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2015, 20(6): 1625-1638. doi: 10.3934/dcdsb.2015.20.1625

Mathematical study of the effects of travel costs on optimal dispersal in a two-patch model

1. 

212A Williams Hall, 953 Danby Road, Ithaca, NY 14850, United States

Received  November 2013 Revised  April 2014 Published  June 2015

The theoretical dispersal of organisms has been widely studied. It is well known for single species dispersal in a spatially heterogeneous and temporally constant environment that ``balanced dispersal'' is an evolutionarily stable strategy [36,10]. This assumes that organisms do not pay a cost to move from one part of the environment to another. We begin this paper by proving that the optimal strategy for organisms constrained by perceptual limitations, described by [19], is evolutionarily stable. Then, we extend this idea of optimal dispersal to a situation where constrained organisms pay a cost to move between two patches in a heterogeneous environment. For moderate travel costs, we find a convergent stable strategy that suggests an extension of the balanced dispersal concept. Furthermore, we show for high costs that the best strategy is to ignore information about the environment.
Citation: Theodore E. Galanthay. Mathematical study of the effects of travel costs on optimal dispersal in a two-patch model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1625-1638. doi: 10.3934/dcdsb.2015.20.1625
References:
[1]

P. A. Abrams, Implications of flexible foraging for interspecific interactions: Lessons from simple models,, Functional Ecology, 24 (2010), 7. doi: 10.1111/j.1365-2435.2009.01621.x.

[2]

P. A. Abrams, H. Matsuda and Y. Harada, Evolutionarily unstable fitness maxima and stable fitness minima of continuous traits,, Evolutionary Ecology, 7 (1993), 465. doi: 10.1007/BF01237642.

[3]

I. Averill, Y. Lou and D. Munther, On several conjectures from evolution of dispersal,, Journal of Biological Dynamics, 6 (2012), 117. doi: 10.1080/17513758.2010.529169.

[4]

F. Belgacem and C. Cosner, The effects of dispersal along environmental gradients on the dynamics of populations in heterogeneous environment,, Canadian Applied Math Quarterly, 3 (1995), 379.

[5]

R. S. Cantrell, C. Cosner, D. L. DeAngelis and V. Padron, The ideal free distribution as an evolutionarily stable strategy,, Journal of Biological Dynamics, 1 (2007), 249. doi: 10.1080/17513750701450227.

[6]

R. S. Cantrell, C. Cosner and Y. Lou, Approximating the ideal free distribution via reaction-diffusion-advection equations,, Journal of Differential Equations, 245 (2008), 3687. doi: 10.1016/j.jde.2008.07.024.

[7]

R. S. Cantrell, C. Cosner and Y. Lou, Evolution of dispersal and the ideal free distribution,, Mathematical Biosciences and Engineering, 7 (2010), 17. doi: 10.3934/mbe.2010.7.17.

[8]

R. S. Cantrell, C. Cosner and Y. Lou, Evolutionary stability of ideal free dispersal strategies in patchy environments,, Journal of Mathematical Biology, 65 (2012), 943. doi: 10.1007/s00285-011-0486-5.

[9]

C. Cosner, A dynamic model for the ideal-free distribution as a partial differential equation,, Theoretical Population Biology, 67 (2005), 101. doi: 10.1016/j.tpb.2004.09.002.

[10]

R. Cressman and V. Křivan, The ideal free distribution as an evolutionarily stable state in density-dependent population games,, Oikos, 119 (2010), 1231. doi: 10.1111/j.1600-0706.2010.17845.x.

[11]

D. L. DeAngelis, G. S. K. Wolkowicz, Y. Lou, Y. Jiang, M. Novak, R. Svanback, M. S. Araujo, Y. Jo and E. A. Cleary, The effect of travel loss on evolutionarily stable distributions of populations in space,, American Naturalist, 178 (2011), 15. doi: 10.1086/660280.

