Resident-invader dynamics of similar strategies in fluctuating environments
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Mathematical Biology
Resident-invader dynamics of similar strategies in fluctuating environments Yuhua Cai1
· Stefan A. H. Geritz1
Received: 8 November 2019 / Revised: 30 June 2020 © The Author(s) 2020
Abstract We study resident-invader dynamics in fluctuating environments when the invader and the resident have close but distinct strategies. First we focus on a class of continuous-time models of unstructured populations of multi-dimensional strategies, which incorporates environmental feedback and environmental stochasticity. Then we generalize our results to a class of structured population models. We classify the generic population dynamical outcomes of an invasion event when the resident population in a given environment is non-growing on the long-run and stochastically persistent. Our approach is based on the series expansion of a model with respect to the small strategy difference, and on the analysis of a stochastic fast-slow system induced by time-scale separation. Theoretical and numerical analyses show that the total size of the resident and invader population varies stochastically and dramatically in time, while the relative size of the invader population changes slowly and asymptotically in time. Thereby the classification is based on the asymptotic behavior of the relative population size, and which is shown to be fully determined by invasion criteria (i.e., without having to study the full generic dynamical system). Our results extend and generalize previous results for a stable resident equilibrium (particularly, Geritz in J Math Biol 50(1):67–82, 2005; Dercole and Geritz in J Theor Biol 394:231-254, 2016) to non-equilibrium resident population dynamics as well as resident dynamics with stochastic (or deterministic) drivers. Keywords Adaptive dynamics · Invasion dynamics · Environmental feedback · Environmental stochasticity · Stochastic fast-slow systems Mathematics Subject Classification 34F05 · 37N25 · 92D15 · 92D25
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Yuhua Cai [email protected] Stefan A. H. Geritz [email protected]
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Department of Mathematics and Statistics, University of Helsinki, PO Box 68, 00014 Helsinki, Finland
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Y. Cai, S. A. H. Geritz
1 Introduction Two important issues in the framework of adaptive dynamics are which mutant strategies can invade a population of given resident strategies, and what would be the population dynamical outcomes of an invasion event. The long-term growth rate of a newly arrived and initially rare mutant with strategy y in the environment generated by a population of resident strategy x (i.e., invasion fitness of mutant y in resident x) determines whether an invasion event may occur or not (Metz et al. 1992, 1996; Dieckmann and Law 1996; Geritz et al. 1997, 1998). If y can invade x but not vice versa, does it mean that y will eventually take over x and becomes the new resident? In general, this need not happen as examples of unprotected coexistence and the “resident strikes back” phenomenon show (see e.g., Doebeli 1998; Parvinen 1999; Mylius and Diekmann 2001; Dercole et a
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