Skip to main content
SpringerLink
Log in

Evolution of dispersal in spatial population models with multiple timescales

  • Published:
Journal of Mathematical Biology Aims and scope Submit manuscript

Abstract

We study the evolutionary stability of dispersal strategies, including but not limited to those that can produce ideal free population distributions (that is, distributions where all individuals have equal fitness and there is no net movement of individuals at equilibrium). The environment is assumed to be variable in space but constant in time. We assume that there is a separation of times scales, so that dispersal occurs on a fast timescale, evolution occurs on a slow timescale, and population dynamics and interactions occur on an intermediate timescale. Starting with advection–diffusion models for dispersal without population dynamics, we use the large time limits of profiles for population distributions together with the distribution of resources in the environment to calculate growth and interaction coefficients in logistic and Lotka–Volterra ordinary differential equations describing population dynamics. We then use a pairwise invasibility analysis approach motivated by adaptive dynamics to study the evolutionary and/or convergence stability of strategies determined by various assumptions about the advection and diffusion terms in the original advection–diffusion dispersal models. Among other results we find that those strategies which can produce an ideal free distribution are evolutionarily stable.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Aronson DG (1985) The role of diffusion in mathematical population biology: Skellam revisited. In: Capasso V, Grosso E, Paveri-Fontana SL (eds) Mathematics in biology and medicine. Lecture notes in biomathematics, vol 57. Springer, Berlin, pp 2–6

    Google Scholar 

  • Auger P, Poggiale J-C, Charles S (2000) Emergence of individual behaviour at the population level. Effects of density-dependent migration on population dynamics. Comptes Rendus de l’Academie des Sciences - Series III - Sciences de la Vie 323:119–127

    Google Scholar 

  • Auger P, Poggiale JC, Sánchez E (2012) A review on spatial aggregation methods involving several time scales. Ecol Complex 10:12–25

    Google Scholar 

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

    Google Scholar 

  • Bolker BM, Pacala SW (1999) Spatial moment equations for plant competition: understanding spatial strategies and the advantages of short dispersal. Am Nat 53:575–602

    Google Scholar 

  • Brännström Å, Johansson J, von Festenberg N (2013) The Hitchhiker’s guide to adaptive dynamics. Games 4:304–328

    MathSciNet  MATH  Google Scholar 

  • Bravo de la Parra R, Sánchez E, Auger P (1997) Time scales in density dependent discrete models. J Biol Syst 5:111–129

    MATH  Google Scholar 

  • Bravo de la Parra R, Sánchez E, Arino O, Auger P (1999) A discrete model with density dependent fast migration. Math Biosci 157:91–110

    MathSciNet  Google Scholar 

  • Bravo de la Parra R, Marvá M, Sánchez E, Sanz L (2013) Reduction of discrete dynamical systems with applications to dynamics population models. Math Model Nat Phenom 8:107–129

    MathSciNet  MATH  Google Scholar 

  • Bravo de la Parra R, Marvá M, Sansegundo F (2016) Fast dispersal in semelparous populations. Math Model Nat Phenom 11:120–134

    MathSciNet  MATH  Google Scholar 

  • Cantrell RS, Cosner C (2003) Spatial ecology via reaction–diffusion equations. Wiley, Chichester

    MATH  Google Scholar 

  • Cantrell RS, Cosner C, Lou Y (2006) Movement toward better environments and the evolution of rapid diffusion. Math Biosci 204:199–214

    MathSciNet  MATH  Google Scholar 

  • Cantrell RS, Cosner C, Lou Y (2007) Advection-mediated coexistence of competing species. Proc R Soc Edinb A 37:497–518

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  • Cantrell RS, Cosner C, Lou Y (2012a) Evolutionary stability of ideal free dispersal strategies in patchy environments. J Math Biol 65:943–965

    MathSciNet  MATH  Google Scholar 

  • Cantrell RS, Cosner C, Lou Y, Ryan D (2012b) Evolutionary stability of ideal free dispersal in spatial population models with nonlocal dispersal. Can Appl Math Q 20:15–38

    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

    MathSciNet  MATH  Google Scholar 

  • Chesson P (2009) Scale transition theory with special reference to species coexistence in a variable environment. J Biol Dyn 3:149–163

    MathSciNet  MATH  Google Scholar 

  • Chesson P (2012) Scale transition theory: its aims, motivations and predictions. Ecol Complex 10:52–68

    Google Scholar 

  • Chesson P, Donahue MJ, Melbourne BA, Sears ALW (2005) Scale transition theory for understanding mechanisms in metacommunities. In: Holyoke M, Leibold MA, Holt RD (eds) Metacommunities: spatial dynamics and ecological communities. The University of Chicago Press, Chicago, pp 279–306

    Google Scholar 

  • Constable GWA (2014) Fast timescales in stochastic population dynamics. Dissertation, University of Manchester

  • Cosner C (2014) Reaction–diffusion–advection models for the effects and evolution of dispersal. Discret Contin Dyn Syst A 34:1701–1745

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  • Fagan WF, Gurarie E, Bewick S, Howard A, Cantrell RS, Cosner C (2017) Perceptual ranges, information gathering, and foraging success in dynamic landscapes. Am Nat 189:474–489

    Google Scholar 

  • Farnsworth KD, Beecham JA (1999) How do grazers achieve their distribution? A continuum of models from random diffusion to the ideal free distribution using biased random walks. Am Nat 153:509–526

    Google Scholar 

  • Geritz S, Metz JAJ, Kisdi E, Meszéna G (1997) Dynamics of adaptation and evolutionary branching. Phys Rev Lett 78:2024–2027

