Large time behavior in a nonlinear age-dependent population dynamics problem with spatial diffusion

In this work we analyze the large time behavior in a nonlinear model of population dynamics with age-dependence and spatial diffusion. We show that when t→+∞ either the solution of our problem goes to 0 or it stabilizes to a nontrivial stationary solution. We give two typical examples where the stationary solutions can be evaluated upon solving very simple partial differential equations. As a by-product of the extinction case we find a necessary condition for a nontrivial periodic solution to exist. Numerical computations not described below show a rapid stabilization.

[1]  G. Webb.  Theory of Nonlinear Age-Dependent Population Dynamics , 1985 .

[2]  Morton E. Gurtin,et al.  Product solutions and asymptotic behavior for age-dependent, dispersing populations , 1982 .

[3]  P. Lions.  On the Existence of Positive Solutions of Semilinear Elliptic Equations , 1982 .

[4]  K. Kunisch,et al.  Nonlinear age-dependent population dynamics with random diffusion , 1985 .

[5]  Henri Berestycki,et al.  Le nombre de solutions de certains problèmes semi-linéaires elliptiques , 1981 .

[6]  William Feller,et al.  On the Integral Equation of Renewal Theory , 1941 .

[7]  R. MacCamy,et al.  A population model with nonlinear diffusion , 1981 .

[8]  M. Langlais,et al.  Stabilization of solutions of nonlinear and degenerate evolution equations , 1985 .

[9]  Michel Langlais,et al.  A Nonlinear Problem in Age-Dependent Population Diffusion , 1985 .

[10]  S. Busenberg,et al.  Separable models in age-dependent population dynamics , 1985 .

[11]  Constantine M. Dafermos,et al.  Asymptotic Behavior of Solutions of Evolution Equations , 1978 .

[12]  F. C. Hoppensteadt.  Mathematical theories of populations : demographics, genetics and epidemics , 1975 .

[13]  Pierangelo Marcati.  Asymptotic Behavior in Age-Dependent Population Dynamics with Hereditary Renewal Law , 1981 .

[14]  O. Ladyženskaja.  Linear and Quasilinear Equations of Parabolic Type , 1968 .

[15]  A. J. Lotka.  The Stability of the Normal Age Distribution. , 1922, Proceedings of the National Academy of Sciences of the United States of America.

[16]  Mimmo Iannelli,et al.  A class of nonlinear diffusion problems in age-dependent population dynamics☆ , 1983 .

[17]  Frank Hoppenstaedt.  Mathematical Theories of Populations: Demographics, Genetics and Epidemics , 1975 .

[18]  M. Gurtin,et al.  Letter: A system of equations for age-dependent population diffusion. , 1973, Journal of theoretical biology.

[19]  Age-dependent population diffusion with external constraint , 1982 .

[20]  Morton E. Gurtin,et al.  Non-linear age-dependent population dynamics , 1974 .