Stochastic models in population biology and their deterministic analogs.

We introduce a class of stochastic population models based on "patch dynamics." The size of the patch may be varied, and this allows one to quantify the departures of these stochastic models from various mean-field theories, which are generally valid as the patch size becomes very large. These models may be used to formulate a broad range of biological processes in both spatial and nonspatial contexts. Here, we concentrate on two-species competition. We present both a mathematical analysis of the patch model, in which we derive the precise form of the competition mean-field equations (and their first-order corrections in the nonspatial case), and simulation results. These mean-field equations differ, in some important ways, from those which are normally written down on phenomenological grounds. Our general conclusion is that mean-field theory is more robust for spatial models than for a single isolated patch. This is due to the dilution of stochastic effects in a spatial setting resulting from repeated rescue events mediated by interpatch diffusion. However, discrete effects due to modest patch sizes lead to striking deviations from mean-field theory even in a spatial setting.

[1]  S. R. Bernard.  An urn model study of variability within a compartment , 1977 .

[2]  Peter Donnelly,et al.  The asymptotic behavior of an urn model arising in population genetics , 1996 .

[3]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1967 .

[4]  W. Wilson.  Lotka's game in predator-prey theory: linking populations to individuals. , 1996, Theoretical population biology.

[5]  Kenneth C. W. Kammeyer,et al.  An introduction to population , 1974 .

[6]  Y. Fu,et al.  Probability of a segregating pattern in a sample of DNA sequences. , 1998, Theoretical population biology.

[7]  David R. Appleton,et al.  Modelling Biological Populations in Space and Time , 1993 .

[8]  Boris M. Smirnov,et al.  Computer simulation studies in condensed matter physics , 1989 .

[9]  The mathematical beauty of physics , 1997 .

[10]  Ricard V Solé,et al.  Self-organized instability in complex ecosystems. , 2002, Philosophical transactions of the Royal Society of London. Series B, Biological sciences.

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

[12]  Per Arne Rikvold,et al.  Fluctuations and correlations in an individual-based model of biological coevolution , 2004 .

[13]  David P. Landau,et al.  Computer Simulation Studies in Condensed Matter Physics , 1988 .

[14]  Simon A. Levin,et al.  Stochastic Spatial Models: A User's Guide to Ecological Applications , 1994 .

[15]  F. Hoppe.  Pólya-like urns and the Ewens' sampling formula , 1984 .

[16]  R. Solé,et al.  Mean-field stochastic theory for species-rich assembled communities. , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[17]  R. Zia,et al.  Nonequilibrium phase transitions in a simple three-state lattice gas , 1997 .

[18]  F. Brauer,et al.  Mathematical Models in Population Biology and Epidemiology , 2001 .

[19]  L. Shenton.  A reinforcement-depletion urn problem—I. Basic theory , 1981 .

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

[21]  Simplest dynamic models of the global vegetation pattern , 1999 .

[22]  L. Berec.  Techniques of spatially explicit individual-based models: construction, simulation, and mean-field analysis , 2002 .

[23]  W. Ebeling.  Stochastic Processes in Physics and Chemistry , 1995 .

[24]  S. Orszag,et al.  "Critical slowing down" in time-to-extinction: an example of critical phenomena in ecology. , 1998, Journal of theoretical biology.

[25]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1951 .

[26]  Sebastian J. Schreiber,et al.  Urn Models, Replicator Processes, and Random Genetic Drift , 2001, SIAM J. Appl. Math..

[27]  R. Zia,et al.  FLUCTUATIONS AND CORRELATIONS IN POPULATION MODELS WITH AGE STRUCTURE , 2000, cond-mat/0001172.

[28]  P. Donnelly,et al.  Partition structures, Polya urns, the Ewens sampling formula, and the ages of alleles. , 1986, Theoretical population biology.

[29]  Beate Schmittmann,et al.  Onset of Spatial Structures in Biased Diffusion of Two Species , 1992 .

[30]  D. Landau,et al.  Computer Simulation Studies in Condensed-Matter Physics XIV , 2002 .

[31]  Rick Durrett,et al.  Stochastic Spatial Models , 1999, SIAM Rev..

[32]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[33]  R. Solé,et al.  Scaling in a network model of a multispecies ecosystem , 2000 .

[34]  F. Hoppe,et al.  The sampling theory of neutral alleles and an urn model in population genetics , 1987, Journal of mathematical biology.

[35]  M. Kimura,et al.  An introduction to population genetics theory , 1971 .

[36]  E. C. Pielou.  An introduction to mathematical ecology , 1970 .

[37]  Philippe Robert,et al.  On generalized Pólya urn models , 2000 .

[38]  W. Wilson.  Resolving Discrepancies between Deterministic Population Models and Individual‐Based Simulations , 1998, The American Naturalist.

[39]  A. McKane,et al.  Extinction dynamics in mainland-island metapopulations: An N-patch stochastic model , 2002, Bulletin of mathematical biology.

[40]  S. Orszag,et al.  Nucleation and relaxation from meta-stability in spatial ecological models. , 1999, Journal of theoretical biology.

[41]  Norman L. Johnson,et al.  Urn models and their application , 1977 .

[42]  M. Burrows,et al.  Modelling biological populations in space and time: Cambridge studies in mathematical biology: 11 , 1993 .

[43]  L. Peliti.  Path integral approach to birth-death processes on a lattice , 1985 .

[44]  David A. Rand,et al.  Invasion, stability and evolution to criticality in spatially extended, artificial host—pathogen ecologies , 1995, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[45]  W. Wilson,et al.  Dynamics of Age-Structured and Spatially Structured Predator-Prey Interactions: Individual-Based Models and Population-Level Formulations , 1993, The American Naturalist.

[46]  William Gurney,et al.  Modelling fluctuating populations , 1982 .

[47]  BruceE. Dunham.  Fluctuation Theory for Moran's Genetics Model☆ , 1997 .

[48]  John Maynard Smith.  Models in ecology , 1974 .

[49]  R. Blin-stoyle.  Elementary Particle Physics , 1965, Nature.

[50]  Andrew G. Glen,et al.  APPL , 2001 .

[51]  J. Izen,et al.  Elementary Particle Physics , 2018 .

[52]  William G. Wilson,et al.  Mobility versus density-limited predator-prey dynamics on different spatial scales , 1991, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[53]  N. Kampen,et al.  Stochastic processes in physics and chemistry , 1981 .

[54]  B. M. Fulk.  MATH , 1992 .

[55]  L. Shenton.  A reinforcement depletion urn problem—II. Application and generalization , 1983 .