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.
Citation: Gonzalo Galiano, Julián Velasco. Finite element approximation of a population spatial adaptation model[J]. Mathematical Biosciences and Engineering, 2013, 10(3): 637-647. doi: 10.3934/mbe.2013.10.637
Related Papers:
| [1] |
Alina Chertock, Pierre Degond, Sophie Hecht, Jean-Paul Vincent .
Incompressible limit of a continuum model of tissue growth with segregation for two cell populations. Mathematical Biosciences and Engineering, 2019, 16(5): 5804-5835.
doi: 10.3934/mbe.2019290
|
| [2] |
Maher Alwuthaynani, Raluca Eftimie, Dumitru Trucu .
Inverse problem approaches for mutation laws in heterogeneous tumours with local and nonlocal dynamics. Mathematical Biosciences and Engineering, 2022, 19(4): 3720-3747.
doi: 10.3934/mbe.2022171
|
| [3] |
Cornel M. Murea, H. G. E. Hentschel .
A finite element method for growth in biological development. Mathematical Biosciences and Engineering, 2007, 4(2): 339-353.
doi: 10.3934/mbe.2007.4.339
|
| [4] |
Moritz Schäfer, Peter Heidrich, Thomas Götz .
Modelling the spatial spread of COVID-19 in a German district using a diffusion model. Mathematical Biosciences and Engineering, 2023, 20(12): 21246-21266.
doi: 10.3934/mbe.2023940
|
| [5] |
Z. Jackiewicz, B. Zubik-Kowal, B. Basse .
Finite-difference and pseudo-spectral methods
for the numerical simulations of in vitro human tumor cell
population kinetics. Mathematical Biosciences and Engineering, 2009, 6(3): 561-572.
doi: 10.3934/mbe.2009.6.561
|
| [6] |
Ahmed Alshehri, Saif Ullah .
A numerical study of COVID-19 epidemic model with vaccination and diffusion. Mathematical Biosciences and Engineering, 2023, 20(3): 4643-4672.
doi: 10.3934/mbe.2023215
|
| [7] |
Raimund Bürger, Gerardo Chowell, Elvis Gavilán, Pep Mulet, Luis M. Villada .
Numerical solution of a spatio-temporal predator-prey model with infected prey. Mathematical Biosciences and Engineering, 2019, 16(1): 438-473.
doi: 10.3934/mbe.2019021
|
| [8] |
Carolina Mendoza, Jean Bragard, Pier Luigi Ramazza, Javier Martínez-Mardones, Stefano Boccaletti .
Pinning control of spatiotemporal chaos in the LCLV device. Mathematical Biosciences and Engineering, 2007, 4(3): 523-530.
doi: 10.3934/mbe.2007.4.523
|
| [9] |
Benedetto Bozzini, Deborah Lacitignola, Ivonne Sgura .
Morphological spatial patterns in a reaction diffusion model for metal growth. Mathematical Biosciences and Engineering, 2010, 7(2): 237-258.
doi: 10.3934/mbe.2010.7.237
|
| [10] |
Hongyong Zhao, Qianjin Zhang, Linhe Zhu .
The spatial dynamics of a zebrafish model with cross-diffusions. Mathematical Biosciences and Engineering, 2017, 14(4): 1035-1054.
doi: 10.3934/mbe.2017054
|
Abstract
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.
References
|
[1]
|
Numer. Math., 98 (2004), 195-221.
|
|
[2]
|
SIAM J. Math. Anal., 36 (2004), 301-322.
|
|
[3]
|
Commun. Part. Diff. Eqs., 32 (2007), 127-148.
|
|
[4]
|
Math. Z., 194 (1987), 375-396.
|
|
[5]
|
Math. Nachr., 195 (1998), 77-114.
|
|
[6]
|
Appl. Math. Comput., 218 (2011), 4587-4594.
|
|
[7]
|
Comput. Math. Appl., 64 (2012), 1927-1936.
|
|
[8]
|
RACSAM Rev. R. Acad. Cienc. Exactas Fs. Nat. Ser. A. Mat., 95 (2001), 281-295.
|
|
[9]
|
Numer. Math., 93 (2003), 655-673.
|
|
[10]
|
Banach Center Publ., 63 (2004), 209-216.
|
|
[11]
|
SIAM J. Math. Anal., 35 (2003), 561-578.
|
|
[12]
|
Nonlinear Anal., 12 (2011), 2826-2838.
|
|
[13]
|
Appl. Numer. Math., 59 (2009), 1059-1074.
|
|
[14]
|
Nonlinear Analysis TMA, 8 (1984), 1121-1144.
|
|
[15]
|
J. Diff. Eqs., 131 (1996), 79-131.
|
|
[16]
|
Adv. Math., Beijing, 25 (1996), 283-284.
|
|
[17]
|
J. Math. Biol., 9 (1980), 49-64.
|
|
[18]
|
J. Theor. Biol., 79 (1979), 83-99.
|
|
[19]
|
Nonlinear Analysis TMA, 21 (1993), 603-630.
|
DownLoad: