A model for the diffusion of populations in annular patchy environments
Authors:
G. Cui and H. I. Freedman
Journal:
Quart. Appl. Math. 57 (1999), 339-354
MSC:
Primary 92D25; Secondary 35Q80, 92D40
DOI:
https://doi.org/10.1090/qam/1686193
MathSciNet review:
MR1686193
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Abstract: A system of reaction-diffusion differential equations is utilized to model the diffusion of a population through annular patches with different carrying capacities. In the case of continuous solutions with continuous flux, it is shown that a unique, positive steady-state solution exists. In the case of radially symmetric initial conditions, it is shown that all solutions of the Cauchy problem approach this steady-state solution. Models involving nonsymmetric initial conditions are also considered.
- Shmuel Agmon, Lectures on elliptic boundary value problems, Van Nostrand Mathematical Studies, No. 2, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London, 1965. Prepared for publication by B. Frank Jones, Jr. with the assistance of George W. Batten, Jr. MR 0178246
- S. W. Ali and C. Cosner, On the uniqueness of the positive steady state for Lotka-Volterra models with diffusion, J. Math. Anal. Appl. 168 (1992), no. 2, 329–341. MR 1175992, DOI https://doi.org/10.1016/0022-247X%2892%2990161-6
S. W. Ali and C. Cosner, Methods for the effects of individual size and spatial scale on competition between species in heterogeneous environment, Math. Biosci. 127, 45–76 (1995)
G. Butkovskiy, Green’s Function and Transfer Functions Handbook, Ellis Horwood Ltd., West Sussex, 1982
- Robert Stephen Cantrell and Chris Cosner, Diffusive logistic equations with indefinite weights: population models in disrupted environments, Proc. Roy. Soc. Edinburgh Sect. A 112 (1989), no. 3-4, 293–318. MR 1014659, DOI https://doi.org/10.1017/S030821050001876X
- Robert Stephen Cantrell and Chris Cosner, Diffusive logistic equations with indefinite weights: population models in disrupted environments. II, SIAM J. Math. Anal. 22 (1991), no. 4, 1043–1064. MR 1112065, DOI https://doi.org/10.1137/0522068
R. S. Cantrell and C. Cosner, Insular biogeographic theory and diffusion models in population dynamics, Theor. Popu. Biol. 45, 177–202 (1994)
- Herbert I. Freedman, Deterministic mathematical models in population ecology, Monographs and Textbooks in Pure and Applied Mathematics, vol. 57, Marcel Dekker, Inc., New York, 1980. MR 586941
- H. I. Freedman and T. Krisztin, Global stability in models of population dynamics with diffusion. I. Patchy environments, Proc. Roy. Soc. Edinburgh Sect. A 122 (1992), no. 1-2, 69–84. MR 1190232, DOI https://doi.org/10.1017/S0308210500020977
- H. I. Freedman, J. B. Shukla, and Y. Takeuchi, Population diffusion in a two-patch environment, Math. Biosci. 95 (1989), no. 1, 111–123. MR 1001295, DOI https://doi.org/10.1016/0025-5564%2889%2990055-2
- H. I. Freedman and J. Wu, Steady-state analysis in a model for population diffusion in a multi-patch environment, Nonlinear Anal. 18 (1992), no. 6, 517–542. MR 1154478, DOI https://doi.org/10.1016/0362-546X%2892%2990208-V
- Avner Friedman, Partial differential equations, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1969. MR 0445088
- Afshin Ghoreishi and Roger Logan, Positive solutions of a class of biological models in a heterogeneous environment, Bull. Austral. Math. Soc. 44 (1991), no. 1, 79–94. MR 1120396, DOI https://doi.org/10.1017/S0004972700029488
- B. Gidas, Wei Ming Ni, and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), no. 3, 209–243. MR 544879
- Daniel Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin-New York, 1981. MR 610244
- Fritz John, Partial differential equations, 4th ed., Applied Mathematical Sciences, vol. 1, Springer-Verlag, New York, 1982. MR 831655
Z. Jiang and F. Yang, Numerical Solutions of the Outer Boundary Problem of Partial Differential Equations (Chinese), Tianjin Univ. Press, Tianjin, 1992
P. Kareiva, Population dynamics in spatially complex environments: Theory and data, Philos. Trans. Roy. Soc. London B330, 175 190 (1990)
- T. Krisztin and H. I. Freedman, Global stability in models of population dynamics with diffusion. II. Continuously varying environments, Rocky Mountain J. Math. 24 (1994), no. 1, 155–163. 20th Midwest ODE Meeting (Iowa City, IA, 1991). MR 1270033, DOI https://doi.org/10.1216/rmjm/1181072458
- Orlando Lopes, Radial and nonradial minimizers for some radially symmetric functionals, Electron. J. Differential Equations (1996), No. 03, approx. 14 pp.}, review=\MR{1375123},.
