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December  2014, 19(10): 3087-3104. doi: 10.3934/dcdsb.2014.19.3087

Effects of dispersal in a non-uniform environment on population dynamics and competition: A patch model approach

Donald L. DeAngelis 1, and  Bo Zhang 2,

1. 

U.S. Geological Survey and Department of Biology, University of Miami, 1301 Memorial Drive, Coral Gables, Florida 33143, United States
2. 

Department of Biology, University of Miami, 1301 Memorial Drive, Coral Gables, Florida 33143, United States

Received  June 2013 Revised  August 2013 Published  October 2014

Partial differential equation models of diffusion and advection are fundamental to understanding population behavior and interactions in space, but can be difficult to analyze when space is heterogeneous. As a proxy for partial differential equation models, and to provide some insight into a few questions regarding growth and movement patterns of a single population and two competing populations, a simple three-patch system is used. For a single population it is shown that diffusion rates occur for which the total biomass supported on a heterogeneous landscape exceeds total carrying capacity, confirming previous studies of partial differential equations and other models. It is also shown that the total population supported can increase indefinitely as the sharpness of the heterogeneity increases. For two competing species, it is shown that adding advection to a reaction-diffusion system can potentially reverse the general rule that the species with smaller diffusion rates always wins, or lead to coexistence. Competitive dominance is also favored for the species for which the sharpness of spatial heterogeneity in growth rate is greater. The results are consistent with analyses of partial differential equations, but the patch approach has some advantages in being more intuitively understandable.
Keywords: advection-diffusion, diffusion, discrete spatial model., Inhomogeneous environment.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.
Citation: Donald L. DeAngelis, Bo Zhang. Effects of dispersal in a non-uniform environment on population dynamics and competition: A patch model approach. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3087-3104. doi: 10.3934/dcdsb.2014.19.3087
References:
[1]

P. Amarasekare and R. M. Nisbet, Spatial heterogeneity, source-sink dynamics, and the local coexistence of competing species, The American Naturalist, 158 (2001), 572-584. doi: 10.1086/323586.  Google Scholar

[2]

B. M. Bolker, Combining endogenous and exogenous spatial variability in analytical population models, Theoretical Population Biology, 64 (2003), 255-270. doi: 10.1016/S0040-5809(03)00090-X.  Google Scholar

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R. S. Cantrell, and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations,, Wiley Series in Mathematical and Computational Biology, ().  doi: 10.1002/0470871296.  Google Scholar

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R. S. Cantrell, C. Cosner, D. L. DeAngelis, and V. Padron, The ideal free distribution as an evolutionarily stable strategy, Journal of Biological Dynamics, 3 (2007), 249-271. doi: 10.1080/17513750701450227.  Google Scholar

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X. Chen, K.-Y. Lam, and Y. Lou, Dynamics of a reaction-diffusion-advection model for two competing species, Discrete and Continuous Dynamic Systems A, 32 (2012), 3841-3859. doi: 10.3934/dcds.2012.32.3841.  Google Scholar

[6]

P. Chesson, Coexistence of competitors in spatially and temporally varying environments: A look at the combined effects of different sorts of variability, Theoretical Population Biology, 28 (1985), 263-287. doi: 10.1016/0040-5809(85)90030-9.  Google Scholar

[7]

P. Chesson, General theory of competitive coexistence in spatially-varying environments, Theoretical Population Biology, 58 (2000), 211-237. doi: 10.1006/tpbi.2000.1486.  Google Scholar

[8]

R. Cressman, V. Křivan, and J. Garay, Ideal free distributions, evolutionary games, and population dynamics in multiple species environments, The American Naturalist, 164 (2004), 473-489. doi: 10.1086/423827.  Google Scholar

[9]

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

[10]

S. D. Fretwell, and H. R. Lucas, On territorial behavior and other factors influencing habitat distribution in birds. I, Theoretical development. Acta Biotheoretica, 19 (1970), 16-36. doi: 10.1007/BF01601953.  Google Scholar

[11]

T. Hara, Effects of variation in individual growth on plant species coexistence, Journal of Vegetarian Science, 4 (1993), 409-416. doi: 10.2307/3235600.  Google Scholar

[12]

R. Hambrock and Y. Lou, The evolution of conditional dispersal strategies in spatially heterogeneous habitats, Bulletin of Mathematical Biology, 71 (2009), 1793-1817. doi: 10.1007/s11538-009-9425-7.  Google Scholar

