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Mathematical Biosciences and Engineering

2016, Volume 13, Issue 1: 19-41. doi: 10.3934/mbe.2016.13.19
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Discrete or distributed delay? Effects on stability of population growth

  • 1.
    CIMAB, University of Milano, via C. Saldini 50, I20133 Milano
  • 2.
    Department of Mathematics and Computer Science, University of Udine, via delle Scienze 206, I33100 Udine
  • Received: 01 March 2015 Accepted: 29 June 2018 Published: 01 October 2015

  • MSC : Primary: 34K21, 92D25; Secondary: 34K18, 34K20.

  • 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.

    Citation: Edoardo Beretta, Dimitri Breda. Discrete or distributed delay? Effects on stability of population growth[J]. Mathematical Biosciences and Engineering, 2016, 13(1): 19-41. doi: 10.3934/mbe.2016.13.19

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  • 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.


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