×

Population diffusion in a two-patch environment. (English) Zbl 0672.92021

Summary: A model of a single-species population diffusing in a two-patch environment is proposed. It is shown that there exists a positive, monotonic, continuous steady-state solution with continuous flux, in the cases of both reservoir and no-flux boundary conditions, that is asymptotically stable. In the case of patches with equal carrying capacities, it is shown that the uniform steady state is globally asymptotically stable.

MSC:

92D40 Ecology
35Q99 Partial differential equations of mathematical physics and other areas of application
35B35 Stability in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Allen, L.J., Persistence, extinction and critical patch number for island populations, J. math. biol., 24, 617-625, (1987) · Zbl 0603.92019
[2] Bailey, P.; Shampine, L.F.; Waltman, P.E., Nonlinear two point boundary value problems, (1968), Academic New York · Zbl 0169.10502
[3] Bernfeld, S.R.; Lakshmikantham, V., An introduction to nonlinear boundary value problems, (1974), Academic New York · Zbl 0286.34018
[4] Cantrell, R.S.; Cosner, C., Diffusive logistic equations with indefinite weights: population models in disrupted environments, (1989), (preprint) · Zbl 0711.92020
[5] Freedman, H.I., Deterministic mathematical models in population ecology, (1987), HIFR Consulting Ltd Edmonton · Zbl 0448.92023
[6] Freedman, H.I.; Shukla, J.B., The effect of a predator resource on a diffusive predator prey system, Natural resources modeling, (1989), (in press) · Zbl 0850.92061
[7] Hallam, T.G., A temporal study of diffusion effects on a population modelled by quadratic growth, Nonlin. anal. theory methods appl., 3, 123-133, (1979) · Zbl 0419.35048
[8] Kierstead, H.; Slobodkin, L.B., The size of water masses containing plankton blooms, J. mar. res., 12, 141-147, (1953)
[9] Levin, S.A., Population models and community structure in heterogeneous environments, (), 295-321
[10] McMurtrie, R., Persistence and stability of single-species and prey-predator systems in spatially heterogeneous environments, Math. biosci., 39, 11-51, (1978) · Zbl 0384.92011
[11] Pacala, S.W.; Roughgarden, J., Spatial heterogeneity and interspecific competition, Theor. pop. biol., 21, 92-113, (1982) · Zbl 0492.92017
[12] Pao, C.V., Coexistence and stability of a competition-diffusion system in population dynamics, J. math. anal. appl., 83, 54-76, (1981) · Zbl 0479.92013
[13] Shigesada, N.; Roughgarden, J., The role of rapid dispersal in the population dynamics of competition, Theor. pop. biol., 21, 353-372, (1982) · Zbl 0494.92021
[14] Skellam, J.G., Random dispersal in theoretical populations, Biometrika, 36, 199-218, (1951) · Zbl 0043.14401
[15] Smith, O.L., The influence of environmental gradients on ecosystem stability, Am. nat., 116, 1-24, (1980)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.