Spatially-explicit matrix models

Abstract.This paper is concerned with mathematical analysis of the ‘critical domain-size’ problem for structured populations. Space is introduced explicitly into matrix models for stage-structured populations. Movement of individuals is described by means of a dispersal kernel. The mathematical analysis investigates conditions for existence, stability and uniqueness of equilibrium solutions as well as some bifurcation behaviors. These mathematical results are linked to species persistence or extinction in connected habitats of different sizes or fragmented habitats; hence the framework is given for application of such models to ecology. Several approximations which reduce the complexity of integrodifference equations are given. A simple example is worked out to illustrate the analytical results and to compare the behavior of the integrodifference model to that of the approximations.

[1]  Chris Cosner,et al.  On the effects of spatial heterogeneity on the persistence of interacting species , 1998 .

[2]  M. Kot,et al.  Discrete-time travelling waves: Ecological examples , 1992, Journal of mathematical biology.

[3]  C. M. Peters,et al.  Population ecology and management of forest fruit trees in Peruvian Amazonia. , 1990 .

[4]  J. G. Skellam.  Random dispersal in theoretical populations , 1951, Biometrika.

[5]  M. A. Lewis,et al.  Integrodifference models for persistence in fragmented habitats , 1997 .

[6]  M. A. Krasnoselʹskii,et al.  Geometrical Methods of Nonlinear Analysis , 1984 .

[7]  Hal Caswell,et al.  DEMOGRAPHY AND DISPERSAL: CALCULATION AND SENSITIVITY ANALYSIS OF INVASION SPEED FOR STRUCTURED POPULATIONS , 2000 .

[8]  Sadri Hassani,et al.  Nonlinear Dynamics and Chaos , 2000 .

[9]  Otso Ovaskainen,et al.  The metapopulation capacity of a fragmented landscape , 2000, Nature.

[10]  Melvyn S. Berger,et al.  Geometrical Methods of Nonlinear Analysis (M. A. Krasnosel’skii and P. P. Zabreiko) , 1985 .

[11]  Chris Cosner,et al.  Should a Park Be an Island? , 1993, SIAM J. Appl. Math..

[12]  Jim M Cushing,et al.  Phase switching in population cycles , 1998, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[13]  Donald L. DeAngelis,et al.  Stability and return times of Leslie matrices with density-dependent survival: applications to fish populations☆ , 1980 .

[14]  S. R. Broadbent,et al.  The Random Walk of Trichostrongylus retortaeformis , 1953 .

[15]  Mark Kot,et al.  Dispersal and Pattern Formation in a Discrete-Time Predator-Prey Model , 1995 .

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

[17]  M. Kot,et al.  Discrete-time growth-dispersal models , 1986 .

[18]  Roger Lui,et al.  Existence and stability of travelling wave solutions of a nonlinear integral operator , 1983 .

[19]  M. Andersen.  Properties of some density-dependent integrodifference equation population models. , 1991, Mathematical biosciences.

[20]  M. Kot,et al.  Integrodifference equations, Allee effects, and invasions , 2002, Journal of mathematical biology.

[21]  P. H. Leslie.  On the use of matrices in certain population mathematics. , 1945, Biometrika.

[22]  V. V. Krishnan,et al.  The Effects of Edge Permeability and Habitat Geometry on Emigration from Patches of Habitat , 1987, The American Naturalist.

[23]  L. Lefkovitch.  The study of population growth in organisms grouped by stages , 1965 .

[24]  Douglas P. Hardin,et al.  Asymptotic properties of a continuous-space discrete-time population model in a random environment , 1988 .

[25]  Botsford,et al.  Dependence of sustainability on the configuration of marine reserves and larval dispersal distance , 2001 .

[26]  Mark Kot,et al.  Elements of Mathematical Ecology: Frontmatter , 2001 .

[27]  Leo F. Boron,et al.  Positive solutions of operator equations , 1964 .

[28]  Richard R. Veit,et al.  Dispersal, Population Growth, and the Allee Effect: Dynamics of the House Finch Invasion of Eastern North America , 1996, The American Naturalist.

[29]  D. Hardin,et al.  Dispersion population models discrete in time and continuous in space , 1990, Journal of mathematical biology.

[30]  Chris Cosner,et al.  How Habitat Edges Change Species Interactions , 1999, The American Naturalist.

[31]  H. Caswell,et al.  Density-dependent vital rates and their population dynamic consequences , 2000, Journal of mathematical biology.

[32]  Hans F. Weinberger,et al.  Long-Time Behavior of a Class of Biological Models , 1982 .

[33]  P. Holgate,et al.  Matrix Population Models. , 1990 .

[34]  Douglas P. Hardin,et al.  A comparison of dispersal strategies for survival of spatially heterogeneous populations , 1988 .

[35]  R. Goodland.  Alternatives to deforestation: edited by A.B. Anderson, Columbia University Press, 1992. $24.00 pbk (xiv + 281 pages) ISBN 0 231 06893 X , 1993 .