On a cross-diffusion population model deduced from mutation and splitting of a single species

We deduce a particular case of the population cross-diffusion model introduced by Shigesada et al. (1979) [1] by using the ideas of mutation and splitting from a single species, as described by Sanchez-Palencia for ODE's systems Sanchez-Palencia (2011) [21]. The resulting equations of the PDE system only differ in the cross-diffusion terms, the corresponding diffusion matrix being self-diffusion dominated, which implies that the well known population segregation patterns of the Shigesada et al. model do not appear in this case. We prove existence and uniqueness of solutions of the PDE system and use a finite element approximation to discuss, numerically, stability properties of solutions with respect to the parameters in comparison with related models.

[1]  Ansgar Jüngel,et al.  Analysis of a Multidimensional Parabolic Population Model with Strong Cross-Diffusion , 2004, SIAM J. Math. Anal..

[2]  Morton E. Gurtin,et al.  On the diffusion of biological populations , 1977 .

[3]  Martin Rumpf,et al.  Nonnegativity preserving convergent schemes for the thin film equation , 2000, Numerische Mathematik.

[4]  Gonzalo Galiano,et al.  Analysis and numerical solution of a nonlinear cross-diffusion system arising in population dynamics , 2001 .

[5]  Yuan Lou,et al.  Diffusion, Self-Diffusion and Cross-Diffusion , 1996 .

[6]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[7]  M. C. Lombardo,et al.  A velocity--diffusion method for a Lotka--Volterra system with nonlinear cross and self-diffusion , 2009 .

[8]  H. Bhadeshia.  Diffusion , 1995, Theory of Transformations in Steels.

[9]  Pierre-Louis Lions,et al.  Une méthode particulaire déterministe pour des équations diffusives non linéaires , 2001 .

[10]  Yi Wang,et al.  The Global Existence of Solutions for a Cross-diffusion System , 2005 .

[11]  M E Gurtin,et al.  On interacting populations that disperse to avoid crowding: preservation of segregation , 1985, Journal of mathematical biology.

[12]  Hideki Murakawa,et al.  A linear scheme to approximate nonlinear cross-diffusion systems , 2011 .

[13]  Roman Cherniha,et al.  New exact solutions of a nonlinear cross-diffusion system , 2008 .

[14]  Shengmao Fu,et al.  Global solutions to a class of multi-species reaction-diffusion systems with cross-diffusions arising in population dynamics , 2009 .

[15]  C C Travis,et al.  Epidemic models with spatial spread due to population migration , 1983, Journal of mathematical biology.

[16]  Ansgar Jüngel,et al.  A Parabolic Cross-Diffusion System for Granular Materials , 2003, SIAM J. Math. Anal..

[17]  Gonzalo Galiano,et al.  Competing through altering the environment: A cross-diffusion population model coupled to transport–Darcy flow equations , 2011 .

[18]  P. Degond,et al.  The weighted particle method for convection-diffusion equations , 1989 .

[19]  Phan Van Tuoc,et al.  On global existence of solutions to a cross-diffusion system , 2008 .

[20]  Mostafa Bendahmane,et al.  Weak and classical solutions to predator―prey system with cross-diffusion , 2010 .

[21]  Masayasu Mimura,et al.  Spatial segregation in competitive interaction-diffusion equations , 1980 .

[22]  John W. Barrett,et al.  Finite element approximation of a nonlinear cross-diffusion population model , 2004, Numerische Mathematik.

[23]  N. Shigesada,et al.  Spatial segregation of interacting species. , 1979, Journal of theoretical biology.

[24]  Andrea L. Bertozzi,et al.  Positivity-Preserving Numerical Schemes for Lubrication-Type Equations , 1999, SIAM J. Numer. Anal..

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

[26]  Paul Deuring,et al.  An initial-boundary-value problem for a certain density-dependent diffusion system , 1987 .

[27]  Jesús Ildefonso Díaz Díaz,et al.  Energy Methods for Free Boundary Problems , 2002 .

[28]  Pierre Degond,et al.  A Deterministic Approximation of Diffusion Equations Using Particles , 1990, SIAM J. Sci. Comput..

[29]  Gonzalo Galiano,et al.  Semi-discretization in time and numerical convergence of solutions of a nonlinear cross-diffusion population model , 2003, Numerische Mathematik.

[30]  Ansgar Jüngel,et al.  Cross Diffusion Preventing Blow-Up in the Two-Dimensional Keller-Segel Model , 2011, SIAM J. Math. Anal..

[31]  Danielle Hilhorst,et al.  A NONLINEAR PARABOLIC-HYPERBOLIC SYSTEM FOR CONTACT INHIBITION OF CELL-GROWTH , 2012 .

[32]  D. Aronson.  The porous medium equation , 1986 .

[33]  Gonzalo Galiano.  Modeling spatial adaptation of populations by a time non-local convection cross-diffusion evolution problem , 2011, Appl. Math. Comput..

[34]  J. U. Kim,et al.  Smooth solutions to a quasi-linear system of diffusion equations for a certain population model☆ , 1984 .

[35]  Atsushi Yagi.  Global solution to some quasilinear parabolic system in population dynamics , 1993 .