Optimization and phenotype allocation.
We study the phenotype allocation problem for the stochastic evolution of a multitype population in a random environment. Our underlying model is a multitype Galton–Watson branching process in a random environment. In the multitype branching model, different types denote different phenotypes of offspring, and offspring distributions denote the allocation strategies. Two possible optimization targets are considered: the long-term growth rate of the population conditioned on nonextinction, and the extinction probability of the lineage. In a simple and biologically motivated case, we derive an explicit formula for the long-term growth rate using the random Perron–Frobenius theorem, and we give an approximation to the extinction probability by a method similar to that developed by Wilkinson. Then we obtain the optimal strategies that maximize the long-term growth rate or minimize the approximate extinction probability, respectively, in a numerical example. It turns out that different optimality criteria can lead to different strategies.
Jürgen
J
Jost
Ying
Y
Wang
D000818:Animals; D005075:Biological Evolution; D004777:Environment; D053476:Extinction, Biological; D006801:Humans; D055641:Mathematical Concepts; D008954:Models, Biological; D010641:Phenotype; D011158:Population Growth; D013269:Stochastic Processes