[12]

O. Diekmann, A beginner's guide to adaptive dynamics,, Banach center publications, 63 (2004), 47.

[13]

J. E. Diffendorfer, Testing models of source-sink dynamics and balanced dispersal,, Oikos, 81 (1998), 417. doi: 10.2307/3546763.

[14]

J. Dockery, V. Hutson, K. Mischaikow and M. Pernarowski, The evolution of slow dispersal rates: A reaction diffusion model,, Journal of Mathematical Biology, 37 (1998), 61. doi: 10.1007/s002850050120.

[15]

S. M. Flaxman and Y. Lou, Tracking prey or tracking the prey's resource? Mechanisms of movement and optimal habitat selection by predators,, Journal of Theoretical Biology, 256 (2009), 187. doi: 10.1016/j.jtbi.2008.09.024.

[16]

H. I. Freedman, B. Rai and P. Waltman, Mathematical models of population interactions with dispersal. II: Differential survival in a change of habitat,, Journal of Mathematical Analysis and Applications, 115 (1986), 140. doi: 10.1016/0022-247X(86)90029-6.

[17]

H. I. Freedman and P. Waltman, Mathematical models of population interactions with dispersal. I: Stability of two habitats with and without a predator,, SIAM Journal of Applied Mathematics, 32 (1977), 631. doi: 10.1137/0132052.

[18]

S. D. Fretwell and H. L. Lucas, On territorial behavior and other factors influencing habitat distribution in birds,, Acta Biotheoretica, 19 (1969), 37. doi: 10.1007/BF01601954.

[19]

T. E. Galanthay and S. M. Flaxman, Generalized movement strategies for constrained consumers: Ignoring fitness can be adaptive,, American Naturalist, 179 (2012), 475. doi: 10.1086/664625.

[20]

S. A. H. Geritz, É. Kisdi, G. Meszéna and J. A. J. Metz, Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree,, Evolutionary Ecology, 12 (1998), 35.

[21]

B. S. Goh, Global stability in 2 species interactions,, Journal of Mathematical Biology, 3 (1976), 313. doi: 10.1007/BF00275063.

[22]

H. Hakoyama and K. Iguchi, The information of food distribution realizes an ideal free distribution: Support of perceptual limitation,, Journal of Ethology, 15 (1997), 69. doi: 10.1007/BF02769391.

[23]

A. Hastings, Dynamics of a single species in a spatially varying environment: The stabilizing role of high dispersal rates,, Journal of Mathematical Biology, 16 (1982), 49. doi: 10.1007/BF00275160.

[24]

A. Hastings, Can spatial variation alone lead to selection for dispersal?,, Theoretical Population Biology, 24 (1983), 244. doi: 10.1016/0040-5809(83)90027-8.

[25]

R. D. Holt, Population dynamics in two-patch environments: Some anomalous consequences of an optimal habitat distribution,, Theoretical Population Biology, 28 (1985), 181. doi: 10.1016/0040-5809(85)90027-9.

[26]

D. M. Hugie and T. C. Grand, Movement between patches, unequal competitors and the ideal free distribution,, Evolutionary Ecology, 12 (1998), 1.

[27]

M. Kennedy and R. D. Gray, Can ecological theory predict the distribution of foraging animals? A critical analysis of experiments on the ideal free distribution,, Oikos, 68 (1993), 158. doi: 10.2307/3545322.

[28]

M. Kennedy and R. D. Gray, Habitat choice, habitat matching and the effect of travel distance,, Behaviour, 134 (1997), 905. doi: 10.1163/156853997X00223.

[29]

S. Kirkland, C.-K. Li and S. J. Schreiber, On the evolution of dispersal in patchy landscapes,, SIAM Journal of Applied Mathematics, 66 (2006), 1366. doi: 10.1137/050628933.

[30]

R. Korona, Travel costs and ideal free distribution of ovipositing female flour beetles, Tribolium confusum,, Animal Behaviour, 40 (1990), 186. doi: 10.1016/S0003-3472(05)80680-3.

