Abstract
We consider effects of competition for space in a heterogeneous environment, making use of nonlinear interaction-diffusion equations. Competition for space is assumed to mean mutual repulsive interactions that force other individuals to disperse from a crowded region. In other words, we are concerned with density-dependent dispersal forced by population pressures. Spatial heterogeneity is incorporated in the growth rates, and the environment is assumed to have a favorable habitat for two populations surrounded by largely hostile regions. Space-independent migration rates are assumed. We ignore the usual density-dependence in the growth rates to focus our attention on density-dependence in the migration rates. Our main conclusion is that two populations can coexist if the interspecific repulsive forces are weaker than the intraspecific ones. It is also emphasized that density-dependent dispersal in a heterogeneous environment is not always a stabilizing agent, and that either of two populations may become extinct by competition for space. Finally, the resemblance of our results to those from Lotka-Volterra competition equations is suggested.
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Aronson, D. G.: Some properties of the interface for a gas flow in porous media. In: Fasano, A., Primicerio, M. (eds.) Free boundary problems: theory and applications, vol. 1, pp. 150–159. Boston London: Pitman 1983
Aronson, D. G.: Bifurcation phenomena associated with nonlinear diffusion mechanisms. In: Fitzgibbon III., W. E. (ed.) Partial differential equations and dynamical systems (Research Notes in Mathematics, vol. 101, pp. 16–33) Boston London: Pitman 1984
Bertsch, M., Gurtin, M. E., Hilhorst, D., Peletier, L. A.: On interacting populations that disperse to avoid crowding; the effect of a sedentary colony. J. Math. Biol. 19, 1–12 (1984)
Bertsch, M., Gurtin, M. E., Hilhorst, D., Peletier, L. A.: On interacting populations that disperse to avoid crowding; preservation of segregation. J. Math. Biol. 23, 1–13 (1985)
Busenberg, S. N., Travis, C. C.: Epidemic models with spatial spread due to population migration. J. Math. Biol. 16, 181–198 (1983)
Gurney, W. S. C., Nisbet, R. M.: The regulation of inhomogeneous populations. J. theor. Biol. 52, 441–457 (1975).
Gurtin, M. E., MacCamy, R. C.: On the diffusion of biological populations. Math. Biosci. 33, 35–49 (1977)
Gurtin, M. E., Pipkin, A. C.: A note on interacting populations that disperse to avoid crowding. Q. Appl. Math. 42, 87–94 (1984)
Hosono, Y.: Traveling wave solutions for some density dependent diffusion equations. Jap. J. Appl. Math. 3, 163–196 (1986)
Hosono, Y.: Traveling waves for some biological systems with density dependent diffusion. Jap. J. Appl. Math. 4, 297–359 (1987)
Kawasaki, K., Teramoto, E.: Spatial pattern formation of prey-predator populations. J. theor. Biol. 8, 33–46 (1979)
Levin, S. A.: Dispersion and population interactions. Am. Nat. 108, 207–228 (1974)
Levin, S. A.: Models of population dispersal. In: Busenberg, S. N., Cooke, K. (eds.) Differential equations and applications in ecology, epidemics, and population problems, pp. 1–18. New York: Academic Press 1981
Mimura, M.: Stationary pattern of some density-dependent diffusion systems with competitive dynamics. Hiroshima Math. J. 11, 621–635 (1981)
Mimura, M., Kawasaki, K.: Spatial segregation in competitive interaction-diffusion equations. J. Math. Biol. 9, 49–64 (1980)
Mimura, M., Nishiura, Y., Tesei, A., Tsujikawa, T.: Coexistence problem for two competing species models with density-dependent diffusion. Hiroshima Math. J. 14, 425–449 (1984)
Namba, T.: Density-dependent dispersal and spatial distribution of a population. J. theor. Biol. 86, 351–363 (1980)
Namba, T., Mimura, M.: Spatial distributions of competing populations. J. theor. Biol. 87, 795–814 (1980)
Nisbet, R. M., Gurney, W. S. C.: Modeling fluctuating populations. Chichester: Wiley 1982
Okubo, A.: Diffusion and ecological problems: mathematical models (Biomathematics, vol. 10), Berlin Heidelberg New York: Springer 1980
Pacala, S., Roughgarden, J.: Spatial heterogeneity and interspecific competition. Theor. Popul. Biol. 21, 92–113 (1982)
Peletier, L. A., Tesei, A.: Diffusion in inhomogeneous media: localization and positivity. Ann. Mat. Pura Appl. 141, 307–330 (1985)
Pozio, M. A., Tesei, A.: On some nonlinear diffusion models in population dynamics. In; Capasso, V., Grosso, E., Paveri-Fontana, S. L. (eds.) Mathematics in biology and medicine (Lect. Notes Biomath., vol. 57, pp. 82–86) Berlin Heidelberg New York: Springer 1985
Pozio, M. A., Tesei, A.: Degenerate parabolic problems in population dynamics. Jap. J. Appl. Math. 2, 351–380 (1985)
Pozio, M. A., Tesei, A.: Support properties of solutions for a class of degenerate parabolic problems. Commun. Partial Differ. Equations 12, 47–76 (1987)
Shigesada, N.: Spatial distribution of dispersing animals. J. Math. Biol. 9, 85–96 (1980)
Shigesada, N.: Spatial distribution of rapidly dispersing animals in heterogeneous environments. In: Levin, S. A., Hallam, T. G. (eds.) Mathematical ecology (Lect. Notes Biomath., vol. 54, pp. 478–491) Berlin Heidelberg New York: Springer 1984
Shigesada, N., Kawasaki, K., Teramoto, E.: Spatial segregation of interacting species. J. theor. Biol. 79, 83–99 (1979)
Shigesada, N., Kawasaki, K., Teramoto, E.: Traveling periodic waves in heterogeneous environments. Theor. Popul. Biol. 30, 143–160 (1986)
Shigesada, N., Roughgarden, J.: The role of rapid dispersal in the population dynamics of competition. Theor. Pop. Biol. 21, 353–373 (1982)
Taylor, L. R., Taylor, R. A. J.: Insect migration as a paradigm for survival by movement. In: Swingland, I. R., Greenwood, P. J. (eds.) The ecology of animal movement, pp. 181–214. Oxford: Clarendon Press 1983
Teramoto, E., Seno, H.: Modeling of biological aggregation patterns. In: Ricciardi, L. M. (ed.) Biomathematics and related computational problems. London New York: Kluwer 1988
Yodzis, P.: Competition, mortality, and community structure. In: Diamond, J., Case, T. J. (eds.) Community ecology, pp. 480–491. New York: Harper & Row 1986
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Namba, T. Competition for space in a heterogeneous environment. J. Math. Biology 27, 1–16 (1989). https://doi.org/10.1007/BF00276077
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DOI: https://doi.org/10.1007/BF00276077