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Demographic stochasticity and evolution of dispersion II: Spatially inhomogeneous environments

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Abstract

Demographic stochasticity, the random fluctuations arising from the intrinsic discreteness of populations and the uncertainty of individual birth and death events, is an essential feature of population dynamics. Nevertheless theoretical investigations often neglect this naturally occurring noise due to the mathematical complexity of stochastic models. This paper reports the results of analytical and computational investigations of models of competitive population dynamics, specifically the competition between species in heterogeneous environments with different phenotypes of dispersal, fully accounting for demographic stochasticity. A novel asymptotic approximation is introduced and applied to derive remarkably simple analytical forms for key statistical quantities describing the populations’ dynamical evolution. These formulas characterize the selection processes that determine which (if either) competitor has an evolutionary advantage. The theory is verified by large-scale numerical simulations. We discover that the fluctuations can (1) break dynamical degeneracies, (2) support polymorphism that does not exist in deterministic models, (3) reverse the direction of the weak selection and cause shifts in selection regimes, and (4) allow for the emergence of evolutionarily stable dispersal rates. Dynamical mechanisms and time scales of the fluctuation-induced phenomena are identified within the theoretical approach.

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Notes

  1. Hamilton first proposed the concept of a certain trait being “evolutionarily stable” to discuss the stability of natural sex ratios (Hamilton 1967), and soon applied the same philosophy to a biological dispersal problem (Hamilton and May 1977).

  2. We only consider the initial conditions with order \(\mathcal {O}(\sigma ^1)\) because in the stochastic models, the demographic fluctuations are of order \(\mathcal {O}(1/\sqrt{K})\) (Lin et al. 2013) and we are interested in the cases when the two effects are comparable; see Sect. 3.

  3. The claim is verified by numerical observations in Fig. 2.

  4. The analytical conclusion is not limited by restriction, however, as verified by numerical simulations in Sect. 4.

  5. The only exceptions being when \(\mu _x\) and \(\mu _Y \ll 1\), as pointed out in previous work (Lin et al. 2013); with such parameters the patches are almost decoupled. The asymptotic analysis breaks down because the separation of time scale no longer holds true.

  6. Details of the derivations are documented in Lin’s dissertation (2013).

  7. It is because the reduced velocity field converges to the corresponding field in the homogeneous many patch model Lin et al. (2013) as \(\sigma \rightarrow 0\) and the corresponding field in deterministic model in Sect. 2 as \(\varLambda \rightarrow \infty \).

  8. Fundamentally there are several distinct model settings between our, Hamilton and May’s (1977), and Comins et al. (1980) models: (1) Hamilton and May’s and Comins’ models have discrete generations; as a consequence, the competition across generations does not exist. On the contrary, the lifespans of individuals overlap in our model and the cross-generation competition is considered. (2) Hamilton and May’s and Comins’ models incorporated dispersal costs, which are absent in our model. (3) There are catastrophic events in Comins’ model, which is again neglected in our study. As a consequence, both Hamilton and May’s and Comins’ models featured non-zero evolutionarily stable dispersal rates (when the population is finite) even in the homogeneous environments—whereas in the first part of the study we showed that there exists no finite evolutionarily stable dispersal rate in in homogeneous environments (Lin et al. 2013).

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Acknowledgments

This work was supported in part by NSF Awards DMS-0927587 and PHY-1205219. One of us (HK) also acknowledges support from the NSF’s Institute for Mathematics and Its Applications.

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  1. Yen Ting Lin

    Present address: Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Straße 38, 01187 , Dresden, Germany

Authors and Affiliations

  1. Department of Physics, University of Michigan, Ann Arbor, MI, 48109-1120, USA

    Yen Ting Lin

  2. Department of Mathematics, Michigan State University, East Lansing, MI, 48824, USA

    Hyejin Kim

  3. Department of Mathematics, University of Michigan, Ann Arbor, MI, 48109-1043, USA

    Charles R. Doering

  4. Department of Mathematics, Center for the Study of Complex Systems, University of Michigan, Ann Arbor, MI, 48109-1107, USA

    Charles R. Doering

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  1. Yen Ting Lin
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Correspondence to Yen Ting Lin.

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Lin, Y.T., Kim, H. & Doering, C.R. Demographic stochasticity and evolution of dispersion II: Spatially inhomogeneous environments. J. Math. Biol. 70, 679–707 (2015). https://doi.org/10.1007/s00285-014-0756-0

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  • DOI: https://doi.org/10.1007/s00285-014-0756-0

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