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August  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

Theodore E. Galanthay 1,

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.
Keywords: evolutionarily stable strategy., optimal dispersal strategy, cost of dispersal, Dispersal, global stability.
Mathematics Subject Classification: Primary: 34D23; Secondary: 92D2.
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), 7-17. doi: 10.1111/j.1365-2435.2009.01621.x.  Google Scholar

[2]

P. A. Abrams, H. Matsuda and Y. Harada, Evolutionarily unstable fitness maxima and stable fitness minima of continuous traits, Evolutionary Ecology, 7 (1993), 465-487. doi: 10.1007/BF01237642.  Google Scholar

[3]

I. Averill, Y. Lou and D. Munther, On several conjectures from evolution of dispersal, Journal of Biological Dynamics, 6 (2012), 117-130. doi: 10.1080/17513758.2010.529169.  Google Scholar

[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), 379-397.  Google Scholar

[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), 249-271. doi: 10.1080/17513750701450227.  Google Scholar

[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), 3687-3703. doi: 10.1016/j.jde.2008.07.024.  Google Scholar

[7]

R. S. Cantrell, C. Cosner and Y. Lou, Evolution of dispersal and the ideal free distribution, Mathematical Biosciences and Engineering, 7 (2010), 17-36. doi: 10.3934/mbe.2010.7.17.  Google Scholar

[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), 943-965. doi: 10.1007/s00285-011-0486-5.  Google Scholar

[9]

C. Cosner, A dynamic model for the ideal-free distribution as a partial differential equation, Theoretical Population Biology, 67 (2005), 101-108. doi: 10.1016/j.tpb.2004.09.002.  Google Scholar

[10]

R. Cressman and V. Křivan, The ideal free distribution as an evolutionarily stable state in density-dependent population games, Oikos, 119 (2010), 1231-1242. doi: 10.1111/j.1600-0706.2010.17845.x.  Google Scholar

[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), 15-29. doi: 10.1086/660280.  Google Scholar

[12]

O. Diekmann, A beginner's guide to adaptive dynamics, Banach center publications, 63 (2004), 47-86.  Google Scholar

[13]

J. E. Diffendorfer, Testing models of source-sink dynamics and balanced dispersal, Oikos, 81 (1998), 417-433. doi: 10.2307/3546763.  Google Scholar

[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), 61-83. doi: 10.1007/s002850050120.  Google Scholar

[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), 187-200. doi: 10.1016/j.jtbi.2008.09.024.  Google Scholar

[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), 140-154. doi: 10.1016/0022-247X(86)90029-6.  Google Scholar

[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), 631-648. doi: 10.1137/0132052.  Google Scholar

[18]

S. D. Fretwell and H. L. Lucas, On territorial behavior and other factors influencing habitat distribution in birds, Acta Biotheoretica, 19 (1969), 37-44. doi: 10.1007/BF01601954.  Google Scholar

[19]

T. E. Galanthay and S. M. Flaxman, Generalized movement strategies for constrained consumers: Ignoring fitness can be adaptive, American Naturalist, 179 (2012), 475-489. doi: 10.1086/664625.  Google Scholar

[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), 35-37. Google Scholar

[21]

B. S. Goh, Global stability in 2 species interactions, Journal of Mathematical Biology, 3 (1976), 313-318. doi: 10.1007/BF00275063.  Google Scholar

[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), 69-78. doi: 10.1007/BF02769391.  Google Scholar

[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), 49-55. doi: 10.1007/BF00275160.  Google Scholar

[24]

A. Hastings, Can spatial variation alone lead to selection for dispersal?, Theoretical Population Biology, 24 (1983), 244-251. doi: 10.1016/0040-5809(83)90027-8.  Google Scholar

[25]

R. D. Holt, Population dynamics in two-patch environments: Some anomalous consequences of an optimal habitat distribution, Theoretical Population Biology, 28 (1985), 181-208. doi: 10.1016/0040-5809(85)90027-9.  Google Scholar

[26]

D. M. Hugie and T. C. Grand, Movement between patches, unequal competitors and the ideal free distribution, Evolutionary Ecology, 12 (1998), 1-19. Google Scholar

[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), 158-166. doi: 10.2307/3545322.  Google Scholar

[28]

