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Evolution of conditional dispersal: evolutionarily stable strategies in spatial models

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Abstract

We consider a two-species competition model in which the species have the same population dynamics but different dispersal strategies. Both species disperse by a combination of random diffusion and advection along environmental gradients, with the same random dispersal rates but different advection coefficients. Regarding these advection coefficients as movement strategies of the species, we investigate their course of evolution. By applying invasion analysis we find that if the spatial environmental variation is less than a critical value, there is a unique evolutionarily singular strategy, which is also evolutionarily stable. If the spatial environmental variation exceeds the critical value, there can be three or more evolutionarily singular strategies, one of which is not evolutionarily stable. Our results suggest that the evolution of conditional dispersal of organisms depends upon the spatial heterogeneity of the environment in a subtle way.

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References

  • Amarasekare P (2010) Effects of non-random dispersal strategies on spatial coexistence mechanisms. J Anim Ecol 79:282–293

    Google Scholar 

  • Averill I, Munther D, Lou Y (2012) On several conjectures from evolution of dispersal. J Biol Dyn 6:117–130

    Article  Google Scholar 

  • Belgacem F, Cosner C (1995) The effects of dispersal along environmental gradients on the dynamics of populations in heterogeneous environment. Canad Appl Math Quart 3:379–397

    Google Scholar 

  • Berestycki H, Hamel F, Roques L (2005) Analysis of the periodic patch model:I - The effects of heterogeneous environment on species conservation. J Math Biol 51:75–113

    Article  MATH  MathSciNet  Google Scholar 

  • Berestycki H, Hamel F, Roques L (2005) Analysis of the periodic patch model: II—biological invasions and pulsating travelling fronts. J Math Pures Appl 84:1101–1146

    Article  MATH  MathSciNet  Google Scholar 

  • Bezugly A, Lou Y (2010) Reaction-diffusion models with large advection coefficients. Appl Anal 89:983–1004

    Article  MathSciNet  Google Scholar 

  • Bouin E, Calveza V, Meunier N, Mirrahimi S, Perthame B, Raoul G, Voituriez R (2012) Invasion fronts with variable motility: phenotype selection, spatial sorting and wave acceleration. Comptes Rendus Mathematiques de l’Academie des Sciences de Paris 350:761–766

    Google Scholar 

  • Cantrell RS, Cosner C (2003) Spatial ecology via reaction-diffusion equations. In: Series in mathematical and computational biology. Wiley, New York

  • Cantrell RS, Cosner C, Lou Y (2006) Movement towards better environments and the evolution of rapid diffusion. Math. Biosci. 240:199–214

    Article  MathSciNet  Google Scholar 

  • Cantrell RS, Cosner C, Lou Y (2007) Advection-mediated coexistence of competing species. Proc Roy Soc Edinb Sect A 137:497–518

    Article  MATH  MathSciNet  Google Scholar 

  • Cantrell RS, Cosner C, Lou Y (2009) Evolution of dispersal in heterogeneous landscape. In: Cantrell RS, Cosner C, Ruan S (eds) Spatial ecology, mathematical and computational biology series. Chapman Hall/CRC Press, Boca Raton, FL, pp 213–229

  • Cantrell RS, Cosner C, Lou Y (2010) Evolution of dispersal and ideal free distribution. Math Biosci Eng 7:17–36

    Article  MATH  MathSciNet  Google Scholar 

  • Chen X, Hambrock R, Lou Y (2008) Evolution of conditional dispersal, a reaction-diffusion-advection model. J Math Biol 57:361–386

    Article  MATH  MathSciNet  Google Scholar 

  • Chen X, Lam K-Y, Lou Y (2012) Dynamics of a reaction-diffusion-advection model for two competing species. Discret Contin Dyn Syst A 32:3841–3859

    Article  MATH  MathSciNet  Google Scholar 

  • Chen X, Lou Y (2008) Principal eigenvalue and eigenfunctions of an elliptic operator with large advection and its application to a competition model. Indiana Univ Math J 57:627–657

    Article  MATH  MathSciNet  Google Scholar 

  • Clobert J, Danchin E, Dhondt AA, Nichols JD (eds) (2001) Dispersal. Oxford University Press, Oxford

    Google Scholar 

  • Dieckmann U, Law R (1996) The dynamical theory of coevolution: a derivation from stochastic ecological processes. J Math Biol 34:579–612

    Article  MATH  MathSciNet  Google Scholar 

  • Dieckmann O (2003) A beginner’s guide to adaptive dynamics. Banach Center Publ 63:47–86

    Article  Google Scholar 

  • Dockery J, Hutson V, Mischaikow K, Pernarowski M (1998) The evolution of slow dispersal rates: a reaction-diffusion model. J Math Biol 37:61–83

    Article  MATH  MathSciNet  Google Scholar 

  • Fretwell SD, Lucas HL (1969) On territorial behavior and other factors influencing habitat distribution in birds, I. Theoretical development. Acta Biotheoretica 19:16–36

