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2012, 9(1): 27-60. doi: 10.3934/mbe.2012.9.27

The implications of model formulation when transitioning from spatial to landscape ecology

Robert Stephen Cantrell 1, , Chris Cosner 1, and  William F. Fagan 2,

1. 

Department of Mathematics, The University of Miami, Coral Gables, FL 33124, United States, United States
2. 

Department of Biology, The University of Maryland, College Park, MD 20742, United States

Received  January 2011 Revised  May 2011 Published  December 2011

In this article we compare and contrast the predictions of some spatially explicit and implicit models in the context of a thought problem at the interface of spatial and landscape ecology. The situation we envision is a one-dimensional spatial universe of infinite extent in which there are two disjoint focal patches of a habitat type that is favorable to some specified species. We assume that neither patch is large enough by itself to sustain the species in question indefinitely, but that a single patch of size equal to the combined sizes of the two focal patches provides enough contiguous favorable habitat to sustain the given species indefinitely. When the two patches are separated by a patch of unfavorable matrix habitat, the natural expectation is that the species should persist indefinitely if the two patches are close enough to each other but should go extinct over time when the patches are far enough apart. Our focus here is to examine how different mathematical regimes may be employed to model this situation, with an eye toward exploring the trade-off between the mathematical tractability of the model on one hand and the suitability of its predictions on the other. In particular, we are interested in seeing how precisely the predictions of mathematically rich spatially explicit regimes (reaction-diffusion models, integro-difference models) can be matched by those of ostensibly mathematically simpler spatially implicit patch approximations (discrete-diffusion models, average dispersal success matrix models).
Keywords: spatially explicit models, spatial and landscape ecology, spatially distributed population dynamical patch models., Habitat fragmentation.
Mathematics Subject Classification: Primary: 92D40; Secondary: 92D2.
Citation: Robert Stephen Cantrell, Chris Cosner, William F. Fagan. The implications of model formulation when transitioning from spatial to landscape ecology. Mathematical Biosciences & Engineering, 2012, 9 (1) : 27-60. doi: 10.3934/mbe.2012.9.27
References:
[1]

A. Berman and J. Plemmons, "Nonnegative Matrices in the Mathematical Sciences," Computer Science and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1979.  Google Scholar

[2]

K. J. Brown, C. Cosner and J. Fleckinger, Principal eigenvalues for problems with indefinite weight function on $\mathbbR^n$, Proceedings of the American Mathematical Society, 109 (1990), 147-155. doi: 10.2307/2048374.  Google Scholar

[3]

R. S. Cantrell and C. Cosner, "Spatial Ecology via Reaction-Diffusion Equations," Wiley Series in Mathematical and Computational Biology, John Wiley & Sons, Ltd., Chichester, 2003.  Google Scholar

[4]

C. Cosner, Reaction-diffusion equations and ecological modeling, in "Tutorials in Mathematical Biosciences. IV" (ed. A. Friedman), Lecture Notes in Mathematics, 1922, Springer, Berlin,(2008), 77-115.  Google Scholar

[5]

W. F. Fagan and F. Lutscher, Average dispersal success: Linking home range, dispersal, and metapopulation dynamics to refuge design, Ecological Applications, 16 (2006), 820-828. doi: 10.1890/1051-0761(2006)016[0820:ADSLHR]2.0.CO;2.  Google Scholar

[6]

I. Hanski, Predictive and practical metapopualtion models: The incidence function approach, in "Spatial Ecology" (eds. D. Tilman and P. Kareiva), Princton University Press, Princeton, NJ, (1997), 21-45. Google Scholar

[7]

I. Hanski, "Metapopulation Ecology," Oxford University Press, Oxford, UK, 1999. Google Scholar

[8]

I. Hanski and O. Ovaskainen, The metapopulation capacity of a fragmented landscape, Nature, 404 (2000), 755-758. doi: 10.1038/35008063.  Google Scholar

[9]

R. Levins, Some demographic and genetic consequences of environmental heterogeneity for biological control, Bulletin of the Entomological Society of America, 15 (1969), 237-240. Google Scholar

[10]

F. Lutscher and M. A. Lewis, Spatially-explicit matrix models, Journal of Mathematical Biology, 48 (2004), 293-324. doi: 10.1007/s00285-003-0234-6.  Google Scholar

[11]

O. Ovaskainen and I. Hanski, Spatially structured metapopulation models: Global and local assessment of metapopulation capacity, Theoretical Population Biology, 60 (2001), 281-304. doi: 10.1006/tpbi.2001.1548.  Google Scholar

[12]

R. Van Kirk and M. A. Lewis, Integro-difference models for persistence in fragmented habitats, Bulletin of Mathematical Biology, 59 (1997), 107-137. Google Scholar

show all references

References:
[1]

A. Berman and J. Plemmons, "Nonnegative Matrices in the Mathematical Sciences," Computer Science and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1979.  Google Scholar

[2]

K. J. Brown, C. Cosner and J. Fleckinger, Principal eigenvalues for problems with indefinite weight function on $\mathbbR^n$, Proceedings of the American Mathematical Society, 109 (1990), 147-155. doi: 10.2307/2048374.  Google Scholar

[3]

R. S. Cantrell and C. Cosner, "Spatial Ecology via Reaction-Diffusion Equations," Wiley Series in Mathematical and Computational Biology, John Wiley & Sons, Ltd., Chichester, 2003.  Google Scholar

[4]

C. Cosner, Reaction-diffusion equations and ecological modeling, in "Tutorials in Mathematical Biosciences. IV" (ed. A. Friedman), Lecture Notes in Mathematics, 1922, Springer, Berlin,(2008), 77-115.  Google Scholar

[5]

W. F. Fagan and F. Lutscher, Average dispersal success: Linking home range, dispersal, and metapopulation dynamics to refuge design, Ecological Applications, 16 (2006), 820-828. doi: 10.1890/1051-0761(2006)016[0820:ADSLHR]2.0.CO;2.  Google Scholar

[6]

I. Hanski, Predictive and practical metapopualtion models: The incidence function approach, in "Spatial Ecology" (eds. D. Tilman and P. Kareiva), Princton University Press, Princeton, NJ, (1997), 21-45. Google Scholar

[7]

I. Hanski, "Metapopulation Ecology," Oxford University Press, Oxford, UK, 1999. Google Scholar

[8]

I. Hanski and O. Ovaskainen, The metapopulation capacity of a fragmented landscape, Nature, 404 (2000), 755-758. doi: 10.1038/35008063.  Google Scholar

[9]

R. Levins, Some demographic and genetic consequences of environmental heterogeneity for biological control, Bulletin of the Entomological Society of America, 15 (1969), 237-240. Google Scholar

[10]

F. Lutscher and M. A. Lewis, Spatially-explicit matrix models, Journal of Mathematical Biology, 48 (2004), 293-324. doi: 10.1007/s00285-003-0234-6.  Google Scholar

[11]

O. Ovaskainen and I. Hanski, Spatially structured metapopulation models: Global and local assessment of metapopulation capacity, Theoretical Population Biology, 60 (2001), 281-304. doi: 10.1006/tpbi.2001.1548.  Google Scholar

[12]

R. Van Kirk and M. A. Lewis, Integro-difference models for persistence in fragmented habitats, Bulletin of Mathematical Biology, 59 (1997), 107-137. Google Scholar

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