M. Kennedy and R. D. Gray, Habitat choice, habitat matching and the effect of travel distance, Behaviour, 134 (1997), 905-920. doi: 10.1163/156853997X00223.  Google Scholar

[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), 1366-1382. doi: 10.1137/050628933.  Google Scholar

[30]

R. Korona, Travel costs and ideal free distribution of ovipositing female flour beetles, Tribolium confusum, Animal Behaviour, 40 (1990), 186-187. doi: 10.1016/S0003-3472(05)80680-3.  Google Scholar

[31]

V. Křivan, Dynamic ideal free distribution: Effects of optimal patch choice on predator-prey dynamics, American Naturalist, 149 (1997), 164-178. Google Scholar

[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), 403-425. Google Scholar

[33]

Y. Lou, Some challenging mathematical problems in evolution of dispersal and population dynamics, in Tutorials in Mathematical Biosciences IV (ed. A. Friedman), Lecture Notes in Math., 1922, Springer, Berlin, 2008, 171-205. doi: 10.1007/978-3-540-74331-6_5.  Google Scholar

[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), 1801-1820. doi: 10.1137/100817954.  Google Scholar

[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), 1469-1483. doi: 10.1111/j.1600-0706.2010.18196.x.  Google Scholar

[36]

M. A. McPeek and R. D. Holt, The evolution of dispersal in spatially and temporally varying environments, American Naturalist, 140 (1992), 1010-1027. doi: 10.1086/285453.  Google Scholar

[37]

J. D. Meiss, Differential Dynamical Systems, Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania, 2007. doi: 10.1137/1.9780898718232.  Google Scholar

[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), 078105. doi: 10.1103/PhysRevLett.95.078105.  Google Scholar

[39]

M. Milinski, An evolutionarily stable feeding strategy in sticklebacks, Journal of comparative ethology, 51 (1979), 36-40. doi: 10.1111/j.1439-0310.1979.tb00669.x.  Google Scholar

[40]

M. Milinski, Ideal free theory predicts more than only matching - a critique of Kennedy and Gray's review, Oikos, 71 (1994), 163-166. doi: 10.2307/3546183.  Google Scholar

[41]

D. W. Morris, Spatial scale and the cost of density-dependent habitat selection, Evolutionary Ecology, 1 (1987), 379-388. doi: 10.1007/BF02071560.  Google Scholar

[42]

R. Nathan, An emerging movement ecology paradigm, in Proceedings of the National Academy of Sciences of the U.S.A., 105 (2008), 19050-19051. doi: 10.1073/pnas.0808918105.  Google Scholar

[43]

V. Padron and M. C. Trevisan, Environmentally induced dispersal under heterogenous logistic growth, Mathematical Biosciences, 199 (2006), 160-174. doi: 10.1016/j.mbs.2005.11.004.  Google Scholar

[44]

G. A. Parker, Searching for mates, in Behavioural Ecology: An Evolutionary Approach (eds. J. R. Krebs and N. B. Davies), Blackwell, 1978, 214-244. Google Scholar

[45]

K. Parvinen, Evolution of migration in a metapopulation, Bulletin of Mathematical Biology, 61 (1999), 531-550. Google Scholar

[46]

H. R. Pulliam, Sources, sinks, and population regulation, American Naturalist, 132 (1988), 652-661. doi: 10.1086/284880.  Google Scholar

[47]

E. Ranta, P. Lundberg and V. Kaitala, Resource matching with limited knowledge, Oikos, 86 (1999), 383-385. doi: 10.2307/3546456.  Google Scholar

[48]

M. L. Rosenzweig, A theory of habitat selection, Ecology, 62 (1981), 327-335. doi: 10.2307/1936707.  Google Scholar

[49]

S. J. Schreiber, Interactive effects of temporal correlations, spatial heterogeneity and dispersal on population persistence, Proceedings of the Royal Society B, 277 (2010), 1907-1914. doi: 10.1098/rspb.2009.2006.  Google Scholar

[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), 331-339. doi: 10.1007/BF01237721.  Google Scholar

[51]

T. Tregenza, Building on the ideal free distribution, Advances in Ecological Research, 26 (1995), 253-307. doi: 10.1016/S0065-2504(08)60067-7.  Google Scholar

[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), 646-670. Google Scholar

[53]