    Article  Google Scholar 

  • Geritz SAH, Gyllenberg M (2008) The mathematical theory of adaptive dynamics. Cambridge University Press, Cambridge

    Google Scholar 

  • Geritz SAH, Kisdi E, Meszena G, Metz JAJ (1998) Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree. Evol Ecol 12:35–57

    Article  Google Scholar 

  • Gilbarg D, Trudinger NS (1983) Elliptic partial differential equations of second order. In: Grundlehren der Mathematischen Wissenschaften, vol 224. Springer, Berlin

  • Hambrock R, Lou Y (2009) The evolution of conditional dispersal strategy in spatially heterogeneous habitats. Bull Math Biol 71:1793–1817

    Article  MATH  MathSciNet  Google Scholar 

  • Hastings A (1983) Can spatial variation alone lead to selection for dispersal? Theor Popul Biol 24:244–251

    Article  MATH  MathSciNet  Google Scholar 

  • Hutson V, Lou Y, Mischaikow K (2005) Convergence in competition models with small diffusion coefficients. J Differ Eqs 211:135–161

    Article  MATH  MathSciNet  Google Scholar 

  • Hutson V, Mischaikow K, Polacik P (2001) The evolution of dispersal rates in a heterogeneous time-periodic environment. J Math Biol 43:501–533

    Article  MATH  MathSciNet  Google Scholar 

  • Kao CY, Lou Y, Shen WX (2012) Evolution of mixed dispersal in periodic environments. Discre Cont Dys Syst B 17:2047–2072

    Article  MATH  MathSciNet  Google Scholar 

  • Lam K-Y (2011) Concentration phenomena of a semilinear elliptic equation with large advection in an ecological model. J Differ Eqs 250:161–181

    Google Scholar 

  • Lam K-Y (2012) Limiting profiles of semilinear elliptic equations with large advection in population dynamics II. SIAM J Math Anal 44:1808–1830

    Article  MATH  MathSciNet  Google Scholar 

  • Lam K-Y, Ni W-M (2010) Limiting profiles of semilinear elliptic equations with large advection in population dynamics. Discret Contin Dyn Syst A 28:1051–1067

    Article  MATH  MathSciNet  Google Scholar 

  • Levin SA (1976) Population dynamic models in heterogeneous environments. Annu Rev Ecol Syst 7:287–310

    Article  Google Scholar 

  • Levin SA, Cohen D, Hastings A (1984) Dispersal strategies in patchy environments. Theor Popul Biol 26:165–191

    Article  MATH  MathSciNet  Google Scholar 

  • Lutscher F, Lewis MA, McCauley E (2006) Effects of heterogeneity on spread and persistence in rivers. Bull Math Biol 68:2129–2160

    Article  MathSciNet  Google Scholar 

  • Lutscher F, McCauley E, Lewis MA (2007) Spatial patterns and coexistence mechanisms in systems with unidirectional flow. Theor Popul Biol 71:267–277

    Article  MATH  Google Scholar 

  • Maynard Smith J, Price G (1973) The logic of animal conflict. Nature 246:15–18

    Article  Google Scholar 

  • McPeek MA, Holt RD (1992) The evolution of dispersal in spatially and temporally varying environments. Am Nat 140:1010–1027

    Article  Google Scholar 

  • Ni W-M (2011) The mathematics of diffusion. In: CBMS Reg Conf Ser Appl Math, vol 82. SIAM, Philadelphia

  • Okubo A, Levin SA (2001) Diffusion and ecological problems: modern perspectives. In: Interdisciplinary applied mathematics, vol 14, 2nd edn Springer, Berlin

  • Payne LE, Weinberger HF (1960) An optimal Poincare inequality for convex domains. Arch Rational Mech Anal 5:286–292

    Article  MATH  MathSciNet  Google Scholar 

  • Ronce O (2007) How does it feel to be like a rolling stone? Ten questions about dispersal evolution. Annu Rev Ecol Syst 38:231–253

    Article  Google Scholar 

  • Shigesada N, Kawasaki K (1997) Biological invasions: theory and practice. In: Oxford series in ecology and evolution. Oxford University Press, Oxford

Download references

Acknowledgments

The authors wish to thank the anonymous referees for their careful reading of the manuscript and helpful suggestions. This research was partially supported by the NSF grant DMS-1021179 and has been supported in part by the Mathematical Biosciences Institute and the National Science Foundation under grant DMS-0931642.

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Authors and Affiliations

  1. Mathematical Biosciences Institute, Ohio State University, Columbus, OH, 43210, USA

    King-Yeung Lam

  2. Department of Mathematics, Ohio State University, Columbus, OH, 43210, USA

    Yuan Lou

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  1. King-Yeung Lam
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  2. Yuan Lou
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Correspondence to King-Yeung Lam.

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Lam, KY., Lou, Y. Evolution of conditional dispersal: evolutionarily stable strategies in spatial models. J. Math. Biol. 68, 851–877 (2014). https://doi.org/10.1007/s00285-013-0650-1

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  • DOI: https://doi.org/10.1007/s00285-013-0650-1

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