SPATIAL DYNAMICS OF THE DIFFUSIVE LOGISTIC EQUATION WITH A SEDENTARY COMPARTMENT

We study an extension of the diffusive logistic equation or Fisher’s equation for a situation where one part of the population is sedentary and reproducing, and the other part migrating and subject to mortality. We show that this system is essentially equivalent to a semi-linear wave equation with viscous damping. With respect to persistence in bounded domains with absorbing boundary conditions and with respect to the rate of spread of a locally introduced population, there are two distinct scenarios, depending on the choice of parameters. In the first scenario the population can survive in sufficiently large domains and the linearization at the leading edge of the front yields a unique candidate for the spread rate. In the second scenario the population can survive in arbitrarily small domains and there are two possible candidates for the spread rate. Analysis shows it is the larger candidate which gives the correct spread rate. The phenomenon of spread is also investigated using travelling wave theory. Here the minimal speed of possible travelling front solutions equals the previously calculated spread rate. The results are explained in biological terms.

[1]  H. Amann.  On the Existence of Positive Solutions of Nonlinear Elliptic Boundary value Problems , 1971 .

[2]  D. Aronson,et al.  Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation , 1975 .

[3]  K. P. Hadeler,et al.  Travelling fronts in nonlinear diffusion equations , 1975 .

[4]  L. Peletier,et al.  Nonlinear diffusion in population genetics , 1977 .

[5]  P. Hess.  On uniqueness of positive solutions of nonlinear elliptic boundary value problems , 1977 .

[6]  F. Rothe.  Convergence to travelling fronts in semilinear parabolic equations , 1978, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[7]  O. Diekmann,et al.  Thresholds and travelling waves for the geographical spread of infection , 1978, Journal of mathematical biology.

[8]  Horst R. Thieme,et al.  Asymptotic estimates of the solutions of nonlinear integral equations and asymptotic speeds for the spread of populations. , 1979 .

[9]  K. Schumacher.  Travelling-front solutions for integro-differential equations. I. , 1980 .

[10]  K. Schumacher.  Travelling-Front Solutions for Integrodifferential Equations II , 1980 .

[11]  Odo Diekmann,et al.  The velocity of spatial population expansion , 1990 .

[12]  Steven R. Dunbar.  A branching random evolution and a nonlinear hyperbolic equation , 1988 .

[13]  R. Lui,et al.  Biological growth and spread modeled by systems of recursions. II. Biological theory. , 1989, Mathematical biosciences.

[14]  D Mollison,et al.  Dependence of epidemic and population velocities on basic parameters. , 1991, Mathematical biosciences.

[15]  Hal L. Smith,et al.  Monotone Dynamical Systems: An Introduction To The Theory Of Competitive And Cooperative Systems (Mathematical Surveys And Monographs) By Hal L. Smith , 1995 .

[16]  G. Schmitz,et al.  Biological Invasion of an Organism with Separate Mobile and Stationary States : Modeling and Analysis , 1996 .

[17]  M. Hadeler.  Reaction Transport Systems in Biological , 1997 .

[18]  K. P. Hadeler,et al.  Reaction transport systems in biological modelling , 1999 .

[19]  Thierry Gallay,et al.  Scaling Variables and Stability of Hyperbolic Fronts , 1998, SIAM J. Math. Anal..

[20]  H. Schwetlick.  Travelling fronts for multidimensional nonlinear transport equations , 2000 .

[21]  Bingtuan Li,et al.  Spreading speed and linear determinacy for two-species competition models , 2002, Journal of mathematical biology.

[22]  T. Hillen.  Transport equations with resting phases , 2003, European Journal of Applied Mathematics.

[23]  K. Hadeler.  9. The Role of Migration and Contact Distributions in Epidemic Spread , 2003 .

[24]  Xiao-Qiang Zhao,et al.  Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models , 2003 .

[25]  U. an der Heiden,et al.  Generation of the nervous impulse and periodic oscillations , 1976, Biological Cybernetics.

[26]  M A Lewis,et al.  Persistence, spread and the drift paradox. , 2005, Theoretical population biology.

[27]  M. Hirsch,et al.  4. Monotone Dynamical Systems , 2005 .