American Institute of Mathematical Sciences

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2013, 10(3): 637-647. doi: 10.3934/mbe.2013.10.637

Finite element approximation of a population spatial adaptation model

Gonzalo Galiano 1, and  Julián Velasco 1,

1. 

Dpto. de Matemáticas, Universidad de Oviedo, c/ Calvo Sotelo, 33007-Oviedo, Spain, Spain

Received  May 2012 Revised  October 2012 Published  April 2013

In [18], Sighesada, Kawasaki and Teramoto presented a system of partial differential equations for modeling spatial segregation of interacting species. Apart from competitive Lotka-Volterra (reaction) and population pressure (cross-diffusion) terms, a convective term modeling the populations attraction to more favorable environmental regions was included. In this article, we study numerically a modification of their convective term to take account for the notion of spatial adaptation of populations. After describing the model, in which a time non-local drift term is considered, we propose a numerical discretization in terms of a mass-preserving time semi-implicit finite element method. Finally, we provied the results of some biologically inspired numerical experiments showing qualitative differences between the original model of [18] and the model proposed in this article.
Keywords: finite element, Population dynamics, segregation., evolution problem, spatial adaptation, time non-local convection, cross-diffusion.
Mathematics Subject Classification: 35K55, 65M60, 92D2.
Citation: Gonzalo Galiano, Julián Velasco. Finite element approximation of a population spatial adaptation model. Mathematical Biosciences & Engineering, 2013, 10 (3) : 637-647. doi: 10.3934/mbe.2013.10.637
References:
[1]

J. W. Barrett and J. F. Blowey, Finite element approximation of a nonlinear cross-diffusion population model, Numer. Math., 98 (2004), 195-221. doi: 10.1007/s00211-004-0540-y.  Google Scholar

[2]

L. Chen and A. Jüngel, Analysis of a multidimensional parabolic population model with strong cross-diffusion, SIAM J. Math. Anal., 36 (2004), 301-322. doi: 10.1137/S0036141003427798.  Google Scholar

[3]

L. Chen and A. Jüngel, Analysis of a parabolic cross-diffusion semiconductor model with electron-hole scattering, Commun. Part. Diff. Eqs., 32 (2007), 127-148. doi: 10.1080/03605300601088815.  Google Scholar

[4]

P. Deuring, An initial-boundary value problem for a certain density-dependent diffusion system, Math. Z., 194 (1987), 375-396. doi: 10.1007/BF01162244.  Google Scholar

[5]

H. Gajewski and K. Zacharias, Global behaviour of a reaction-diffusion system modelling chemotaxis, Math. Nachr., 195 (1998), 77-114. doi: 10.1002/mana.19981950106.  Google Scholar

[6]

G. Galiano, Modeling spatial adaptation of populations by a time non-local convection cross-diffusion evolution problem, Appl. Math. Comput., 218 (2011), 4587-4594. doi: 10.1016/j.amc.2011.10.041.  Google Scholar

[7]

G. Galiano, On a cross-diffusion population model deduced from mutation and splitting of a single species, Comput. Math. Appl., 64 (2012), 1927-1936. doi: 10.1016/j.camwa.2012.03.045.  Google Scholar

[8]

G. Galiano, M. L. Garzón and A. Jüngel, Analysis and numerical solution of a nonlinear cross-diffusion system arising in population dynamics, RACSAM Rev. R. Acad. Cienc. Exactas Fs. Nat. Ser. A. Mat., 95 (2001), 281-295.  Google Scholar

[9]

G. Galiano, M. L. Garzón and A. Jüngel, Semi-discretization in time and numerical convergence of solutions of a nonlinear cross-diffusion population model, Numer. Math., 93 (2003), 655-673. doi: 10.1007/s002110200406.  Google Scholar

[10]

G. Galiano and A. Jüngel, Global existence of solutions for a strongly coupled population system, Banach Center Publ., 63 (2004), 209-216.  Google Scholar

[11]

G. Galiano, A. Jüngel and J. Velasco, A parabolic cross-diffusion system for granular materials, SIAM J. Math. Anal., 35 (2003), 561-578. doi: 10.1137/S0036141002409386.  Google Scholar

[12]

G. Galiano and J. Velasco, Competing through altering the environment: A cross-diffusion population model coupled to transport Darcy flow equations, Nonlinear Anal., 12 (2011), 2826-2838. doi: 10.1016/j.nonrwa.2011.04.009.  Google Scholar

[13]

G. Gambino, M. C. Lombardo and M. Sammartino, A velocity-diffusion method for a Lotka-Volterra system with nonlinear cross and self-diffusion, Appl. Numer. Math., 59 (2009), 1059-1074. doi: 10.1016/j.apnum.2008.05.002.  Google Scholar

[14]

J. U. Kim, Smooth solutions to a quasi-linear system of diffusion equations for a certain population model, Nonlinear Analysis TMA, 8 (1984), 1121-1144. doi: 10.1016/0362-546X(84)90115-9.  Google Scholar

[15]

Y. Lou and W.-M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Diff. Eqs., 131 (1996), 79-131. doi: 10.1006/jdeq.1996.0157.  Google Scholar

[16]