J. H. Vandermeer, The community matrix and the number of species in a community, American Naturalist, 104 (1970), 73-83. Google Scholar

show all references

References:
[1]

P. A. Abrams, Implications of flexible foraging for interspecific interactions: Lessons from simple models, Functional Ecology, 24 (2010), 7-17. doi: 10.1111/j.1365-2435.2009.01621.x.  Google Scholar

[2]

P. A. Abrams, H. Matsuda and Y. Harada, Evolutionarily unstable fitness maxima and stable fitness minima of continuous traits, Evolutionary Ecology, 7 (1993), 465-487. doi: 10.1007/BF01237642.  Google Scholar

[3]

I. Averill, Y. Lou and D. Munther, On several conjectures from evolution of dispersal, Journal of Biological Dynamics, 6 (2012), 117-130. doi: 10.1080/17513758.2010.529169.  Google Scholar

[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), 379-397.  Google Scholar

[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), 249-271. doi: 10.1080/17513750701450227.  Google Scholar

[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), 3687-3703. doi: 10.1016/j.jde.2008.07.024.  Google Scholar

[7]

R. S. Cantrell, C. Cosner and Y. Lou, Evolution of dispersal and the ideal free distribution, Mathematical Biosciences and Engineering, 7 (2010), 17-36. doi: 10.3934/mbe.2010.7.17.  Google Scholar

[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), 943-965. doi: 10.1007/s00285-011-0486-5.  Google Scholar

[9]

C. Cosner, A dynamic model for the ideal-free distribution as a partial differential equation, Theoretical Population Biology, 67 (2005), 101-108. doi: 10.1016/j.tpb.2004.09.002.  Google Scholar

[10]

R. Cressman and V. Křivan, The ideal free distribution as an evolutionarily stable state in density-dependent population games, Oikos, 119 (2010), 1231-1242. doi: 10.1111/j.1600-0706.2010.17845.x.  Google Scholar

[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), 15-29. doi: 10.1086/660280.  Google Scholar

[12]

O. Diekmann, A beginner's guide to adaptive dynamics, Banach center publications, 63 (2004), 47-86.  Google Scholar

[13]

J. E. Diffendorfer, Testing models of source-sink dynamics and balanced dispersal, Oikos, 81 (1998), 417-433. doi: 10.2307/3546763.  Google Scholar

[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), 61-83. doi: 10.1007/s002850050120.  Google Scholar

[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), 187-200. doi: 10.1016/j.jtbi.2008.09.024.  Google Scholar

[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), 140-154. doi: 10.1016/0022-247X(86)90029-6.  Google Scholar

[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), 631-648. doi: 10.1137/0132052.  Google Scholar

[18]

S. D. Fretwell and H. L. Lucas, On territorial behavior and other factors influencing habitat distribution in birds, Acta Biotheoretica, 19 (1969), 37-44. doi: 10.1007/BF01601954.  Google Scholar

[19]

T. E. Galanthay and S. M. Flaxman, Generalized movement strategies for constrained consumers: Ignoring fitness can be adaptive, American Naturalist, 179 (2012), 475-489. doi: 10.1086/664625.  Google Scholar

[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), 35-37. Google Scholar

[21]

B. S. Goh, Global stability in 2 species interactions, Journal of Mathematical Biology, 3 (1976), 313-318. doi: 10.1007/BF00275063.  Google Scholar

[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), 69-78. doi: 10.1007/BF02769391.  Google Scholar

[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), 49-55. doi: 10.1007/BF00275160.  Google Scholar

[24]

A. Hastings, Can spatial variation alone lead to selection for dispersal?, Theoretical Population Biology, 24 (1983), 244-251. doi: 10.1016/0040-5809(83)90027-8.  Google Scholar

[25]

R. D. Holt, Population dynamics in two-patch environments: Some anomalous consequences of an optimal habitat distribution, Theoretical Population Biology, 28 (1985), 181-208. doi: 10.1016/0040-5809(85)90027-9.  Google Scholar

[26]

D. M. Hugie and T. C. Grand, Movement between patches, unequal competitors and the ideal free distribution, Evolutionary Ecology, 12 (1998), 1-19. Google Scholar

[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), 158-166. doi: 10.2307/3545322.  Google Scholar

[28]

M. Kennedy and R. D. Gray, Habitat choice, habitat matching and the effect of travel distance, Behaviour, 134 (1997), 905-920. doi: 10.1163/156853997X00223.  Google Scholar