    Google Scholar 

  • Hairston NG Jr, Ellner SP, Geber MA, Yoshida T, Fox JA (2005) Rapid evolution and the convergence of ecological and evolutionary time. Ecol Lett 8:1114–1127

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  • Hastings A (2010) Timescales, dynamics, and ecological understanding. Ecology 91:3471–3480

    Google Scholar 

  • Hutson V, Martínez S, Mischaikow K, Vickers GT (2003) The evolution of dispersal. J Math Biol 47:483–517

    MathSciNet  MATH  Google Scholar 

  • Kao C-Y, Lou Y, Shen W (2010) Random dispersal vs. nonlocal dispersal. Discret Contin Dyn Syst A 26:551–596

    MATH  Google Scholar 

  • Korobenko L, Braverman E (2012) On logistic models with a carrying capacity dependent diffusion: stability of equilibria and coexistence with a regularly diffusing population. Nonlinear Anal RWA 13:2648–2658

    MathSciNet  MATH  Google Scholar 

  • Korobenko L, Braverman E (2014) On evolutionary stability of carrying capacity driven dispersal in competition with regularly diffusing populations. J Math Biol 69:1181–1206

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  • Lam K-Y, Lou Y (2014a) Evolutionarily stable and convergent stable strategies in reaction–diffusion models for conditional dispersal. Bull Math Biol 76:261–291

    MathSciNet  MATH  Google Scholar 

  • Lam K-Y, Lou Y (2014b) Evolution of dispersal: ESS in spatial models. J Math Biol 68:851–877

    MathSciNet  MATH  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

    MathSciNet  MATH  Google Scholar 

  • Law R, Leibold MA (2005) Assembly dynamics in metacommunities. In: Holyoke M, Leibold MA, Holt RD (eds) Metacommunities: spatial dynamics and ecological communities. The University of Chicago Press, Chicago, pp 263–278

    Google Scholar 

  • Levin SA (1992) The problem of pattern and scale in ecology: the Robert H. MacArthur award lecture. Ecology 73:1943–1967

    Google Scholar 

  • Levin SA (2000) Multiple scales and the maintenance of biodiversity. Ecosystems 3:498–506

    Google Scholar 

  • López-Gómez J (2013) Linear second order elliptic operators. World Scientific, Singapore

    MATH  Google Scholar 

  • Morozov A, Poggiale J-C (2012) From spatially explicit ecological models to mean-field dynamics: The state of the art and perspectives. Ecol Complex 10:1–11

    Google Scholar 

  • Mose VN, Nguyen-Huu T, Auger P, Western D (2012) Modelling herbivore population dynamics in the Amboseli National Park, Kenya: application of spatial aggregation of variables to derive a master model. Ecol Complex 10:42–51

    Google Scholar 

  • Nguyen-Ngoc D, Nguyen-Huu T, Auger P (2012) Effects of fast density dependent dispersal on pre-emptive competition dynamics. Ecol Complex 10:26–33

    MATH  Google Scholar 

  • Okubo A, Levin SA (2001) Diffusion and ecological problems. Springer, New York

    MATH  Google Scholar 

  • Polechová J, Barton NH (2015) Limits to adaptation along environmental gradients. PNAS 20:6401–6406

    Google Scholar 

  • Potapov A, Schlägel U, Lewis MA (2014) Evolutionarily stable diffusive dispersal. Discret Contin Dyn Syst B 19:3319–3340

    MathSciNet  MATH  Google Scholar 

  • Protter MH, Weinberger HF (1966) On the spectrum of general second order operators. Bull Am Math Soc 72:251–255

    MathSciNet  MATH  Google Scholar 

  • Sanz L, Bravo de la Parra R (2000) Time scales in stochastic multiregional models. Nonlinear Anal RWA 1:89–122

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

Some of the ideas in this paper arose from discussions in workshops at the Banff International Research Station: Multiscale Analysis of Self-Organization in Biology (09w5070) July 12–July 17, 2009, organizers B. Perthame and T. Hillen, and Emerging Challenges at the Interface of Mathematics, Environmental Science and Spatial Ecology (11w5106) July 3–July 08, 2011, organizers R. S. Cantrell, R. Holt, and M. A. Lewis. CC thanks Odo Diekmann for discussions that helped motivate this paper.

Author information

Authors and Affiliations

  1. Institute for Mathematical Sciences, Renmin University of China, Beijing, 100872, People’s Republic of China

    Robert Stephen Cantrell & Yuan Lou

  2. Department of Mathematics, University of Miami, Coral Gables, FL, 33146, USA

    Robert Stephen Cantrell & Chris Cosner

  3. Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, T6G 2G1, Canada

    Mark A. Lewis

  4. Department of Biological Sciences, University of Alberta, Edmonton, AB, T6G 2G1, Canada

    Mark A. Lewis

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

    Yuan Lou

Authors
  1. Robert Stephen Cantrell
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chris Cosner
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mark A. Lewis
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Yuan Lou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chris Cosner.

Additional information

The paper should be listed for the special issue S.I. : In honor of Alan Hastings’ 65th birthday.

Research partially supported by NSF Grants DMS-1118623 and DMS-1514752 (RSC, CC), NSF Grant DMS-1411476 (YL), NSFC Grant No. 11571364 (YL), and an NSERC Discovery Grant (ML)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cantrell, R.S., Cosner, C., Lewis, M.A. et al. Evolution of dispersal in spatial population models with multiple timescales. J. Math. Biol. 80, 3–37 (2020). https://doi.org/10.1007/s00285-018-1302-2

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00285-018-1302-2

Keywords

Mathematics Subject Classification

Search

Navigation