zbMATH — the first resource for mathematics

Diffusion models for population dynamics incorporating individual behavior at boundaries: Applications to refuge design. (English) Zbl 0958.92028
The authors construct models for dispersal of a population which incorporate the response of individuals to interfaces between habitat types. The paper is divided into six sections. The first section is introductory in nature and contains a brief account of diffusion models for population dynamics. The construction of a diffusion model describing the density of the population is discussed in section 2. In section 3, the methods of analysis and some simple conclusions about parameter dependence are discussed. In section 4, the authors consider how the effectiveness of buffer zones depends on their size, quality, and the population’s response to the interface between the buffer zone and the refuge. Section 5 deals with the effect of sensitivity of habitat quality on the behavior of the average growth rate. Discussions and conclusions are discussed in section 6.

92D40 Ecology
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
92D25 Population dynamics (general)
Full Text: DOI
[1] Abramsky, Z.; van Dyne, G.M., Field studies and a simulation model of small mammals inhabiting a patchy environment, Oikos, 35, 80-92, (1980)
[2] Andow, D.A.; Kareiva, P.M.; Levin, S.A.; Okubo, A., Spread of invading organisms, Landscape ecol., 4, 177-188, (1990)
[3] Bach, C.E., Plant spatial pattern and herbivore population dynamics: plant factors affecting the movement patterns of a tropical cucurbit specialist (acalymma innubum), Ecology, 65, 175-190, (1984)
[4] Benson, D.L.; Sherratt, J.A.; Maini, P.K., Diffusion driven instability in an inhomogeneous domain, Bull. math. biol., 55, 365-384, (1993) · Zbl 0758.92003
[5] Cantrell, R.S.; Cosner, C., Diffusive logistic equations with indefinite weights: population models in disrupted environments, Proc. R. soc. Edinburgh sect. A, 112, 293-318, (1989) · Zbl 0711.92020
[6] Cantrell, R.S.; Cosner, C., The effects of spatial heterogeneity in population dynamics, J. math. biol., 29, 315-338, (1991) · Zbl 0722.92018
[7] Cantrell, R.S.; Cosner, C., Diffusive logistic equations with indefinite weights: population models in disrupted environments II, SIAM J. math. anal., 22, 1043-1064, (1991) · Zbl 0726.92024
[8] Cantrell, R.S.; Cosner, C., Should a park be an island?, SIAM J. appl. math., 53, 219-252, (1993) · Zbl 0811.92022
[9] Cantrell, R.S.; Cosner, C., Insular biogeographic theory and diffusion models in population dynamics, Theor. popul. biol., 45, 177-202, (1994) · Zbl 0798.92029
[10] Cantrell, R.S.; Cosner, C.; Hutson, V., Permanence in ecological systems with spatial heterogeneity, Proc. R. soc. Edinburgh sect. A, 123, 533-559, (1993) · Zbl 0796.92026
[11] Cantrell, R.S.; Cosner, C.; Hutson, V., Ecological models, permanence, and spatial heterogeneity, Rocky mountain J. math., 26, 1-36, (1996) · Zbl 0851.92019
[12] Cosner, C., Eigenvalue problems with indefinite weights and reaction – diffusion models in population dynamics, Reaction – diffusion equations, (1990), Clarendon Press Oxford, p. 47-137
[13] Dunning, J.B.; Stewart, D.J.; Danielson, B.J.; Noon, B.R.; Root, T.L.; Lamerson, R.H.; Stevens, E.E., Spatially explicit population models: current forms and future uses, Ecol. appl., 5, 3-11, (1995)
[14] Freedman, H.; Krisztin, T., Global stability in models of population dynamics with diffusion in patchy environments, Proc. R. soc. Edinburgh sect. A, 122, 69-84, (1992) · Zbl 0801.35053