- A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR 710486
- S. W. Pacala and J. Roughgarden, Spatial heterogeneity and interspecific competition, Theoret. Population Biol. 21 (1982), no. 1, 92–113. MR 662524, DOI https://doi.org/10.1016/0040-5809%2882%2990008-9
- Murray H. Protter and Hans F. Weinberger, Maximum principles in differential equations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967. MR 0219861
- H. L. Royden, Real analysis, The Macmillan Co., New York; Collier-Macmillan Ltd., London, 1963. MR 0151555
- Nanako Shigesada and Jonathan Roughgarden, The role of rapid dispersal in the population dynamics of competition, Theoret. Population Biol. 21 (1982), no. 3, 353–372. MR 667337, DOI https://doi.org/10.1016/0040-5809%2882%2990023-5
S. Agmon, Lectures on Elliptic Boundary Value Problems, Van Nostrand, New York, 1965
S. W. Ali and C. Cosner, On the uniqueness of the positive steady-state for Lotka-Volterra models with diffusion, J. Math. Anal. Appl. 168, 329 341 (1992)
S. W. Ali and C. Cosner, Methods for the effects of individual size and spatial scale on competition between species in heterogeneous environment, Math. Biosci. 127, 45–76 (1995)
G. Butkovskiy, Green’s Function and Transfer Functions Handbook, Ellis Horwood Ltd., West Sussex, 1982
R. S. Cantrell and C. Cosner, Diffusive logistic equations with indefinite weights: population models in disrupted environments, Proc. Roy. Soc. Edinburgh A112, 293–318 (1989)
R. S. Cantrell and C. Cosner, Diffusive logistic equations with indefinite weights: population models in disrupted environments II, SIAM J. Math. Anal. 22, 1043–1064 (1991)
R. S. Cantrell and C. Cosner, Insular biogeographic theory and diffusion models in population dynamics, Theor. Popu. Biol. 45, 177–202 (1994)
H. I. Freedman, Deterministic Mathematical Models in Population Ecology, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 57, Marcel Dekker, New York, 1980
H. I. Freedman and T. Krisztin, Global stability in models of population dynamics with diffusion I. Patchy Environments, Proc. Roy. Soc. Edinburgh 122A, 69–84 (1992)
H. I. Freedman, J. B. Shukla, and Y. Takeuchi, Population diffusion in a two-patch environment, Math. Biosci. 95, 111–123 (1989)
H. I. Freedman and J. Wu, Steady-state analysis in a model for population diffusion in a multi-patch environment, Nonlinear Anal. Theor. Meth. Appl. 18, 517–542 (1992)
A. Friedman, Partial Differential Equations, Holt, Reinhart and Winston, New York, 1969
A. Goreiski and R. Logan, Positive solutions of a class of biological models in a heterogeneous environment, Bull. Austral. Math. Soc. 44, 79 94 (1991)
B. Gidas, W. Ni, and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68, 209–243 (1979)
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Heidelberg, 1981
F. John, Partial Differential Equations, Springer-Verlag, Heidelberg, 1982
Z. Jiang and F. Yang, Numerical Solutions of the Outer Boundary Problem of Partial Differential Equations (Chinese), Tianjin Univ. Press, Tianjin, 1992
P. Kareiva, Population dynamics in spatially complex environments: Theory and data, Philos. Trans. Roy. Soc. London B330, 175 190 (1990)
T. Krisztin and H. I. Freedman, Global stability in models of population dynamics with diffusion II: Continuously varying environments, Rocky Mountain J. Math. 24, 155–163 (1994)
O. Lopes, Radial and nonradial minimizers for some radially symmetric functions, Elec. J. Diff. Equations 1996, 1–14 (1996)
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, Heidelberg, 1983
S. W. Pacala and J. Roughgarden, Spatial heterogeneity and interspecific competition, Theor. Pop. Biol. 21, 92–113 (1982)
M. Protter and H. Weinberger, Maximum Principles in Differential Equations, Prentice Hall, New Jersey, 1967
H. Royden, Real Analysis, 3rd Edition, MacMillan, New York, 1988
N. Shigesada and J. Roughgarden, The role of rapid dispersal in the population dynamics of competition, Theor. Pop. Biol. 21, 353–372 (1982)
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© Copyright 1999
American Mathematical Society