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Time scales and wave formation in non-linear spatial public goods games

Journal article
Authors G. J. Kimmel
Philip Gerlee
P. M. Altrock
Published in PLoS Computational Biology
Volume 15
Issue 9
ISSN 1553-734X
Publication year 2019
Published at Department of Mathematical Sciences
Language en
Links dx.doi.org/10.1371/journal.pcbi.100...
Keywords article, cell motion, coevolution, diffusion, disease simulation, growth rate, prediction, travel
Subject categories Bioinformatics (Computational Biology)

Abstract

The co-evolutionary dynamics of competing populations can be strongly affected by frequency- dependent selection and spatial population structure. As co-evolving populations grow into a spatial domain, their initial spatial arrangement and their growth rate differences are important factors that determine the long-term outcome. We here model producer and free-rider co-evolution in the context of a diffusive public good (PG) that is produced by the producers at a cost but evokes local concentration-dependent growth benefits to all. The benefit of the PG can be non-linearly dependent on public good concentration. We consider the spatial growth dynamics of producers and free-riders in one, two and three dimensions by modeling producer cell, free-rider cell and public good densities in space, driven by the processes of birth, death and diffusion (cell movement and public good distribution). Typically, one population goes extinct, but the time-scale of this process varies with initial conditions and the growth rate functions. We establish that spatial variation is transient regardless of dimensionality, and that structured initial conditions lead to increasing times to get close to an extinction state, called ∈-extinction time. Further, we find that uncorrelated initial spatial structures do not influence this ∈-extinction time in comparison to a corresponding well-mixed (non-spatial) system. In order to estimate the ∈-extinction time of either free-riders or producers we derive a slow manifold solution. For invading populations, i.e. for populations that are initially highly segregated, we observe a traveling wave, whose speed can be calculated. Our results provide quantitative predictions for the transient spatial dynamics of cooperative traits under pressure of extinction. © 2019 Kimmel et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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