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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 and 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), 82-106. doi: 10.1006/tpbi.1995.1004.

[2]

J. E. Franke and A. Yakubu, Mutual exclusion versus coexistence for discrete competitive systems, Journal of Mathematical Biology, 30 (1991), 161-168. doi: 10.1007/BF00160333.

[3]

J. E. Franke and A. Yakubu, Geometry of exclusion principles in discrete systems, Journal of Mathematical Analysis and Applications, 168 (1992), 385-400. doi: 10.1016/0022-247X(92)90167-C.

[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), 746-761. doi: 10.1006/jmaa.1996.0410.

[5]

J. E. Franke and A. Yakubu, Diffusion between patches in multi-species discrete competitive systems, in Advances in Difference Equations (eds. S. Elaydi, G. Ladas, and I. Gyori), Gordon and Breach, Amsterdam, 1997, 205-212. doi: 10.1207/s15374424jccp2602_9.

[6]

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.

[7]

A. Hastings, Complex interactions between dispersal and dynamics: Lessons from coupled logistic equations, Ecology, 74 (1993), 1362-1372. doi: 10.2307/1940066.

[8]

A. Hastings and C. L. Wolin, Within-patch dynamics in a metapopulation, Ecology, 70 (1989), 1261-1266. doi: 10.2307/1938184.

[9]

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.

[10]

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

[11]

C. M. Taylor and A. Hastings, Allee effects in biological invasions, Ecology Letters, 8 (2005), 895-908. doi: 10.1111/j.1461-0248.2005.00787.x.

[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), 89-101. doi: 10.1016/S0006-3207(00)00210-X.

[13]

A. Yakubu and C. Castillo-Chavez, Interplay between local dynamics and dispersal in discrete-time metapopulation models, Journal of Theoretical Biology, 218 (2002), 273-288. doi: 10.1006/jtbi.2002.3075.

show all references

References:
[1]

M. Doebeli, Dispersal and dynamics, Theoretical Population Biology, 47 (1995), 82-106. doi: 10.1006/tpbi.1995.1004.

[2]

J. E. Franke and A. Yakubu, Mutual exclusion versus coexistence for discrete competitive systems, Journal of Mathematical Biology, 30 (1991), 161-168. doi: 10.1007/BF00160333.

[3]

J. E. Franke and A. Yakubu, Geometry of exclusion principles in discrete systems, Journal of Mathematical Analysis and Applications, 168 (1992), 385-400. doi: 10.1016/0022-247X(92)90167-C.

[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), 746-761. doi: 10.1006/jmaa.1996.0410.

[5]

J. E. Franke and A. Yakubu, Diffusion between patches in multi-species discrete competitive systems, in Advances in Difference Equations (eds. S. Elaydi, G. Ladas, and I. Gyori), Gordon and Breach, Amsterdam, 1997, 205-212. doi: 10.1207/s15374424jccp2602_9.

[6]

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.

[7]

A. Hastings, Complex interactions between dispersal and dynamics: Lessons from coupled logistic equations, Ecology, 74 (1993), 1362-1372. doi: 10.2307/1940066.

[8]

A. Hastings and C. L. Wolin, Within-patch dynamics in a metapopulation, Ecology, 70 (1989), 1261-1266. doi: 10.2307/1938184.

[9]

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.

[10]

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

[11]

C. M. Taylor and A. Hastings, Allee effects in biological invasions, Ecology Letters, 8 (2005), 895-908. doi: 10.1111/j.1461-0248.2005.00787.x.

[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), 89-101. doi: 10.1016/S0006-3207(00)00210-X.

[13]

A. Yakubu and C. Castillo-Chavez, Interplay between local dynamics and dispersal in discrete-time metapopulation models, Journal of Theoretical Biology, 218 (2002), 273-288. doi: 10.1006/jtbi.2002.3075.

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