[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), 1366-1382. doi: 10.1137/050628933.  Google Scholar

[30]

R. Korona, Travel costs and ideal free distribution of ovipositing female flour beetles, Tribolium confusum, Animal Behaviour, 40 (1990), 186-187. doi: 10.1016/S0003-3472(05)80680-3.  Google Scholar

[31]

V. Křivan, Dynamic ideal free distribution: Effects of optimal patch choice on predator-prey dynamics, American Naturalist, 149 (1997), 164-178. Google Scholar

[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), 403-425. Google Scholar

[33]

Y. Lou, Some challenging mathematical problems in evolution of dispersal and population dynamics, in Tutorials in Mathematical Biosciences IV (ed. A. Friedman), Lecture Notes in Math., 1922, Springer, Berlin, 2008, 171-205. doi: 10.1007/978-3-540-74331-6_5.  Google Scholar

[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), 1801-1820. doi: 10.1137/100817954.  Google Scholar

[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), 1469-1483. doi: 10.1111/j.1600-0706.2010.18196.x.  Google Scholar

[36]

M. A. McPeek and R. D. Holt, The evolution of dispersal in spatially and temporally varying environments, American Naturalist, 140 (1992), 1010-1027. doi: 10.1086/285453.  Google Scholar

[37]

J. D. Meiss, Differential Dynamical Systems, Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania, 2007. doi: 10.1137/1.9780898718232.  Google Scholar

[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), 078105. doi: 10.1103/PhysRevLett.95.078105.  Google Scholar

[39]

M. Milinski, An evolutionarily stable feeding strategy in sticklebacks, Journal of comparative ethology, 51 (1979), 36-40. doi: 10.1111/j.1439-0310.1979.tb00669.x.  Google Scholar

[40]

M. Milinski, Ideal free theory predicts more than only matching - a critique of Kennedy and Gray's review, Oikos, 71 (1994), 163-166. doi: 10.2307/3546183.  Google Scholar

[41]

D. W. Morris, Spatial scale and the cost of density-dependent habitat selection, Evolutionary Ecology, 1 (1987), 379-388. doi: 10.1007/BF02071560.  Google Scholar

[42]

R. Nathan, An emerging movement ecology paradigm, in Proceedings of the National Academy of Sciences of the U.S.A., 105 (2008), 19050-19051. doi: 10.1073/pnas.0808918105.  Google Scholar

[43]

V. Padron and M. C. Trevisan, Environmentally induced dispersal under heterogenous logistic growth, Mathematical Biosciences, 199 (2006), 160-174. doi: 10.1016/j.mbs.2005.11.004.  Google Scholar

[44]

G. A. Parker, Searching for mates, in Behavioural Ecology: An Evolutionary Approach (eds. J. R. Krebs and N. B. Davies), Blackwell, 1978, 214-244. Google Scholar

[45]

K. Parvinen, Evolution of migration in a metapopulation, Bulletin of Mathematical Biology, 61 (1999), 531-550. Google Scholar

[46]

H. R. Pulliam, Sources, sinks, and population regulation, American Naturalist, 132 (1988), 652-661. doi: 10.1086/284880.  Google Scholar

[47]

E. Ranta, P. Lundberg and V. Kaitala, Resource matching with limited knowledge, Oikos, 86 (1999), 383-385. doi: 10.2307/3546456.  Google Scholar

[48]

M. L. Rosenzweig, A theory of habitat selection, Ecology, 62 (1981), 327-335. doi: 10.2307/1936707.  Google Scholar

[49]

S. J. Schreiber, Interactive effects of temporal correlations, spatial heterogeneity and dispersal on population persistence, Proceedings of the Royal Society B, 277 (2010), 1907-1914. doi: 10.1098/rspb.2009.2006.  Google Scholar

[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), 331-339. doi: 10.1007/BF01237721.  Google Scholar

[51]

T. Tregenza, Building on the ideal free distribution, Advances in Ecological Research, 26 (1995), 253-307. doi: 10.1016/S0065-2504(08)60067-7.  Google Scholar

[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), 646-670. Google Scholar

[53]

J. H. Vandermeer, The community matrix and the number of species in a community, American Naturalist, 104 (1970), 73-83. Google Scholar

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