessary and sufficient conditions are provided by analyzing the relevant characteristic equations. It is shown that for any choice of parameter values for which the discrete delay model presents stability switches there exists a maximum delay variance beyond which no switch occurs for the continuous delay model: the delay variance has a stabilizing effect. Moreover, it is illustrated how, in the presence of switches, the unstable delay domain is as larger as lower is the ratio between the juveniles and the adults mortality rates.