American Institute of Mathematical Sciences

Advanced
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.  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.  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.  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.  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.  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.  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.  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.  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, 9 (1970), 16.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  doi: 10.1086/424969.  Google Scholar

[25]

D. Tilman, Competition and biodiversity in spatially structures habitats,, Ecology, 75 (1994), 2.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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, 9 (1970), 16.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  doi: 10.1086/424969.  Google Scholar

[25]

D. Tilman, Competition and biodiversity in spatially structures habitats,, Ecology, 75 (1994), 2.  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.  doi: 10.2307/2679816.  Google Scholar

[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 & Continuous Dynamical Systems - B, 2012, 17 (4) : 1261-1287. doi: 10.3934/dcdsb.2012.17.1261
[3]

Danhua Jiang, Zhi-Cheng Wang, Liang Zhang. A reaction-diffusion-advection SIS epidemic model in a spatially-temporally heterogeneous environment. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4557-4578. doi: 10.3934/dcdsb.2018176
[4]

Alexandre Caboussat, Roland Glowinski. A Numerical Method for a Non-Smooth Advection-Diffusion Problem Arising in Sand Mechanics. Communications on Pure & Applied Analysis, 2009, 8 (1) : 161-178. doi: 10.3934/cpaa.2009.8.161
[5]

Patrick Henning, Mario Ohlberger. A-posteriori error estimate for a heterogeneous multiscale approximation of advection-diffusion problems with large expected drift. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1393-1420. doi: 10.3934/dcdss.2016056
[6]

Patrick Henning, Mario Ohlberger. The heterogeneous multiscale finite element method for advection-diffusion problems with rapidly oscillating coefficients and large expected drift. Networks & Heterogeneous Media, 2010, 5 (4) : 711-744. doi: 10.3934/nhm.2010.5.711
[7]

Lena-Susanne Hartmann, Ilya Pavlyukevich. Advection-diffusion equation on a half-line with boundary Lévy noise. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 637-655. doi: 10.3934/dcdsb.2018200
[8]

Xiaoyan Zhang, Yuxiang Zhang. Spatial dynamics of a reaction-diffusion cholera model with spatial heterogeneity. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2625-2640. doi: 10.3934/dcdsb.2018124
[9]

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
[10]

Xinfu Chen, King-Yeung Lam, Yuan Lou. Corrigendum: Dynamics of a reaction-diffusion-advection model for two competing species. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4989-4995. doi: 10.3934/dcds.2014.34.4989
[11]

Xinfu Chen, King-Yeung Lam, Yuan Lou. Dynamics of a reaction-diffusion-advection model for two competing species. Discrete & Continuous Dynamical Systems - A, 2012, 32 (11) : 3841-3859. doi: 10.3934/dcds.2012.32.3841
[12]

Xueying Wang, Drew Posny, Jin Wang. A reaction-convection-diffusion model for cholera spatial dynamics. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2785-2809. doi: 10.3934/dcdsb.2016073
[13]

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
[14]

Claude-Michel Brauner, Danaelle Jolly, Luca Lorenzi, Rodolphe Thiebaut. Heterogeneous viral environment in a HIV spatial model. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 545-572. doi: 10.3934/dcdsb.2011.15.545
[15]

Qi Wang. On steady state of some Lotka-Volterra competition-diffusion-advection model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2019193
[16]

Chengxia Lei, Jie Xiong, Xinhui Zhou. Qualitative analysis on an SIS epidemic reaction-diffusion model with mass action infection mechanism and spontaneous infection in a heterogeneous environment. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 81-98. doi: 10.3934/dcdsb.2019173
[17]

Yongli Cai, Weiming Wang. Dynamics of a parasite-host epidemiological model in spatial heterogeneous environment. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 989-1013. doi: 10.3934/dcdsb.2015.20.989
[18]

Shuling Yan, Shangjiang Guo. Dynamics of a Lotka-Volterra competition-diffusion model with stage structure and spatial heterogeneity. Discrete & Continuous Dynamical Systems - B, 2018, 23 (4) : 1559-1579. doi: 10.3934/dcdsb.2018059
[19]

Ihab Haidar, Alain Rapaport, Frédéric Gérard. Effects of spatial structure and diffusion on the performances of the chemostat. Mathematical Biosciences & Engineering, 2011, 8 (4) : 953-971. doi: 10.3934/mbe.2011.8.953
[20]

Chris Cosner. Reaction-diffusion-advection models for the effects and evolution of dispersal. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1701-1745. doi: 10.3934/dcds.2014.34.1701

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (17)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences