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Optimization and Phenotype Allocation

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Abstract

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.

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Acknowledgements

We would like to thank Professor Rüdiger Frey for helpful discussions and comments. We would like to thank the referees for their thoughtful comments and suggestions, especially for drawing our attention to Brennan and Lo (2011).

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Authors and Affiliations

  1. Max-Planck-Institute for Mathematics in the Sciences, Inselstr. 22-26, 04103, Leipzig, Germany

    Jürgen Jost

  2. Department of Mathematics and Computer Science, Leipzig University, 04091, Leipzig, Germany

    Jürgen Jost

  3. Santa Fe Institute for the Sciences of Complexity, Santa Fe, NM, 87501, USA

    Jürgen Jost

  4. Warwick Systems Biology Centre, The University of Warwick, Coventry, CV4 7AL, UK

    Ying Wang

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  1. Jürgen Jost
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  2. Ying Wang
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Correspondence to Ying Wang.

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Jost, J., Wang, Y. Optimization and Phenotype Allocation. Bull Math Biol 76, 184–200 (2014). https://doi.org/10.1007/s11538-013-9915-5

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  • DOI: https://doi.org/10.1007/s11538-013-9915-5

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