Spatial Inhomogenity Due to Turing Instability in a Capital-Labour Model

Affiliation(s)

Department of Mathematics, Faculty of Science, King Khalid University, Abha, Saudi Arabia.

Department of Mathematics, Faculty of Science, King Khalid University, Abha, Saudi Arabia.

ABSTRACT

A cross-diffusion system is set up modelling the distribution of capital and labour over the land of two identical patches (cites, markets or countries) in which the per capita migration rate of each species (investment capital or labour force) is influenced not only by its own but also by the other one’s density, i.e. there is cross-diffusion present. Numerical studies show that at a critical value of the bifurcation parameter the system undergoes a Turing bifurcation and the cross-migration response is an important factor that should not be ignored when pattern emerges.

A cross-diffusion system is set up modelling the distribution of capital and labour over the land of two identical patches (cites, markets or countries) in which the per capita migration rate of each species (investment capital or labour force) is influenced not only by its own but also by the other one’s density, i.e. there is cross-diffusion present. Numerical studies show that at a critical value of the bifurcation parameter the system undergoes a Turing bifurcation and the cross-migration response is an important factor that should not be ignored when pattern emerges.

Cite this paper

S. Aly, "Spatial Inhomogenity Due to Turing Instability in a Capital-Labour Model,"*Applied Mathematics*, Vol. 3 No. 2, 2012, pp. 172-176. doi: 10.4236/am.2012.32027.

S. Aly, "Spatial Inhomogenity Due to Turing Instability in a Capital-Labour Model,"

References

[1] A. M. Turing, “The Chemical Basis of Morphogensis,” Philosophical Transactions of the Royal Society B, Vol. 237, No. 641, 1953, pp. 37-72. doi:10.1098/rstb.1952.0012

[2] M. Farkas, “Two Ways of Modeling cross Diffusion,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 30, No. 2, 1997, pp. 1225-1233. doi:10.1016/S0362-546X(96)00161-7

[3] M. Farkas, “Dynamical Models in Biology,” Academic Press, Cambridge, 2001.

[4] Y. Huang and O. Diekmann, “Interspecific Influence on Mobility and Turing Instability,” Bulletin of athematical Biology, Vol. 65, No. 1, 2003, pp. 143-156. doi:10.1006/bulm.2002.0328

[5] J. D. Murray, “Mathematical Biology,” Springer-Verlag, Berlin, 1989.

[6] S. Aly and M. Farkas, “Competition in Patchy Environment with cross Diffusion,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 5, No. 4, 2004, pp. 589-595. doi:10.1016/j.nonrwa.2003.10.001

[7] S. Aly and M. Farkas, “Bifurcation in a Predator-Prey Model in Patchy Environment with Diffusion,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 5, No. 3, 2004, pp. 519-526. doi:10.1016/j.nonrwa.2003.11.004

[8] S. Aly, “Bifurcations in a Predator-Prey Model with Diffusion and Memory,” International Journal of Bifurcation and Chaos, Vol. 16 No. 6, 2006, pp. 1855-1863. doi:10.1142/S0218127406015751

[9] Y. Takeuchi, “Global Dynamical Properties of Lotka-Volterra System,” World Scientific, Hackensack, 1996.

[10] M. Farkas, “On the Distribution of Capital and Labour in a Closed Economy,” Southeast Asian Bulletin of Mathematics, Vol. 19 No. 2, 1995, pp. 27-37.

[1] A. M. Turing, “The Chemical Basis of Morphogensis,” Philosophical Transactions of the Royal Society B, Vol. 237, No. 641, 1953, pp. 37-72. doi:10.1098/rstb.1952.0012

[2] M. Farkas, “Two Ways of Modeling cross Diffusion,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 30, No. 2, 1997, pp. 1225-1233. doi:10.1016/S0362-546X(96)00161-7

[3] M. Farkas, “Dynamical Models in Biology,” Academic Press, Cambridge, 2001.

[4] Y. Huang and O. Diekmann, “Interspecific Influence on Mobility and Turing Instability,” Bulletin of athematical Biology, Vol. 65, No. 1, 2003, pp. 143-156. doi:10.1006/bulm.2002.0328

[5] J. D. Murray, “Mathematical Biology,” Springer-Verlag, Berlin, 1989.

[6] S. Aly and M. Farkas, “Competition in Patchy Environment with cross Diffusion,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 5, No. 4, 2004, pp. 589-595. doi:10.1016/j.nonrwa.2003.10.001

[7] S. Aly and M. Farkas, “Bifurcation in a Predator-Prey Model in Patchy Environment with Diffusion,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 5, No. 3, 2004, pp. 519-526. doi:10.1016/j.nonrwa.2003.11.004

[8] S. Aly, “Bifurcations in a Predator-Prey Model with Diffusion and Memory,” International Journal of Bifurcation and Chaos, Vol. 16 No. 6, 2006, pp. 1855-1863. doi:10.1142/S0218127406015751

[9] Y. Takeuchi, “Global Dynamical Properties of Lotka-Volterra System,” World Scientific, Hackensack, 1996.

[10] M. Farkas, “On the Distribution of Capital and Labour in a Closed Economy,” Southeast Asian Bulletin of Mathematics, Vol. 19 No. 2, 1995, pp. 27-37.