[31]

V. Křivan, Dynamic ideal free distribution: Effects of optimal patch choice on predator-prey dynamics,, American Naturalist, 149 (1997), 164.

[32]

V. Křivan, R. Cressman and C. Schneider, The ideal free distribution: A review and synthesis of the game-theoretic perspective,, Theoretical Population Biology, 73 (2008), 403.

[33]

Y. Lou, Some challenging mathematical problems in evolution of dispersal and population dynamics,, in Tutorials in Mathematical Biosciences IV (ed. A. Friedman), (1922), 171. doi: 10.1007/978-3-540-74331-6_5.

[34]

Y. Lou and C.-H. Wu, Global dynamics of a tritrophic model for two patches with cost of dispersal,, SIAM Journal of Applied Mathematics, 71 (2011), 1801. doi: 10.1137/100817954.

[35]

S. Matsumura, R. Arlinghaus and U. Dieckmann, Foraging on spatially distributed resources with sub-optimal movement, imperfect information, and travelling costs: Departures from the ideal free distribution,, Oikos, 119 (2010), 1469. doi: 10.1111/j.1600-0706.2010.18196.x.

[36]

M. A. McPeek and R. D. Holt, The evolution of dispersal in spatially and temporally varying environments,, American Naturalist, 140 (1992), 1010. doi: 10.1086/285453.

[37]

J. D. Meiss, Differential Dynamical Systems,, Society for Industrial and Applied Mathematics, (2007). doi: 10.1137/1.9780898718232.

[38]

G. Meszéna, M. Gyllenberg, F. J. Jacobs and J. A. J. Metz, Link between population dynamics and dynamics of Darwinian evolution,, Physical Review Letters, 95 (2005). doi: 10.1103/PhysRevLett.95.078105.

[39]

M. Milinski, An evolutionarily stable feeding strategy in sticklebacks,, Journal of comparative ethology, 51 (1979), 36. doi: 10.1111/j.1439-0310.1979.tb00669.x.

[40]

M. Milinski, Ideal free theory predicts more than only matching - a critique of Kennedy and Gray's review,, Oikos, 71 (1994), 163. doi: 10.2307/3546183.

[41]

D. W. Morris, Spatial scale and the cost of density-dependent habitat selection,, Evolutionary Ecology, 1 (1987), 379. doi: 10.1007/BF02071560.

[42]

R. Nathan, An emerging movement ecology paradigm,, in Proceedings of the National Academy of Sciences of the U.S.A., 105 (2008), 19050. doi: 10.1073/pnas.0808918105.

[43]

V. Padron and M. C. Trevisan, Environmentally induced dispersal under heterogenous logistic growth,, Mathematical Biosciences, 199 (2006), 160. doi: 10.1016/j.mbs.2005.11.004.

[44]

G. A. Parker, Searching for mates,, in Behavioural Ecology: An Evolutionary Approach (eds. J. R. Krebs and N. B. Davies), (1978), 214.

[45]

K. Parvinen, Evolution of migration in a metapopulation,, Bulletin of Mathematical Biology, 61 (1999), 531.

[46]

H. R. Pulliam, Sources, sinks, and population regulation,, American Naturalist, 132 (1988), 652. doi: 10.1086/284880.

[47]

E. Ranta, P. Lundberg and V. Kaitala, Resource matching with limited knowledge,, Oikos, 86 (1999), 383. doi: 10.2307/3546456.

[48]

M. L. Rosenzweig, A theory of habitat selection,, Ecology, 62 (1981), 327. doi: 10.2307/1936707.

[49]

S. J. Schreiber, Interactive effects of temporal correlations, spatial heterogeneity and dispersal on population persistence,, Proceedings of the Royal Society B, 277 (2010), 1907. doi: 10.1098/rspb.2009.2006.