[13]

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

[14]

X. He and W.-M. Ni, The effects of diffusion and spatial variation in Lotka-Volterra competition-diffusion system. I: Heterogeneity vs. homogeneity, Journal of Differential Equations, 254 (2013), 528-546. doi: 10.1016/j.jde.2012.08.032.  Google Scholar

[15]

K.-Y. Lam and Y. Lou., Evolution of conditional dispersal: Evolutionarily stable strategy in spatial models,, In press, ().  doi: 10.1007/s00285-013-0650-1.  Google Scholar

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K.-Y. Lam and Y. Lou., Evolutionarily stable and convergent stable strategies in reaction-diffusion models for conditional dispersal,, Submitted., ().  doi: 10.1007/s11538-013-9901-y.  Google Scholar

[17]

J. Latore, P. Gould, and A. M. Mortimer, Effects of habitat heterogeneity and dispersal strategies on population persistence in annual plants, Ecological Modelling, 123 (1999), 127-139. doi: 10.1016/S0304-3800(99)00132-5.  Google Scholar

[18]

Y. Lou, On the effects of migration and spatial heterogeneity on single and multiple species, Journal of Differential Equations, 223 (2006), 400-426. doi: 10.1016/j.jde.2005.05.010.  Google Scholar

[19]

F. Lutscher, E. McCauley, and M. A. Lewis, Spatial patterns and coexistence mechanism in systems with unidirectional flow, Theoretical Population Biology, 71 (2007), 267-277. doi: 10.1016/j.tpb.2006.11.006.  Google Scholar

[20]

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

[21]

S. Muko and Y. Iwasa, Species coexistence by permanent spatial heterogeneity in a lottery model, Theoretical Population Biology, 57 (2000), 273-284. doi: 10.1006/tpbi.2000.1456.  Google Scholar

[22]

J. Silvertown and R. Law, Do plants need niches? Some recent developments in plant community ecology, Trends in Ecology and Evolution, 2 (1987), 24-26. doi: 10.1016/0169-5347(87)90197-2.  Google Scholar

[23]

R. E. Snyder and P. Chesson, Local dispersal can facilitate coexistence in the presence of permanent spatial heterogeneity, Ecology Letters, 6 (2003), 301-309. doi: 10.1046/j.1461-0248.2003.00434.x.  Google Scholar

[24]

R. E. Snyder and P. Chesson, How the spatial scales of dispersal, competition, and environmental heterogeneity interact to affect coexistence, The American Naturalist, 164 (2004), 633-650. doi: 10.1086/424969.  Google Scholar

[25]

D. Tilman, Competition and biodiversity in spatially structures habitats, Ecology, 75 (1994), 2-16. doi: 10.2307/1939377.  Google Scholar

[26]

D. W. Yu, H. B. Wilson and N. E. Pierce, An empirical model of species coexistence in a spatially structured environment, Ecology, 82 (2001), 1761-1771. doi: 10.2307/2679816.  Google Scholar

show all references

References:
[1]

P. Amarasekare and R. M. Nisbet, Spatial heterogeneity, source-sink dynamics, and the local coexistence of competing species, The American Naturalist, 158 (2001), 572-584. doi: 10.1086/323586.  Google Scholar

[2]

B. M. Bolker, Combining endogenous and exogenous spatial variability in analytical population models, Theoretical Population Biology, 64 (2003), 255-270. doi: 10.1016/S0040-5809(03)00090-X.  Google Scholar

[3]

R. S. Cantrell, and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations,, Wiley Series in Mathematical and Computational Biology, ().  doi: 10.1002/0470871296.  Google Scholar

[4]

R. S. Cantrell, C. Cosner, D. L. DeAngelis, and V. Padron, The ideal free distribution as an evolutionarily stable strategy, Journal of Biological Dynamics, 3 (2007), 249-271. doi: 10.1080/17513750701450227.  Google Scholar

[5]

X. Chen, K.-Y. Lam, and Y. Lou, Dynamics of a reaction-diffusion-advection model for two competing species, Discrete and Continuous Dynamic Systems A, 32 (2012), 3841-3859. doi: 10.3934/dcds.2012.32.3841.  Google Scholar

[6]