Y. Lou, W.-M. Ni and Y. Wu, The global existence of solutions for a cross-diffusion system, Adv. Math., Beijing, 25 (1996), 283-284. Google Scholar

[17]

M. Mimura and K. Kawasaki, Spatial segregation in competitive interaction-diffusion equations, J. Math. Biol., 9 (1980), 49-64. doi: 10.1007/BF00276035.  Google Scholar

[18]

N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Theor. Biol., 79 (1979), 83-99. doi: 10.1016/0022-5193(79)90258-3.  Google Scholar

[19]

A. Yagi, Global solution to some quasilinear parabolic system in population dynamics, Nonlinear Analysis TMA, 21 (1993), 603-630. doi: 10.1016/0362-546X(93)90004-C.  Google Scholar

show all references

References:
[1]

J. W. Barrett and J. F. Blowey, Finite element approximation of a nonlinear cross-diffusion population model, Numer. Math., 98 (2004), 195-221. doi: 10.1007/s00211-004-0540-y.  Google Scholar

[2]

L. Chen and A. Jüngel, Analysis of a multidimensional parabolic population model with strong cross-diffusion, SIAM J. Math. Anal., 36 (2004), 301-322. doi: 10.1137/S0036141003427798.  Google Scholar

[3]

L. Chen and A. Jüngel, Analysis of a parabolic cross-diffusion semiconductor model with electron-hole scattering, Commun. Part. Diff. Eqs., 32 (2007), 127-148. doi: 10.1080/03605300601088815.  Google Scholar

[4]

P. Deuring, An initial-boundary value problem for a certain density-dependent diffusion system, Math. Z., 194 (1987), 375-396. doi: 10.1007/BF01162244.  Google Scholar

[5]

H. Gajewski and K. Zacharias, Global behaviour of a reaction-diffusion system modelling chemotaxis, Math. Nachr., 195 (1998), 77-114. doi: 10.1002/mana.19981950106.  Google Scholar

[6]

G. Galiano, Modeling spatial adaptation of populations by a time non-local convection cross-diffusion evolution problem, Appl. Math. Comput., 218 (2011), 4587-4594. doi: 10.1016/j.amc.2011.10.041.  Google Scholar

[7]

G. Galiano, On a cross-diffusion population model deduced from mutation and splitting of a single species, Comput. Math. Appl., 64 (2012), 1927-1936. doi: 10.1016/j.camwa.2012.03.045.  Google Scholar

[8]

G. Galiano, M. L. Garzón and A. Jüngel, Analysis and numerical solution of a nonlinear cross-diffusion system arising in population dynamics, RACSAM Rev. R. Acad. Cienc. Exactas Fs. Nat. Ser. A. Mat., 95 (2001), 281-295.  Google Scholar

[9]

G. Galiano, M. L. Garzón and A. Jüngel, Semi-discretization in time and numerical convergence of solutions of a nonlinear cross-diffusion population model, Numer. Math., 93 (2003), 655-673. doi: 10.1007/s002110200406.  Google Scholar

[10]

G. Galiano and A. Jüngel, Global existence of solutions for a strongly coupled population system, Banach Center Publ., 63 (2004), 209-216.  Google Scholar

[11]

G. Galiano, A. Jüngel and J. Velasco, A parabolic cross-diffusion system for granular materials, SIAM J. Math. Anal., 35 (2003), 561-578. doi: 10.1137/S0036141002409386.  Google Scholar

[12]

G. Galiano and J. Velasco, Competing through altering the environment: A cross-diffusion population model coupled to transport Darcy flow equations, Nonlinear Anal., 12 (2011), 2826-2838. doi: 10.1016/j.nonrwa.2011.04.009.  Google Scholar

[13]

G. Gambino, M. C. Lombardo and M. Sammartino, A velocity-diffusion method for a Lotka-Volterra system with nonlinear cross and self-diffusion, Appl. Numer. Math., 59 (2009), 1059-1074. doi: 10.1016/j.apnum.2008.05.002.  Google Scholar

[14]

J. U. Kim, Smooth solutions to a quasi-linear system of diffusion equations for a certain population model, Nonlinear Analysis TMA, 8 (1984), 1121-1144. doi: 10.1016/0362-546X(84)90115-9.  Google Scholar

[15]

Y. Lou and W.-M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Diff. Eqs., 131 (1996), 79-131. doi: 10.1006/jdeq.1996.0157.  Google Scholar

[16]

Y. Lou, W.-M. Ni and Y. Wu, The global existence of solutions for a cross-diffusion system, Adv. Math., Beijing, 25 (1996), 283-284. Google Scholar

[17]

M. Mimura and K. Kawasaki, Spatial segregation in competitive interaction-diffusion equations, J. Math. Biol., 9 (1980), 49-64. doi: 10.1007/BF00276035.  Google Scholar

[18]

N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Theor. Biol., 79 (1979), 83-99. doi: 10.1016/0022-5193(79)90258-3.  Google Scholar

[19]

A. Yagi, Global solution to some quasilinear parabolic system in population dynamics, Nonlinear Analysis TMA, 21 (1993), 603-630. doi: 10.1016/0362-546X(93)90004-C.  Google Scholar

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