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

Previous Article
The diffusive competition model with a free boundary: Invasion of a superior or inferior competitor
DCDS-B Home
This Issue
Next Article
Inside dynamics of solutions of integro-differential equations

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

Abstract
Related Papers

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 and 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.
[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.
[3]	R. S. Cantrell, and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley Series in Mathematical and Computational Biology, John Wiley & Sons, Chichester, United Kingdom. doi: 10.1002/0470871296.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[15]	K.-Y. Lam and Y. Lou., Evolution of conditional dispersal: Evolutionarily stable strategy in spatial models, In press, Journal of Mathematical Biology. doi: 10.1007/s00285-013-0650-1.
[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.
[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.
[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.
[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.
[20]	D. W. Morris, Spatial scale and the cost of density-dependent habitat selection, Evolutionary Ecology, 1 (1987), 379-388. doi: 10.1007/BF02071560.
[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.
[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.
[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.
[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.
[25]	D. Tilman, Competition and biodiversity in spatially structures habitats, Ecology, 75 (1994), 2-16. doi: 10.2307/1939377.
[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.

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.
[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.
[3]	R. S. Cantrell, and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley Series in Mathematical and Computational Biology, John Wiley & Sons, Chichester, United Kingdom. doi: 10.1002/0470871296.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[15]	K.-Y. Lam and Y. Lou., Evolution of conditional dispersal: Evolutionarily stable strategy in spatial models, In press, Journal of Mathematical Biology. doi: 10.1007/s00285-013-0650-1.
[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.
[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.
[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.
[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.
[20]	D. W. Morris, Spatial scale and the cost of density-dependent habitat selection, Evolutionary Ecology, 1 (1987), 379-388. doi: 10.1007/BF02071560.
[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.
[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.
[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.
[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.
[25]	D. Tilman, Competition and biodiversity in spatially structures habitats, Ecology, 75 (1994), 2-16. doi: 10.2307/1939377.
[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.

[1]	Assyr Abdulle. Multiscale methods for advection-diffusion problems. Conference Publications, 2005, 2005 (Special) : 11-21. doi: 10.3934/proc.2005.2005.11
[2]	Michael Taylor. Random walks, random flows, and enhanced diffusivity in advection-diffusion equations. Discrete and Continuous Dynamical Systems - B, 2012, 17 (4) : 1261-1287. doi: 10.3934/dcdsb.2012.17.1261
[3]	Qing Tang. On an optimal control problem of time-fractional advection-diffusion equation. Discrete and Continuous Dynamical Systems - B, 2020, 25 (2) : 761-779. doi: 10.3934/dcdsb.2019266
[4]	Danhua Jiang, Zhi-Cheng Wang, Liang Zhang. A reaction-diffusion-advection SIS epidemic model in a spatially-temporally heterogeneous environment. Discrete and Continuous Dynamical Systems - B, 2018, 23 (10) : 4557-4578. doi: 10.3934/dcdsb.2018176
[5]	Alexandre Caboussat, Roland Glowinski. A Numerical Method for a Non-Smooth Advection-Diffusion Problem Arising in Sand Mechanics. Communications on Pure and Applied Analysis, 2009, 8 (1) : 161-178. doi: 10.3934/cpaa.2009.8.161
[6]	Patrick Henning, Mario Ohlberger. The heterogeneous multiscale finite element method for advection-diffusion problems with rapidly oscillating coefficients and large expected drift. Networks and Heterogeneous Media, 2010, 5 (4) : 711-744. doi: 10.3934/nhm.2010.5.711
[7]	Patrick Henning, Mario Ohlberger. A-posteriori error estimate for a heterogeneous multiscale approximation of advection-diffusion problems with large expected drift. Discrete and Continuous Dynamical Systems - S, 2016, 9 (5) : 1393-1420. doi: 10.3934/dcdss.2016056
[8]	Lena-Susanne Hartmann, Ilya Pavlyukevich. Advection-diffusion equation on a half-line with boundary Lévy noise. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 637-655. doi: 10.3934/dcdsb.2018200
[9]	S. Sadeghi, H. Jafari, S. Nemati. Solving fractional Advection-diffusion equation using Genocchi operational matrix based on Atangana-Baleanu derivative. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3747-3761. doi: 10.3934/dcdss.2020435
[10]	Masahiro Yamamoto. Uniqueness for inverse problem of determining fractional orders for time-fractional advection-diffusion equations. Mathematical Control and Related Fields, 2022 doi: 10.3934/mcrf.2022017
[11]	Bo Duan, Zhengce Zhang. A reaction-diffusion-advection two-species competition system with a free boundary in heterogeneous environment. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 837-861. doi: 10.3934/dcdsb.2021067
[12]	Xiaoyan Zhang, Yuxiang Zhang. Spatial dynamics of a reaction-diffusion cholera model with spatial heterogeneity. Discrete and Continuous Dynamical Systems - B, 2018, 23 (6) : 2625-2640. doi: 10.3934/dcdsb.2018124
[13]	M. B. A. Mansour. Computation of traveling wave fronts for a nonlinear diffusion-advection model. Mathematical Biosciences & Engineering, 2009, 6 (1) : 83-91. doi: 10.3934/mbe.2009.6.83
[14]	Xinfu Chen, King-Yeung Lam, Yuan Lou. Corrigendum: Dynamics of a reaction-diffusion-advection model for two competing species. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4989-4995. doi: 10.3934/dcds.2014.34.4989
[15]	Xinfu Chen, King-Yeung Lam, Yuan Lou. Dynamics of a reaction-diffusion-advection model for two competing species. Discrete and Continuous Dynamical Systems, 2012, 32 (11) : 3841-3859. doi: 10.3934/dcds.2012.32.3841
[16]	Xueying Wang, Drew Posny, Jin Wang. A reaction-convection-diffusion model for cholera spatial dynamics. Discrete and Continuous Dynamical Systems - B, 2016, 21 (8) : 2785-2809. doi: 10.3934/dcdsb.2016073
[17]	Benedetto Bozzini, Deborah Lacitignola, Ivonne Sgura. Morphological spatial patterns in a reaction diffusion model for metal growth. Mathematical Biosciences & Engineering, 2010, 7 (2) : 237-258. doi: 10.3934/mbe.2010.7.237
[18]	Yachun Tong, Inkyung Ahn, Zhigui Lin. Effect of diffusion in a spatial SIS epidemic model with spontaneous infection. Discrete and Continuous Dynamical Systems - B, 2021, 26 (8) : 4045-4057. doi: 10.3934/dcdsb.2020273
[19]	Ho-Youn Kim, Yong-Jung Kim, Hyun-Jin Lim. Heterogeneous discrete kinetic model and its diffusion limit. Kinetic and Related Models, 2021, 14 (5) : 749-765. doi: 10.3934/krm.2021023
[20]	Claude-Michel Brauner, Danaelle Jolly, Luca Lorenzi, Rodolphe Thiebaut. Heterogeneous viral environment in a HIV spatial model. Discrete and Continuous Dynamical Systems - B, 2011, 15 (3) : 545-572. doi: 10.3934/dcdsb.2011.15.545

2021 Impact Factor: 1.497

Tools

Metrics

Other articles
by authors

on AIMS
Donald L. DeAngelis Bo Zhang
on Google Scholar
Donald L. DeAngelis Bo Zhang

[Back to Top]