Abstract
Cross diffusion has been widely considered in the mathematical modelling of spatially structured ecological and epidemic systems, either in the mechanical description of diffusion or in the stochastic point process description of interacting populations. In this paper mathematical results recently obtained by the authors about the asymptotic behaviour of reaction-diffusion systems with full matrices of diffusion are applied to classes of biological systems.
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Aris, R.: The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts, vols. I and II. Oxford: Oxford University Press 1975
Auchmuty, J. F. G.: Lyapunov Methods and Equations of Parabolic Type. (Lect. Notes Math., vol. 322) Berlin Heidelberg New York: Springer 1972
Bartlett, M. S.: Deterministic and stochastic models for recurrent epidemics. In: Proc. 3rd Berkeley Symp. Math. Statist. Prob., vol. 4, pp. 81–109. Berkeley Los Angeles: University of California Press 1956
Beretta, E., Capasso, V.: On the general structure of epidemic systems. Global asymptotic stability. Comput. Math. Appl. 12A, 677–694 (1986)
Beretta, E., Capasso, V., Rinaldi, F.: Global stability results for a generalized Lotka-Volterra system with distributed delays. J. Math. Biol. 26, 661–688 (1988)
Britton, N. F.: Reaction-Diffusion Equations and Their Applications to Biology. New York: Academic Press 1986
Capasso, V., Di Liddo, A.: Global attractivity for reaction-diffusion systems. The case of nondiagonal diffusion matrices. J. Math. Anal. Appl. 177, 510–529 (1993)
Fisher, R. A.: The wave of advance of advantageous genes. Ann. Eugen., London 7, 355–369 (1973)
Goh, B. S.: Global stability in a class of prey-predator models. Bull. Math. Biol. 40, 525–533 (1978)
Gurtin, M. E.: Some mathematical models for population dynamics that lead to segregation. Q. J. Appl. Math. 32, 1–9 (1974)
Henry, D.: Geometric Theory of Semilinear Parabolic Equations. Berlin Heidelberg New York: Springer 1981
Hoppensteadt, F.: Mathematical Theories of Populations: Demographics, Genetics and Epidemics. Philadelphia: SIAM 1975
Kendall, D. G.: A form of wave propagation associated with the equation of heat conduction. Proc. Camb. Philos. Soc. 44, 591–593 (1948)
Kendall, D. G.: Mathematical models of the spread of infection. (Math. Comput. Sci. Biol. Med.) London: H.M.S.O. 1965
Kerner, E. H.: Further considerations on the statistical mechanics of biological associations. Bull. Math. Biophys. 21, 217–255 (1959)
Kolmogorov, A., Petrovsky, L, Piscounov, N.: Etude de l'équation de la diffusion avec croissance de la quantité de matière et son application á un problème biologique. Mosc. Univ. Bull. Ser. Inte. Sect. A 1, 1–25 (1937)
Kopell N., Howard, L. N.: Plane wave solutions to reaction-diffusion equations. Stud. Appl. Math. 52, 291–328 (1973)
Levin, S. A.: Spatial patterning and the structure of ecological communities. In: Levin, S. A. (ed.) Some Mathematical Questions in Biology (Lect. Math. Life Sci., vol. 7) Providence, RI: Am. Math. Soc. 1976
Levin, S. A.: Population models and community structure in heterogeneous environments. In: Levin, S. A. (ed.) Mathematical Association America Study in Mathematical Biology; vol. II: Populations and Communities. Washington: Math. Assoc. Am. 1978
Mora, X.: Semilinear parabolic problems define semiflows on C k spaces. Trans. Am. Math. Soc. 278, 21–55 (1983)
Nagel, R.: Operator matrices and reaction-diffusion systems. Rend. Semin. Mat. Fis. Milano 59, 185–196 (1989)
Nagel, R.: Well-posedness and positivity for systems of linear evolution equations. Conf. Semin. Mat. Univ. Bari 203 (l985)
Okubo, A.: Diffusion and Ecological Problems: Mathematical Models. Berlin Heidelberg New York: Springer 1980
Radcliffe, J.: The initial geographical spread of host-vector and carrier-borne epidemics. J. Appl. Probab. 10, 703–717 (1973)
Radcliffe, J.: The effect of the length of incubation period on the velocity of propagation of an epidemic wave. Math. Biosci. 19, 257–262 (1974)
Skellam, J. G.: Random dispersal in theoretical populations. Biometrika 38, 196–218 (1951)
Smeller, J.: Shock Waves and Reaction-Diffusion Equations. Berlin Heidelberg New York: Springer 1983
Walker, J. A.: Dynamical Systems and Evolution Equations. Theory and Applications. New York London: Plenum Press 1980
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Work performed with partial support of the MURST 40% program “Evolution Equations and Applications” and GNFM-CNR
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Capasso, V., Di Liddo, A. Asymptotic behaviour of reaction-diffusion systems in population and epidemic models. J. Math. Biol. 32, 453–463 (1994). https://doi.org/10.1007/BF00160168
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DOI: https://doi.org/10.1007/BF00160168