Asymptotically exact analysis of stochastic metapopulation dynamics with explicit spatial structure.

We describe a mathematically exact method for the analysis of spatially structured Markov processes. The method is based on a systematic perturbation expansion around the deterministic, non-spatial mean-field theory, using the theory of distributions to account for space and the underlying stochastic differential equations to account for stochasticity. As an example, we consider a spatial version of the Levins metapopulation model, in which the habitat patches are distributed in the d-dimensional landscape Rd in a random (but possibly correlated) manner. Assuming that the dispersal kernel is characterized by a length scale L, we examine how the behavior of the metapopulation deviates from the mean-field model for a finite but large L. For example, we show that the equilibrium fraction of occupied patches is given by p(0)+c/L(d)+O(L(-3d/2)), where p(0) is the equilibrium state of the Levins model and the constant c depends on p(0), the dispersal kernel, and the structure of the landscape. We show that patch occupancy can be increased or decreased by spatial structure, but is always decreased by stochasticity. Comparison with simulations show that the analytical results are not only asymptotically exact (as L-->infinity), but a good approximation also when L is relatively small.

[1]  Anthony W. King,et al.  Dispersal success on spatially structured landscapes: when do spatial pattern and dispersal behavior really matter? , 2002 .

[2]  Atte Moilanen,et al.  PATCH OCCUPANCY MODELS OF METAPOPULATION DYNAMICS: EFFICIENT PARAMETER ESTIMATION USING IMPLICIT STATISTICAL INFERENCE , 1999 .

[3]  R. Durrett,et al.  The Importance of Being Discrete (and Spatial) , 1994 .

[4]  Matt J Keeling,et al.  Contact tracing and disease control , 2003, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[5]  Jordi Bascompte,et al.  Extinction thresholds: insights from simple models , 2003 .

[6]  Peter Chesson,et al.  How species with different regeneration niches coexist in patchy habitats with local disturbances , 1995 .

[7]  Lenore Fahrig,et al.  EFFECT OF HABITAT FRAGMENTATION ON THE EXTINCTION THRESHOLD: A SYNTHESIS* , 2002 .

[8]  W. Godwin.  Article in Press , 2000 .

[9]  Otso Ovaskainen,et al.  How much does an individual habitat fragment contribute to metapopulation dynamics and persistence? , 2003, Theoretical population biology.

[10]  J. Matis,et al.  On stochastic logistic population growth models with immigration and multiple births. , 2004, Theoretical population biology.

[11]  P. Turchin.  Quantitative analysis of movement : measuring and modeling population redistribution in animals and plants , 1998 .

[12]  Otso Ovaskainen,et al.  HABITAT-SPECIFIC MOVEMENT PARAMETERS ESTIMATED USING MARK–RECAPTURE DATA AND A DIFFUSION MODEL , 2004 .

[13]  Benjamin M Bolker,et al.  Combining endogenous and exogenous spatial variability in analytical population models. , 2003, Theoretical population biology.

[14]  Ulf Dieckmann,et al.  On moment closures for population dynamics in continuous space. , 2004, Journal of theoretical biology.

[15]  Oscar E. Gaggiotti,et al.  Ecology, genetics, and evolution of metapopulations , 2004 .

[16]  S. Ellner,et al.  Pair approximation for lattice models with multiple interaction scales. , 2001, Journal of theoretical biology.

[17]  B K Szymanski,et al.  Host spatial heterogeneity and the spread of vector-borne infection. , 2001, Theoretical population biology.

[18]  A. Hastings,et al.  Resistance may be futile: dispersal scales and selection for disease resistance in competing plants. , 2003, Journal of theoretical biology.

[19]  B. Bolker,et al.  Using Moment Equations to Understand Stochastically Driven Spatial Pattern Formation in Ecological Systems , 1997, Theoretical population biology.

[20]  Simon A. Levin,et al.  The Geometry of Ecological Interactions: Moment Methods for Ecological Processes in Continuous Space , 2000 .

[21]  Benjamin M. Bolker,et al.  3 – Continuous-Space Models for Population Dynamics , 2004 .

[22]  H. B. Wilson,et al.  Deterministic Limits to Stochastic Spatial Models of Natural Enemies , 2002, The American Naturalist.

[23]  David E. Hiebeler,et al.  Populations on fragmented landscapes with spatially structured heterogeneities : Landscape generation and local dispersal , 2000 .

[24]  Alun L Lloyd,et al.  Estimating variability in models for recurrent epidemics: assessing the use of moment closure techniques. , 2004, Theoretical population biology.

[25]  Otso Ovaskainen,et al.  Extinction threshold in metapopulation models , 2003 .

[26]  Hal Caswell,et al.  Habitat fragmentation and extinction thresholds on fractal landscapes , 1999 .

[27]  J. Gamarra,et al.  Metapopulation Ecology , 2007 .

[28]  H. Hinrichsen.  Non-equilibrium critical phenomena and phase transitions into absorbing states , 2000, cond-mat/0001070.

[29]  R. Lande,et al.  MIGRATION AND SPATIOTEMPORAL VARIATION IN POPULATION DYNAMICS IN A HETEROGENEOUS ENVIRONMENT , 2002 .

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

[31]  M. Whitlock,et al.  The effective size of a subdivided population. , 1997, Genetics.

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

[33]  B M Bolker,et al.  Analytic models for the patchy spread of plant disease , 1999, Bulletin of mathematical biology.

[34]  Otso Ovaskainen,et al.  Biased movement at a boundary and conditional occupancy times for diffusion processes , 2003, Journal of Applied Probability.

[35]  Otso Ovaskainen,et al.  Metapopulation theory for fragmented landscapes. , 2003, Theoretical population biology.

[36]  R. Lande,et al.  Spatial Scale of Population Synchrony: Environmental Correlation versus Dispersal and Density Regulation , 1999, The American Naturalist.

[37]  Ulf Dieckmann,et al.  Relaxation Projections and the Method of Moments , 1999 .

[38]  S. Ellner,et al.  Pair-edge approximation for heterogeneous lattice population models. , 2003, Theoretical population biology.

[39]  Ulf Dieckmann,et al.  The Geometry of Ecological Interactions: Simplifying Spatial Complexity , 2000 .

[40]  Ilkka Hanski,et al.  Long‐Term Dynamics in a Metapopulation of the American Pika , 1998, The American Naturalist.

[41]  B. Charlesworth,et al.  Effects of metapopulation processes on measures of genetic diversity. , 2000, Philosophical transactions of the Royal Society of London. Series B, Biological sciences.

[42]  Jean-Baptiste Ferdy,et al.  Extinction times and moment closure in the stochastic logistic process. , 2004, Theoretical population biology.

[43]  Ali Esmaili,et al.  Probability and Random Processes , 2005, Technometrics.

[44]  C.J.F. ter Braak,et al.  Application of Stochastic Patch Occupancy Models to Real Metapopulations , 2004 .

[45]  Jordi Bascompte,et al.  Metapopulation models for extinction threshold in spatially correlated landscapes. , 2002, Journal of theoretical biology.

[46]  R. Estrada,et al.  Introduction to the Theory of Distributions , 1994 .

[47]  Jacqueline McGlade,et al.  Advanced Ecological Theory , 1999 .

[48]  Steinar Engen,et al.  The Spatial Scale of Population Fluctuations and Quasi‐Extinction Risk , 2002, The American Naturalist.

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

[50]  G. Grimmett,et al.  Probability and random processes , 2002 .