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August  2015, 20(6): 1583-1590. doi: 10.3934/dcdsb.2015.20.1583

Extinction in discrete, competitive, multi-species patch models

David M. Chan 1, , Matt McCombs 1, , Sarah Boegner 1, , Hye Jin Ban 1, and  Suzanne L. Robertson 1,

1. 

Department of Mathematics and Applied Mathematics, Virginia Commonwealth University, Richmond, VA 23284, United States, United States, United States, United States, United States

Received  November 2013 Revised  December 2014 Published  June 2015

In this paper we extend the results of Franke and Yakubu in [5] for extinction in discrete competitive patch models. For a system of $n$ species on $m$ patches, we define conditions under which one species is a ``superior competitor" to another and show that this is sufficient for one species to drive another to extinction. We also illustrate the result with an example for three species on three patches.
Keywords: Competition, difference equations., extinction, patch models, Lyapunov function.
Mathematics Subject Classification: Primary: 92B05, 39A30; Secondary: 39A6.
Citation: David M. Chan, Matt McCombs, Sarah Boegner, Hye Jin Ban, Suzanne L. Robertson. Extinction in discrete, competitive, multi-species patch models. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1583-1590. doi: 10.3934/dcdsb.2015.20.1583
References:
[1]

M. Doebeli, Dispersal and dynamics,, Theoretical Population Biology, 47 (1995), 82.  doi: 10.1006/tpbi.1995.1004.  Google Scholar

[2]

J. E. Franke and A. Yakubu, Mutual exclusion versus coexistence for discrete competitive systems,, Journal of Mathematical Biology, 30 (1991), 161.  doi: 10.1007/BF00160333.  Google Scholar

[3]

J. E. Franke and A. Yakubu, Geometry of exclusion principles in discrete systems,, Journal of Mathematical Analysis and Applications, 168 (1992), 385.  doi: 10.1016/0022-247X(92)90167-C.  Google Scholar

[4]

J. E. Franke and A. Yakubu, Extinction and persistence of species in discrete competitive systems with a safe refuge,, Journal of Mathematical Analysis and Applications, 203 (1996), 746.  doi: 10.1006/jmaa.1996.0410.  Google Scholar

[5]

J. E. Franke and A. Yakubu, Diffusion between patches in multi-species discrete competitive systems,, in Advances in Difference Equations (eds. S. Elaydi, (1997), 205.  doi: 10.1207/s15374424jccp2602_9.  Google Scholar

[6]

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

[7]

A. Hastings, Complex interactions between dispersal and dynamics: Lessons from coupled logistic equations,, Ecology, 74 (1993), 1362.  doi: 10.2307/1940066.  Google Scholar

[8]

A. Hastings and C. L. Wolin, Within-patch dynamics in a metapopulation,, Ecology, 70 (1989), 1261.  doi: 10.2307/1938184.  Google Scholar

[9]

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.  Google Scholar

[10]

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.  Google Scholar

[11]

C. M. Taylor and A. Hastings, Allee effects in biological invasions,, Ecology Letters, 8 (2005), 895.  doi: 10.1111/j.1461-0248.2005.00787.x.  Google Scholar

[12]

J. Verboom, R. Foppen, P. Chardon, P. Opdam and P. Luttikhuizen, Introducing the key patch approach for habitat networks with persistent populations: An example for marshland birds,, Biological Conservation, 100 (2001), 89.  doi: 10.1016/S0006-3207(00)00210-X.  Google Scholar

[13]

A. Yakubu and C. Castillo-Chavez, Interplay between local dynamics and dispersal in discrete-time metapopulation models,, Journal of Theoretical Biology, 218 (2002), 273.  doi: 10.1006/jtbi.2002.3075.  Google Scholar

show all references

References:
[1]

M. Doebeli, Dispersal and dynamics,, Theoretical Population Biology, 47 (1995), 82.  doi: 10.1006/tpbi.1995.1004.  Google Scholar

[2]

J. E. Franke and A. Yakubu, Mutual exclusion versus coexistence for discrete competitive systems,, Journal of Mathematical Biology, 30 (1991), 161.  doi: 10.1007/BF00160333.  Google Scholar

[3]

J. E. Franke and A. Yakubu, Geometry of exclusion principles in discrete systems,, Journal of Mathematical Analysis and Applications, 168 (1992), 385.  doi: 10.1016/0022-247X(92)90167-C.  Google Scholar

[4]

J. E. Franke and A. Yakubu, Extinction and persistence of species in discrete competitive systems with a safe refuge,, Journal of Mathematical Analysis and Applications, 203 (1996), 746.  doi: 10.1006/jmaa.1996.0410.  Google Scholar

[5]

J. E. Franke and A. Yakubu, Diffusion between patches in multi-species discrete competitive systems,, in Advances in Difference Equations (eds. S. Elaydi, (1997), 205.  doi: 10.1207/s15374424jccp2602_9.  Google Scholar

[6]

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

[7]

A. Hastings, Complex interactions between dispersal and dynamics: Lessons from coupled logistic equations,, Ecology, 74 (1993), 1362.  doi: 10.2307/1940066.  Google Scholar

[8]

A. Hastings and C. L. Wolin, Within-patch dynamics in a metapopulation,, Ecology, 70 (1989), 1261.  doi: 10.2307/1938184.  Google Scholar

[9]

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.  Google Scholar

[10]

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.  Google Scholar

[11]

C. M. Taylor and A. Hastings, Allee effects in biological invasions,, Ecology Letters, 8 (2005), 895.  doi: 10.1111/j.1461-0248.2005.00787.x.  Google Scholar

[12]

J. Verboom, R. Foppen, P. Chardon, P. Opdam and P. Luttikhuizen, Introducing the key patch approach for habitat networks with persistent populations: An example for marshland birds,, Biological Conservation, 100 (2001), 89.  doi: 10.1016/S0006-3207(00)00210-X.  Google Scholar

[13]

A. Yakubu and C. Castillo-Chavez, Interplay between local dynamics and dispersal in discrete-time metapopulation models,, Journal of Theoretical Biology, 218 (2002), 273.  doi: 10.1006/jtbi.2002.3075.  Google Scholar

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