×

zbMATH — the first resource for mathematics

Permanence in ecological systems with spatial heterogeneity. (English) Zbl 0796.92026
A basic problem in population dynamics is that of finding criteria for the long-term coexistence of interacting species. An important aspect of this problem is determining how coexistence is affected by spatial dispersal and environmental heterogeneity. The object of this paper is to study the problem of coexistence for two interacting species dispersing through a spatially heterogeneous region.
The authors model the population dynamics of the species with a system of two reaction-diffusion equations which are interpreted as a semi- dynamical system. There are a number of ways of characterising coexistence; here the authors use the criterion of permanence. A system is said to be permanent if any state with all components positive initially must ultimately enter and remain within a fixed set of positive states that are strictly bounded away from zero in each component. The analysis produces conditions that can be interpreted in a natural way in terms of environmental conditions and parameters, by combining the dynamic idea of permanence with the static idea of studying geometric problems via eigenvalue estimation.

MSC:
92D40 Ecology
35K57 Reaction-diffusion equations
35Q92 PDEs in connection with biology, chemistry and other natural sciences
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Cantrell, Houston J. Math. 15 pp 341– (1989)
[2] Cosner, Reaction-Diffusion Equations (1990)
[3] Cantrell, Proc. Roy. Soc. Edinburgh Sect. A 112 pp 293– (1989) · Zbl 0711.92020 · doi:10.1017/S030821050001876X
[4] DOI: 10.1090/S0002-9939-1986-0822433-4 · doi:10.1090/S0002-9939-1986-0822433-4
[5] DOI: 10.1007/978-1-4612-4838-5 · doi:10.1007/978-1-4612-4838-5
[6] DOI: 10.1137/0138002 · Zbl 0511.92019 · doi:10.1137/0138002
[7] Smoller, Shock Waves and Reaction-Diffusion Equations (1983) · Zbl 0508.35002 · doi:10.1007/978-1-4684-0152-3
[8] DOI: 10.1016/0362-546X(87)90035-6 · Zbl 0631.92014 · doi:10.1016/0362-546X(87)90035-6
[9] DOI: 10.1007/BF00280826 · Zbl 0377.35038 · doi:10.1007/BF00280826
[10] Blat, Proc. Roy. Soc. Edinburgh Sect. A 97 pp 21– (1984) · Zbl 0554.92012 · doi:10.1017/S0308210500031802
[11] Bhatia, Stability Theory of Dynamical Systems (1970) · doi:10.1007/978-3-642-62006-5
[12] Protter, Maximum Principles in Differential Equations (1967)
[13] DOI: 10.1515/crll.1985.360.47 · Zbl 0564.35060 · doi:10.1515/crll.1985.360.47
[14] Okubo, Diffusion and Ecological Problems: Mathematical Models (1980)
[15] DOI: 10.1016/0022-0396(79)90088-3 · Zbl 0386.34046 · doi:10.1016/0022-0396(79)90088-3
[16] Mora, Trans. Amer. Math. Soc. 278 pp 21– (1983)
[17] Manes, Boll. Un. Mat. Ital. 7 pp 285– (1973)
[18] Li, J. Differential and Integral Equations 4 pp 817– (1991)
[19] DOI: 10.1016/0025-5564(89)90040-0 · Zbl 0698.92022 · doi:10.1016/0025-5564(89)90040-0
[20] DOI: 10.1090/S0002-9947-1988-0920151-1 · doi:10.1090/S0002-9947-1988-0920151-1
[21] Levin, Mathematical Ecology (1986)
[22] DOI: 10.1016/0025-5564(92)90078-B · Zbl 0783.92002 · doi:10.1016/0025-5564(92)90078-B
[23] DOI: 10.1216/RMJ-1987-17-2-301 · Zbl 0636.35037 · doi:10.1216/RMJ-1987-17-2-301
[24] DOI: 10.1216/rmjm/1181073060 · Zbl 0722.92012 · doi:10.1216/rmjm/1181073060
[25] DOI: 10.1007/BF01540776 · Zbl 0542.34043 · doi:10.1007/BF01540776
[26] Hofbauer, Dynamical Systems and the Theory of Evolution (1988) · Zbl 0678.92010
[27] DOI: 10.1080/03605308008820162 · Zbl 0477.35075 · doi:10.1080/03605308008820162
[28] Henry, Geometric Theory of Semilinear Parabolic Equations, (1981) · Zbl 0456.35001 · doi:10.1007/BFb0089647
[29] DOI: 10.1137/0520025 · Zbl 0692.34053 · doi:10.1137/0520025
[30] Friedman, Partial Differential Equations of Parabolic Type (1964) · Zbl 0144.34903
[31] Friedman, Partial Differential Equations (1969) · Zbl 0224.35002
[32] Fife, Mathematical Aspects of Reacting and Diffusing Systems (1979) · Zbl 0403.92004 · doi:10.1007/978-3-642-93111-6
[33] DOI: 10.1016/0022-0396(86)90044-6 · Zbl 0605.34044 · doi:10.1016/0022-0396(86)90044-6
[34] DOI: 10.1137/0137048 · Zbl 0425.35055 · doi:10.1137/0137048
[35] DOI: 10.1016/0022-0396(85)90115-9 · Zbl 0549.35024 · doi:10.1016/0022-0396(85)90115-9
[36] DOI: 10.1090/S0002-9947-1984-0743741-4 · doi:10.1090/S0002-9947-1984-0743741-4
[37] DOI: 10.1137/0144080 · Zbl 0562.92012 · doi:10.1137/0144080
[38] Cantrell, Houston J. Math. 13 pp 337– (1987)
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.