[50]

H. G. Spencer, M. Kennedy and R. D. Gray, Perceptual constraints on optimal foraging: The effect of variation among foragers,, Evolutionary Ecology, 10 (1996), 331. doi: 10.1007/BF01237721.

[51]

T. Tregenza, Building on the ideal free distribution,, Advances in Ecological Research, 26 (1995), 253. doi: 10.1016/S0065-2504(08)60067-7.

[52]

M. van Baalen and M. W. Sabelis, Coevolution of patch selection strategies of predator and prey and the consequences for ecological stability,, American Naturalist, 142 (1993), 646.

[53]

J. H. Vandermeer, The community matrix and the number of species in a community,, American Naturalist, 104 (1970), 73.

show all references

References:
[1]

P. A. Abrams, Implications of flexible foraging for interspecific interactions: Lessons from simple models,, Functional Ecology, 24 (2010), 7. doi: 10.1111/j.1365-2435.2009.01621.x.

[2]

P. A. Abrams, H. Matsuda and Y. Harada, Evolutionarily unstable fitness maxima and stable fitness minima of continuous traits,, Evolutionary Ecology, 7 (1993), 465. doi: 10.1007/BF01237642.

[3]

I. Averill, Y. Lou and D. Munther, On several conjectures from evolution of dispersal,, Journal of Biological Dynamics, 6 (2012), 117. doi: 10.1080/17513758.2010.529169.

[4]

F. Belgacem and C. Cosner, The effects of dispersal along environmental gradients on the dynamics of populations in heterogeneous environment,, Canadian Applied Math Quarterly, 3 (1995), 379.

[5]

R. S. Cantrell, C. Cosner, D. L. DeAngelis and V. Padron, The ideal free distribution as an evolutionarily stable strategy,, Journal of Biological Dynamics, 1 (2007), 249. doi: 10.1080/17513750701450227.

[6]

R. S. Cantrell, C. Cosner and Y. Lou, Approximating the ideal free distribution via reaction-diffusion-advection equations,, Journal of Differential Equations, 245 (2008), 3687. doi: 10.1016/j.jde.2008.07.024.

[7]

R. S. Cantrell, C. Cosner and Y. Lou, Evolution of dispersal and the ideal free distribution,, Mathematical Biosciences and Engineering, 7 (2010), 17. doi: 10.3934/mbe.2010.7.17.

[8]

R. S. Cantrell, C. Cosner and Y. Lou, Evolutionary stability of ideal free dispersal strategies in patchy environments,, Journal of Mathematical Biology, 65 (2012), 943. doi: 10.1007/s00285-011-0486-5.

[9]

C. Cosner, A dynamic model for the ideal-free distribution as a partial differential equation,, Theoretical Population Biology, 67 (2005), 101. doi: 10.1016/j.tpb.2004.09.002.

[10]

R. Cressman and V. Křivan, The ideal free distribution as an evolutionarily stable state in density-dependent population games,, Oikos, 119 (2010), 1231. doi: 10.1111/j.1600-0706.2010.17845.x.

[11]

D. L. DeAngelis, G. S. K. Wolkowicz, Y. Lou, Y. Jiang, M. Novak, R. Svanback, M. S. Araujo, Y. Jo and E. A. Cleary, The effect of travel loss on evolutionarily stable distributions of populations in space,, American Naturalist, 178 (2011), 15. doi: 10.1086/660280.

[12]

O. Diekmann, A beginner's guide to adaptive dynamics,, Banach center publications, 63 (2004), 47.

[13]

J. E. Diffendorfer, Testing models of source-sink dynamics and balanced dispersal,, Oikos, 81 (1998), 417. doi: 10.2307/3546763.

[14]

J. Dockery, V. Hutson, K. Mischaikow and M. Pernarowski, The evolution of slow dispersal rates: A reaction diffusion model,, Journal of Mathematical Biology, 37 (1998), 61. doi: 10.1007/s002850050120.

