Bang–Bang Growth Rate Optimization in a Coupled McKendrick Model

Philippe Michel

Journal of Optimization Theory and Applications, 2019, vol. 183, issue 1, No 17, 332-351

Abstract: Abstract In this work, we model the time and age evolution of a partially clonal population, i.e., able to reproduce sexually and asexually, in an environment with unlimited resources. The population is divided in two subpopulations, sexual and asexual, whose densities follow a coupled system of McKendrick–Von Foerster equations of evolution. The transition from one subpopulation to another is driven by transition probabilities for newborns to be sexual (resp. asexual) when their parent(s) is(are) in the asexual (sexual) subpopulation. We study the optimization of the growth rate of the whole population, with respect to these transition probabilities. We prove, using a result of the variation of the first eigenvalue (Malthusian growth rate) for this problem, that the maximal eigenvalue is reached when the probabilities are exactly (in time) equal to zero or one. Moreover, depending on birth and death rates of both subpopulations (asexual and sexual), we show that the maximal growth rate is reached when the population newborns switch (completely) from sexual to asexual and then from asexual to sexual (periodically in time) or when a subpopulation disappears.

Keywords: Bang–Bang control; Growth rate; Eigenvalue; McKendrick equation; 65N25; 35Q93; 49J30 (search for similar items in EconPapers)
Date: 2019
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DOI: 10.1007/s10957-019-01556-1

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