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Persistence and extinction in single-species reaction-diffusion models

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Abstract

Single-species reaction-diffusion models are analyzed to determine the effect of various diffusion mechanisms on species persistence or extinction.

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Literature

  • Alikakos, N. D. 1979. “An application of the invariance principle to reaction-diffusion equations.”J. Diff. Eq. 33, 201–225.

    Article  MATH  MathSciNet  Google Scholar 

  • Allen, L. J. 1981. “Applications of differential inequalities to persistence and extinction problems for reaction-diffusion systems.” Ph.D. thesis, The University of Tennessee, Knoxville.

    Google Scholar 

  • Coddington, E. A. and N. Levinson. 1955.Theory of Ordinary Differential Equations. New York: McGraw-Hill.

    Google Scholar 

  • Goh, B. S. 1977. “Global stability in a many-species system.”Am. Nat. 111, 135–143.

    Article  Google Scholar 

  • Gopalsamy, K. and B. D. Aggarwala. 1980. “On the non-existence of periodic solutions of the reactive-diffusive Volterra system of equations.”J. theor. Biol. 82, 537–540.

    Article  MathSciNet  Google Scholar 

  • Gurney, W. S. C. and R. M. Nisbet. 1975. “The regulation of inhomogeneous populations.”J. theor. Biol. 52, 441–457.

    Article  Google Scholar 

  • Gurtin, M. E. and R. C. MacCamy. 1977. “On the diffusion of biological populations.”Math. Biosci. 33, 35–49.

    Article  MATH  MathSciNet  Google Scholar 

  • Hastings, A. 1978. “Global stability in Lotka-Volterra systems with diffusion.”J. math. Biol. 6, 163–168.

    Article  MATH  MathSciNet  Google Scholar 

  • Jorné, J. and S. Carmi. 1977. “Liapunov stability of the diffusive Lotka-Volterra equations.”Math. Biosci. 37, 51–61.

    Article  MATH  MathSciNet  Google Scholar 

  • — and U. N. Safriel. 1979. “Linear and non-linear diffusion models applied to the behavior of a population of intertidal snail.”J. theor. Biol. 79, 367–380.

    Article  Google Scholar 

  • Kierstead, H. and L. B. Slobodkin. 1953. “The size of water masses containing plankton blooms.”J. mar. Res. 12, 141–147.

    Google Scholar 

  • Kolata, G. B. 1974. “Theoretical ecology: beginnings of a predictive science.”Science, N. Y. 183, 401–402, 450.

    MathSciNet  Google Scholar 

  • Lakshmikantham, V. and S. Leela. 1969.Differential and Integral Inequalities Theory and Applications, Vol. 1. New York: Academic Press.

    Google Scholar 

  • Landahl, H. D. 1959. “A note on population growth under random dispersal.”Bull. math. Biophys. 21, 153–159.

    MathSciNet  Google Scholar 

  • Leung, A. 1978. “Limiting behaviour for a prey-predator model with diffusion and crowding effects.”J. math. Biol. 6, 87–93.

    MATH  MathSciNet  Google Scholar 

  • — and D. Clark. 1980. “Bifurcations and large-time asymptotic behavior for prey-predator reaction-diffusion equations with Dirichlet boundary data.”J. Diff. Eq. 35, 113–127.

    Article  MATH  MathSciNet  Google Scholar 

  • Levin, S. A. 1974. “Dispersion and population interactions.”Am. Nat. 108, 207–228.

    Article  Google Scholar 

  • — 1978. “Population models and community structure in heterogeneous environments.” InStudies in Mathematical Biology, Vol. II:Populations and Communities, Ed. S. A. Levin. Washington, D.C.: M.A.A.

    Google Scholar 

  • May, R. M. 1975. “Island biogeography and the design of wildlife preserves.”Nature, Lond. 254, 177–178.

    Article  Google Scholar 

  • McMurtrie, M. 1978. “Persistence and stability of single-species and prey-predator systems in spatially heterogeneous environments.”Math. Biosci. 39, 11–51.

    Article  MATH  MathSciNet  Google Scholar 

  • Murray, J. D. 1977.Lectures on Nonlinear-differential-Equation Models in Biology. Oxford: Clarendon Press.

    Google Scholar 

  • Okubo, A. 1980. “Diffusion and ecological problems: mathematical models.” InBiomathematics, Vol. 10, Eds. K. Krickelerg and S. A. Levin. Berlin: Springer-Verlag.

    Google Scholar 

  • Protter, M. H. and H. F. Weinberger. 1967.Maximum Principles in Differential Equations. Englewood Cliffs, NJ: Prentice-Hall.

    Google Scholar 

  • Rudin, W. 1974.Real and Complex Analysis, Second Edition. New York: McGraw-Hill.

    Google Scholar 

  • Sansone, G. and R. Conti. 1964.Nonlinear Differential Equations. Oxford: Pergamon Press.

    Google Scholar 

  • Simberloff, D. S. and L. G. Abele. 1976. “Island biogeography theory and conservation practice.”Science, N.Y. 191, 285–286.

    Google Scholar 

  • Skellam, J. G. 1951. “Random dispersal in theoretical populations.”Biometrika 38, 196–218.

    Article  MATH  MathSciNet  Google Scholar 

  • Yodzis, P. 1978. “Competition for space and the structure of ecological communities.” InLecture Notes in Biomathematics, Ed. S. Levin. Berlin: Springer-Verlag.

    Google Scholar 

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Allen, L.J.S. Persistence and extinction in single-species reaction-diffusion models. Bltn Mathcal Biology 45, 209–227 (1983). https://doi.org/10.1007/BF02462357

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  • DOI: https://doi.org/10.1007/BF02462357

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