[15]

S. M. Flaxman and Y. Lou, Tracking prey or tracking the prey's resource? Mechanisms of movement and optimal habitat selection by predators,, Journal of Theoretical Biology, 256 (2009), 187. doi: 10.1016/j.jtbi.2008.09.024.

[16]

H. I. Freedman, B. Rai and P. Waltman, Mathematical models of population interactions with dispersal. II: Differential survival in a change of habitat,, Journal of Mathematical Analysis and Applications, 115 (1986), 140. doi: 10.1016/0022-247X(86)90029-6.

[17]

H. I. Freedman and P. Waltman, Mathematical models of population interactions with dispersal. I: Stability of two habitats with and without a predator,, SIAM Journal of Applied Mathematics, 32 (1977), 631. doi: 10.1137/0132052.

[18]

S. D. Fretwell and H. L. Lucas, On territorial behavior and other factors influencing habitat distribution in birds,, Acta Biotheoretica, 19 (1969), 37. doi: 10.1007/BF01601954.

[19]

T. E. Galanthay and S. M. Flaxman, Generalized movement strategies for constrained consumers: Ignoring fitness can be adaptive,, American Naturalist, 179 (2012), 475. doi: 10.1086/664625.

[20]

S. A. H. Geritz, É. Kisdi, G. Meszéna and J. A. J. Metz, Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree,, Evolutionary Ecology, 12 (1998), 35.

[21]

B. S. Goh, Global stability in 2 species interactions,, Journal of Mathematical Biology, 3 (1976), 313. doi: 10.1007/BF00275063.

[22]

H. Hakoyama and K. Iguchi, The information of food distribution realizes an ideal free distribution: Support of perceptual limitation,, Journal of Ethology, 15 (1997), 69. doi: 10.1007/BF02769391.

[23]

A. Hastings, Dynamics of a single species in a spatially varying environment: The stabilizing role of high dispersal rates,, Journal of Mathematical Biology, 16 (1982), 49. doi: 10.1007/BF00275160.

[24]

A. Hastings, Can spatial variation alone lead to selection for dispersal?,, Theoretical Population Biology, 24 (1983), 244. doi: 10.1016/0040-5809(83)90027-8.

[25]

R. D. Holt, Population dynamics in two-patch environments: Some anomalous consequences of an optimal habitat distribution,, Theoretical Population Biology, 28 (1985), 181. doi: 10.1016/0040-5809(85)90027-9.

[26]

D. M. Hugie and T. C. Grand, Movement between patches, unequal competitors and the ideal free distribution,, Evolutionary Ecology, 12 (1998), 1.

[27]

M. Kennedy and R. D. Gray, Can ecological theory predict the distribution of foraging animals? A critical analysis of experiments on the ideal free distribution,, Oikos, 68 (1993), 158. doi: 10.2307/3545322.

[28]

M. Kennedy and R. D. Gray, Habitat choice, habitat matching and the effect of travel distance,, Behaviour, 134 (1997), 905. doi: 10.1163/156853997X00223.

[29]

S. Kirkland, C.-K. Li and S. J. Schreiber, On the evolution of dispersal in patchy landscapes,, SIAM Journal of Applied Mathematics, 66 (2006), 1366. doi: 10.1137/050628933.

[30]

R. Korona, Travel costs and ideal free distribution of ovipositing female flour beetles, Tribolium confusum,, Animal Behaviour, 40 (1990), 186. doi: 10.1016/S0003-3472(05)80680-3.

[31]

V. Křivan, Dynamic ideal free distribution: Effects of optimal patch choice on predator-prey dynamics,, American Naturalist, 149 (1997), 164.

[32]

V. Křivan, R. Cressman and C. Schneider, The ideal free distribution: A review and synthesis of the game-theoretic perspective,, Theoretical Population Biology, 73 (2008), 403.

[33]

Y. Lou, Some challenging mathematical problems in evolution of dispersal and population dynamics,, in Tutorials in Mathematical Biosciences IV (ed. A. Friedman), (1922), 171. doi: 10.1007/978-3-540-74331-6_5.

