The limiting behaviour of a stochastic patch occupancy model

Metapopulation models have been used to better understand the conditions necessary for the persistence of the metapopulation. In this paper, we study a stochastic patch occupancy model that incorporates variation in quality and connectivity of the habitat patches. Two important assumptions are imposed in our analysis. Firstly, the distance between patches has a special form. This amounts to assuming that migrating individuals follow certain pathways. Secondly, the area of the habitat patches is assumed to scale with the number of patches in the metapopulation. Under these assumptions, a deterministic limit is obtained as the number of patches goes to infinity. Using the deterministic limiting process, a condition for persistence of the metapopulation is derived.

[1]  M. Fogarty,et al.  Metapopulation dynamics of coastal decapods , 2006 .

[2]  A D Barbour,et al.  Convergence of a structured metapopulation model to Levins’s model , 2004, Journal of mathematical biology.

[3]  Atte Moilanen,et al.  SIMPLE CONNECTIVITY MEASURES IN SPATIAL ECOLOGY , 2002 .

[4]  Otso Ovaskainen,et al.  Exact asymptotic analysis for metapopulation dynamics on correlated dynamic landscapes. , 2008, Theoretical population biology.

[5]  H. Akçakaya,et al.  Ecological risk analysis for single and multiple populations. , 1991 .

[6]  Daryl J. Daley,et al.  An Introduction to the Theory of Point Processes , 2013 .

[7]  M. Gilpin,et al.  Metapopulation Biology: Ecology, Genetics, and Evolution , 1997 .

[8]  P. Pollett,et al.  Limit theorems for discrete-time metapopulation models , 2010 .

[9]  P K Pollett,et al.  The limiting behaviour of a mainland-island metapopulation , 2012, Journal of mathematical biology.

[10]  Otso Ovaskainen,et al.  The quasistationary distribution of the stochastic logistic model , 2001, Journal of Applied Probability.

[11]  I. Hanski,et al.  Habitat Deterioration, Habitat Destruction, and Metapopulation Persistence in a Heterogenous Landscape , 1997, Theoretical population biology.

[12]  Hugh P. Possingham,et al.  A Stochastic Metapopulation Model with Variability in Patch Size and Position , 1995 .

[13]  P. Pollett,et al.  Interaction between habitat quality and an Allee-like effect in metapopulations , 2013 .

[14]  P. Amarasekare,et al.  Allee Effects in Metapopulation Dynamics , 1998, The American Naturalist.

[15]  Damian Clancy,et al.  A note on quasi-stationary distributions of birth–death processes and the SIS logistic epidemic , 2003, Journal of Applied Probability.

[16]  I. Hanski.  A Practical Model of Metapopulation Dynamics , 1994 .

[17]  J. Gascoigne,et al.  Allee Effects in Ecology and Conservation , 2008 .

[18]  R. Rao.  Relations between Weak and Uniform Convergence of Measures with Applications , 1962 .

[19]  H. Caswell,et al.  The effects of habitat destruction in finite landscapes: a chain‐binomial metapopulation model , 2001 .

[20]  Ross McVinish,et al.  Limits of large metapopulations with patch-dependent extinction probabilities , 2010, Advances in Applied Probability.

[21]  O. Ovaskainen,et al.  Spatially structured metapopulation models: global and local assessment of metapopulation capacity. , 2001, Theoretical population biology.

[22]  E. Seneta.  Non-negative Matrices and Markov Chains , 2008 .

[23]  R. Levins.  Some Demographic and Genetic Consequences of Environmental Heterogeneity for Biological Control , 1969 .

[24]  Atte Moilanen,et al.  SPOMSIM: software for stochastic patch occupancy models of metapopulation dynamics , 2004 .

[25]  E. Meijden,et al.  Tritrophic Metapopulation Dynamics: A Case Study of Ragwort, the Cinnabar Moth, and the Parasitoid Cotesia popularis , 1997 .

[26]  Francesca Arrigoni,et al.  Deterministic approximation of a stochastic metapopulation model , 2003, Advances in Applied Probability.

[27]  H. Andersson,et al.  A threshold limit theorem for the stochastic logistic epidemic , 1998 .

[28]  Morris W. Hirsch,et al.  Monotone maps: A review , 2005 .

[29]  George H. Weiss,et al.  On the asymptotic behavior of the stochastic and deterministic models of an epidemic , 1971 .

[30]  Valerie Isham,et al.  Non‐Negative Matrices and Markov Chains , 1983 .

[31]  R. Lande,et al.  Extinction Thresholds in Demographic Models of Territorial Populations , 1987, The American Naturalist.

[32]  Central limit approximations for Markov population processes with countably many types , 2012, 1201.5538.

[33]  A. D. Barbour,et al.  Asymptotic behavior of a metapopulation model , 2005 .

[34]  O. Kallenberg.  Foundations of Modern Probability , 2021, Probability Theory and Stochastic Modelling.

[35]  M. Sheaves.  Large lutjanid and serranid fishes in tropical estuaries: Are they adults or juveniles? , 1995 .

[36]  Habitat destruction, habitat restoration and eigenvector-eigenvalue relations. , 2003, Mathematical biosciences.

[37]  D. Aldous,et al.  Two Applications of Urn Processes The Fringe Analysis of Search Trees and The Simulation of Quasi-Stationary Distributions of Markov Chains , 1988, Probability in the Engineering and Informational Sciences.

[38]  Otso Ovaskainen,et al.  Transient dynamics in metapopulation response to perturbation. , 2002, Theoretical population biology.

[39]  Otso Ovaskainen,et al.  Asymptotically exact analysis of stochastic metapopulation dynamics with explicit spatial structure. , 2006, Theoretical population biology.

[40]  A. Barbour,et al.  A law of large numbers approximation for Markov population processes with countably many types , 2009, 1001.0044.