[15] Freedman, H.; Wu, J., Steady-state analysis in a model for population diffusion in a multipatch environment, Nonlinear anal. T.M.A., 18, 517-542, (1992) · Zbl 0760.92021
[16] Goszczynski, J., Density estimation for an urban population of the field mouse, Acta theoriol., 24, 417-419, (1979)
[17] Goszczynski, J., Penetration of mammals over urban Green spaces in Warsaw, Acta theoriol., 24, 419-423, (1979)
[18] Harrison, J.M.; Shepp, L.A., On skew Brownian motion, Ann. probab., 9, 309-313, (1981) · Zbl 0462.60076
[19] Holmes, E.E.; Lewis, M.A.; Banks, J.E.; Veit, R.R., Partial differential equations in ecology: spatial interactions and population dynamics, Ecology, 75, 17-29, (1994)
[20] Ito, K.; McKean, H.P., Diffusion processes and their sample paths, (1965), Springer-Verlag New York · Zbl 0127.09503
[21] Janzen, D.H., No park is an island: increase in interference from outside as park side decreases, Oikos, 41, 402-410, (1983)
[22] Janzen, D.H., The eternal external threat, ()
[23] Kareiva, P., Finding and losing host plants by phyllotreta: patch size and surrounding habitat, Ecology, 66, 1809-1816, (1985)
[24] Kareiva, P., Population dynamics in spatially complex environments: theory and data, Philos. trans. R. soc. London ser. B, 330, 175-190, (1990)
[25] Kierstead, H.; Slobodkin, L.B., The size of water masses containing plankton Bloom, J. mar. res., 12, 141-147, (1953)
[26] Ludwig, D.; Aronson, D.G.; Weinberger, H.F., Spatial patterning of the spruce budworm, J. math. biol., 8, 217-258, (1979) · Zbl 0412.92020
[27] Murray, J.D.; Sperb, R.P., Minimum domains for spatial patterns in a class of reaction – diffusion equations, J. math. biol., 18, 169-184, (1983) · Zbl 0526.92013
[28] Okubo, A., Diffusion and ecological problems: mathematical models, (1980), Springer-Verlag Berlin · Zbl 0422.92025
[29] Okubo, A.; Maini, P.K.; Williamson, M.H.; Murray, J.D., On the spatial spread of the grey squirrel in britain, Proc. R. soc. London ser. B, 238, 113-125, (1989)
[30] Pacala, S.; Roughgarden, J., Spatial heterogeneity and interspecific competition, Theor. popul. biol., 21, 92-113, (1982) · Zbl 0492.92017
[31] Pulliam, R., Sources, sinks, and population regulation, Am. natur., 132, 652-661, (1988)
[32] Shigesada, N.; Kawasaki, K.; Teramoto, E., Travelling periodic waves in heterogeneous environments, Theor. popul. biol., 30, 143-160, (1986) · Zbl 0591.92026
[33] Skellam, J.G., Random dispersal in theoretical populations, Biometrika, 38, 196-218, (1951) · Zbl 0043.14401
[34] Stamps, J.A.; Buechner, M.; Krishnan, V.V., The effects of edge permeability and habitat geometry on emigration from patches of habitat, Am. natur., 129, 533-552, (1987)
[35] Stamps, J.A.; Buechner, M.; Krishnan, V.V., The effects of habitat geometry on territorial defense costs: intruder pressure in bounded habitats, Am. zool., 27, 307-325, (1987)
[36] Strauss, W., Partial differential equations: an introduction, (1992), Wiley New York · Zbl 0817.35001
[37] Taira, K., Diffusion processes and partial differential equations, (1988), Harcourt Brace Jovanovich New York · Zbl 0652.35003
[38] Walsh, J.B., A diffusion with discontinuous local time, Astérisque, 52-53, 37-45, (1978)
[39] Wegner, J.F.; Merriam, G., Movements by birds and small mammals between a wood and adjoining farmland habitats, J. appl. ecol., 16, 349-357, (1979)
[40] Wiens, J.A.; Stenseth, N.C.; van Horne, B.; Ims, R.A., Ecological mechanisms and landscape ecology, Oikos, 66, 369-380, (1993)
[41] Yahner, R.H., Population dynamics of small mammals in farmstead shelterbelts, J. mammal., 64, 380-386, (1983)
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.