[34]

Y. Lou and C.-H. Wu, Global dynamics of a tritrophic model for two patches with cost of dispersal,, SIAM Journal of Applied Mathematics, 71 (2011), 1801. doi: 10.1137/100817954.

[35]

S. Matsumura, R. Arlinghaus and U. Dieckmann, Foraging on spatially distributed resources with sub-optimal movement, imperfect information, and travelling costs: Departures from the ideal free distribution,, Oikos, 119 (2010), 1469. doi: 10.1111/j.1600-0706.2010.18196.x.

[36]

M. A. McPeek and R. D. Holt, The evolution of dispersal in spatially and temporally varying environments,, American Naturalist, 140 (1992), 1010. doi: 10.1086/285453.

[37]

J. D. Meiss, Differential Dynamical Systems,, Society for Industrial and Applied Mathematics, (2007). doi: 10.1137/1.9780898718232.

[38]

G. Meszéna, M. Gyllenberg, F. J. Jacobs and J. A. J. Metz, Link between population dynamics and dynamics of Darwinian evolution,, Physical Review Letters, 95 (2005). doi: 10.1103/PhysRevLett.95.078105.

[39]

M. Milinski, An evolutionarily stable feeding strategy in sticklebacks,, Journal of comparative ethology, 51 (1979), 36. doi: 10.1111/j.1439-0310.1979.tb00669.x.

[40]

M. Milinski, Ideal free theory predicts more than only matching - a critique of Kennedy and Gray's review,, Oikos, 71 (1994), 163. doi: 10.2307/3546183.

[41]

D. W. Morris, Spatial scale and the cost of density-dependent habitat selection,, Evolutionary Ecology, 1 (1987), 379. doi: 10.1007/BF02071560.

[42]

R. Nathan, An emerging movement ecology paradigm,, in Proceedings of the National Academy of Sciences of the U.S.A., 105 (2008), 19050. doi: 10.1073/pnas.0808918105.

[43]

V. Padron and M. C. Trevisan, Environmentally induced dispersal under heterogenous logistic growth,, Mathematical Biosciences, 199 (2006), 160. doi: 10.1016/j.mbs.2005.11.004.

[44]

G. A. Parker, Searching for mates,, in Behavioural Ecology: An Evolutionary Approach (eds. J. R. Krebs and N. B. Davies), (1978), 214.

[45]

K. Parvinen, Evolution of migration in a metapopulation,, Bulletin of Mathematical Biology, 61 (1999), 531.

[46]

H. R. Pulliam, Sources, sinks, and population regulation,, American Naturalist, 132 (1988), 652. doi: 10.1086/284880.

[47]

E. Ranta, P. Lundberg and V. Kaitala, Resource matching with limited knowledge,, Oikos, 86 (1999), 383. doi: 10.2307/3546456.

[48]

M. L. Rosenzweig, A theory of habitat selection,, Ecology, 62 (1981), 327. doi: 10.2307/1936707.

[49]

S. J. Schreiber, Interactive effects of temporal correlations, spatial heterogeneity and dispersal on population persistence,, Proceedings of the Royal Society B, 277 (2010), 1907. doi: 10.1098/rspb.2009.2006.

[50]

H. G. Spencer, M. Kennedy and R. D. Gray, Perceptual constraints on optimal foraging: The effect of variation among foragers,, Evolutionary Ecology, 10 (1996), 331. doi: 10.1007/BF01237721.

[51]

T. Tregenza, Building on the ideal free distribution,, Advances in Ecological Research, 26 (1995), 253. doi: 10.1016/S0065-2504(08)60067-7.

[52]

M. van Baalen and M. W. Sabelis, Coevolution of patch selection strategies of predator and prey and the consequences for ecological stability,, American Naturalist, 142 (1993), 646.

[53]

J. H. Vandermeer, The community matrix and the number of species in a community,, American Naturalist, 104 (1970), 73.

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