P. Chesson, Coexistence of competitors in spatially and temporally varying environments: A look at the combined effects of different sorts of variability, Theoretical Population Biology, 28 (1985), 263-287. doi: 10.1016/0040-5809(85)90030-9.  Google Scholar

[7]

P. Chesson, General theory of competitive coexistence in spatially-varying environments, Theoretical Population Biology, 58 (2000), 211-237. doi: 10.1006/tpbi.2000.1486.  Google Scholar

[8]

R. Cressman, V. Křivan, and J. Garay, Ideal free distributions, evolutionary games, and population dynamics in multiple species environments, The American Naturalist, 164 (2004), 473-489. doi: 10.1086/423827.  Google Scholar

[9]

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

[10]

S. D. Fretwell, and H. R. Lucas, On territorial behavior and other factors influencing habitat distribution in birds. I, Theoretical development. Acta Biotheoretica, 19 (1970), 16-36. doi: 10.1007/BF01601953.  Google Scholar

[11]

T. Hara, Effects of variation in individual growth on plant species coexistence, Journal of Vegetarian Science, 4 (1993), 409-416. doi: 10.2307/3235600.  Google Scholar

[12]

R. Hambrock and Y. Lou, The evolution of conditional dispersal strategies in spatially heterogeneous habitats, Bulletin of Mathematical Biology, 71 (2009), 1793-1817. doi: 10.1007/s11538-009-9425-7.  Google Scholar

[13]

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

[14]

X. He and W.-M. Ni, The effects of diffusion and spatial variation in Lotka-Volterra competition-diffusion system. I: Heterogeneity vs. homogeneity, Journal of Differential Equations, 254 (2013), 528-546. doi: 10.1016/j.jde.2012.08.032.  Google Scholar

[15]

K.-Y. Lam and Y. Lou., Evolution of conditional dispersal: Evolutionarily stable strategy in spatial models,, In press, ().  doi: 10.1007/s00285-013-0650-1.  Google Scholar

[16]

K.-Y. Lam and Y. Lou., Evolutionarily stable and convergent stable strategies in reaction-diffusion models for conditional dispersal,, Submitted., ().  doi: 10.1007/s11538-013-9901-y.  Google Scholar

[17]

J. Latore, P. Gould, and A. M. Mortimer, Effects of habitat heterogeneity and dispersal strategies on population persistence in annual plants, Ecological Modelling, 123 (1999), 127-139. doi: 10.1016/S0304-3800(99)00132-5.  Google Scholar

[18]

Y. Lou, On the effects of migration and spatial heterogeneity on single and multiple species, Journal of Differential Equations, 223 (2006), 400-426. doi: 10.1016/j.jde.2005.05.010.  Google Scholar

[19]

F. Lutscher, E. McCauley, and M. A. Lewis, Spatial patterns and coexistence mechanism in systems with unidirectional flow, Theoretical Population Biology, 71 (2007), 267-277. doi: 10.1016/j.tpb.2006.11.006.  Google Scholar

[20]

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

[21]

S. Muko and Y. Iwasa, Species coexistence by permanent spatial heterogeneity in a lottery model, Theoretical Population Biology, 57 (2000), 273-284. doi: 10.1006/tpbi.2000.1456.  Google Scholar

[22]

J. Silvertown and R. Law, Do plants need niches? Some recent developments in plant community ecology, Trends in Ecology and Evolution, 2 (1987), 24-26. doi: 10.1016/0169-5347(87)90197-2.  Google Scholar

[23]

R. E. Snyder and P. Chesson, Local dispersal can facilitate coexistence in the presence of permanent spatial heterogeneity, Ecology Letters, 6 (2003), 301-309. doi: 10.1046/j.1461-0248.2003.00434.x.  Google Scholar

[24]

R. E. Snyder and P. Chesson, How the spatial scales of dispersal, competition, and environmental heterogeneity interact to affect coexistence, The American Naturalist, 164 (2004), 633-650. doi: 10.1086/424969.  Google Scholar

[25]

D. Tilman, Competition and biodiversity in spatially structures habitats, Ecology, 75 (1994), 2-16. doi: 10.2307/1939377.  Google Scholar

[26]

D. W. Yu, H. B. Wilson and N. E. Pierce, An empirical model of species coexistence in a spatially structured environment, Ecology, 82 (2001), 1761-1771. doi: 10.2307/2679816.  Google Scholar

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