A note on interacting populations that disperse to avoid crowding
Authors:
Morton E. Gurtin and A. C. Pipkin
Journal:
Quart. Appl. Math. 42 (1984), 87-94
MSC:
Primary 92A15; Secondary 35Q99
DOI:
https://doi.org/10.1090/qam/736508
MathSciNet review:
736508
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Abstract: In this note we derive partial differential equations for populations that disperse to avoid crowding, paying particular attention to situations in which the ease of dispersal is not uniform among individuals. We develop equations for the dispersal of a finite number of interacting biological groups and for a single age-structured group, and we give conditions under which the latter equations reduce to the former. In all cases the equations generalize the classical porous flow equation—a degenerate parabolic equation that exhibits a myriad of interesting effects. For the special case of two groups we deduce a simple solution in which the species remain segregated for all time.
M. Morisita, Dispersal and population density of a waterstrider, Gerris lacustris $L$., Contribut. Physiol. Ecol. Kyoto Univ. No. 65, 1–149 (Japanese)
Y. Ito, The growth form of populations in some aphids, with special reference to the relation between population density and movements, Res. Popul. Ecol. 1, 36–48 (Japanese with English summary)
T. Kono, Time-dispersion curve: experimental studies on the dispersion of insects (2), Res. Popul. Ecol. 1, 109–118 (Japanese with English summary)
S. Watanabe, S. Utida, and T. Yosida, Dispersion of insect and change of distribution type in its process: experimental studies on the dispersion of insects (1), Res. Popul. Ecol. 1, 94–108 (Japanese with English summary)
G. I. Baranblatt, On certain non-stationary motions of liquids and gases in porous media, Prikl. Mat. Mekh. 16, 67–78
M. Morisita, Dispersion and population pressure: experimental studies on the population density of an ant-lion, Glenuroides japonicus M’L (2), Japanese J. Ecol. 4, 71–79
- R. E. Pattle, Diffusion from an instantaneous point source with a concentration-dependent coefficient, Quart. J. Mech. Appl. Math. 12 (1959), 407–409. MR 114505, DOI https://doi.org/10.1093/qjmam/12.4.407
O. A. Oleinik, On some degenerate quasilinear equations, Istituto Nazionale di Alta Matematica Seminari 1962–1963, Cremonese, Rome, 1965
- D. G. Aronson, Regularity propeties of flows through porous media, SIAM J. Appl. Math. 17 (1969), 461–467. MR 247303, DOI https://doi.org/10.1137/0117045
E. A. Carl, Population control in arctic ground squirrels, Ecology 52, 395–413
W. S. C. Gurney and R. M. Nisbet, The regulation of inhomogeneous populations, J. Theor. Biol. 52, 441–457
S. A. Levin, Population dynamic models in heterogeneous environments, An. Rev. Ecol. Systematics, 7 287–310
- Morton E. Gurtin and Richard C. MacCamy, On the diffusion of biological populations, Math. Biosci. 33 (1977), no. 1-2, 35–49. MR 682594, DOI https://doi.org/10.1016/0025-5564%2877%2990062-1
- Ross McMurtrie, Persistence and stability of single-species and prey-predator systems in spatially heterogeneous environments, Math. Biosci. 39 (1978), no. 1-2, 11–51. MR 490056, DOI https://doi.org/10.1016/0025-5564%2878%2990026-3
- Nanako Shigesada, Kohkichi Kawasaki, and Ei Teramoto, Spatial segregation of interacting species, J. Theoret. Biol. 79 (1979), no. 1, 83–99. MR 540951, DOI https://doi.org/10.1016/0022-5193%2879%2990258-3
- Nanako Shigesada, Spatial distribution of dispersing animals, J. Math. Biol. 9 (1980), no. 1, 85–96. MR 648847, DOI https://doi.org/10.1007/BF00276037
- Akira Okubo, Diffusion and ecological problems: mathematical models, Biomathematics, vol. 10, Springer-Verlag, Berlin-New York, 1980. An extended version of the Japanese edition, Ecology and diffusion; Translated by G. N. Parker. MR 572962
- L. A. Peletier, The porous media equation, Applications of nonlinear analysis in the physical sciences (Bielefeld, 1979), Surveys Reference Works Math., vol. 6, Pitman, Boston, Mass.-London, 1981, pp. 229–241. MR 659697
R. M. Nisbet and W. S. C. Gurney, Modeling fluctuating populations, Wiley, New York
- Stavros N. Busenberg and Curtis C. Travis, Epidemic models with spatial spread due to population migration, J. Math. Biol. 16 (1983), no. 2, 181–198. MR 694493, DOI https://doi.org/10.1007/BF00276056
M. Morisita, Dispersal and population density of a waterstrider, Gerris lacustris $L$., Contribut. Physiol. Ecol. Kyoto Univ. No. 65, 1–149 (Japanese)
Y. Ito, The growth form of populations in some aphids, with special reference to the relation between population density and movements, Res. Popul. Ecol. 1, 36–48 (Japanese with English summary)
T. Kono, Time-dispersion curve: experimental studies on the dispersion of insects (2), Res. Popul. Ecol. 1, 109–118 (Japanese with English summary)
S. Watanabe, S. Utida, and T. Yosida, Dispersion of insect and change of distribution type in its process: experimental studies on the dispersion of insects (1), Res. Popul. Ecol. 1, 94–108 (Japanese with English summary)
G. I. Baranblatt, On certain non-stationary motions of liquids and gases in porous media, Prikl. Mat. Mekh. 16, 67–78
M. Morisita, Dispersion and population pressure: experimental studies on the population density of an ant-lion, Glenuroides japonicus M’L (2), Japanese J. Ecol. 4, 71–79
R. E. Pattle, Diffusion from an instantaneous point source with concentration-dependent coefficient, Q. J. Mech. Appl. Math. 12, 407–409
O. A. Oleinik, On some degenerate quasilinear equations, Istituto Nazionale di Alta Matematica Seminari 1962–1963, Cremonese, Rome, 1965
D. G. Aronson, Regularity properties of flows through porous media, SIAM J. Appl. Math. 17, 461–467
E. A. Carl, Population control in arctic ground squirrels, Ecology 52, 395–413
W. S. C. Gurney and R. M. Nisbet, The regulation of inhomogeneous populations, J. Theor. Biol. 52, 441–457
S. A. Levin, Population dynamic models in heterogeneous environments, An. Rev. Ecol. Systematics, 7 287–310
M. E. Gurtin and R. C. MacCamy, On the diffusion of biological populations, Math. Biosci. 33, 35–49
R. McMurtrie, Persistence and stability of single-species and prey-predator systems in spatially heterogeneous environments, Math. Biosci. 39, 11–51
N. Shigesada, K. Kawasaki, and E. Teramoto, Spatial segregation of interacting species, J. Theor. Biol. 79, 83–99
N. Shigesada, Spatial distribution of dispersing animals, J. Math. Biol. 9, 85–96
A. Okubo, Diffusion and ecological problems: Mathematical models, Biomathematics 10, Springer-Verlag, Berlin
L. A. Peletier, The porous media equation, Application of Nonlinear Analysis in the Physical Sciences (Eds. Amann, Bazley, Kirchgassner), 229–241, Pitman, Belmont
R. M. Nisbet and W. S. C. Gurney, Modeling fluctuating populations, Wiley, New York
S. N. Busenberg and C. C. Travis, Epidemic models with spatial spread due to population migration, J. Math. Biol. 16 (1983) 181–198
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© Copyright 1984
American Mathematical Society