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Travelling waves for the population genetics model with delay

Published online by Cambridge University Press:  17 February 2009

Guojian Lin  and
Rong Yuan
Guojian Lin
Affiliation:
Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Science, Beijing 100080, People's Republic of China; e-mail: [email protected].
Rong Yuan
Affiliation:
School of Mathematics, Beijing Normal University, Beijing, 100875, People's Republic of China; e-mail: [email protected].
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Abstract

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Under the assumptions that the spatial variable is one dimensional and the distributed delay kernel is the general Gamma distributed delay kernel, when the average delay is small, the existence of travelling wave solutions for the population genetics model with distributed delay is obtained by using the linear chain trick and geometric singular perturbation theory. On the other hand, for the population genetics model with small discrete delay, the existence of travelling wave solutions is obtained by employing a technique which is based on a result concerning the existence of the inertial manifold for small discrete delay equations.

Keywords

travelling wavepopulation genetics modelgeometric singular perturbation theoryinertial manifold.
Type
Research Article
Information
The ANZIAM Journal , Volume 48 , Issue 1 , July 2006 , pp. 57 - 71
Copyright
Copyright © Australian Mathematical Society 2006

References

[1]Aronson, D. and Weinberger, H., “Nonlinear diffusion in population genetics, combustion and nerve propagation”, in Partial Differential Equations and Related Topics, Lecture Notes in Mathematics 446, (Springer, New York, 1975) 5–19.CrossRefGoogle Scholar
[2]Aronson, D. and Weinberger, H., “Multidimensional nonlinear diffusion arising in population genetics”, Adv. Math. 30 (1978) 33–76.CrossRefGoogle Scholar
[3]Ashwin, P., Bartuccelli, M., Bridges, T. and Gourley, S., “Travelling fronts for the KPP equation with spatial-temporal delay”, Z. Angew. Math. Phys. 53 (2002) 103–122.CrossRefGoogle Scholar
[4]Britton, N., “Spatial structures and periodic travelling waves in an integrodifferential reaction-diffusion population model”, SIAM J. Appl. Math. 50 (1990) 1663–1688.CrossRefGoogle Scholar
[5]Chicone, C., “Inertial and slow manifolds for delay equations with small delays”, J. Differential Equations 190 (2003) 364–406.CrossRefGoogle Scholar
[6]Chicone, C., “Inertial flows, slow flows, and combinatorial identities for delay equations”, J. Dynam. Differential Equations 16 (2004) 805–831.CrossRefGoogle Scholar
[7]Fenichel, N., “Geometric singular perturbation theory for ordinary differential equations”, J. Differential Equations 31 (1979) 53–98.CrossRefGoogle Scholar
[8]Fisher, R. A., “The advance of advantageous genes”, Ann. Eugenics 7 (1937) 355–369.CrossRefGoogle Scholar
[9]Gardner, R. A. and Jones, C. K. R. T., “Travelling waves of a perturbed diffusion equation arising in a phase field model”, Indiana Univ. Math. J. 38 (1989) 1197–1222.Google Scholar
[10]Gourley, S., “Travelling fronts in the diffusive Nicholson's blowflies equation with distributed delays”, Math. Comput. Modelling 32 (2000) 843–853.CrossRefGoogle Scholar
[11]Gourley, S. and Britton, N., “Instability of travelling wave solutions of a population model with nonlocal effects”, IMA J. Appl. Math. 51 (1993) 299–310.CrossRefGoogle Scholar
[12]Gourley, S. and Britton, N., “A predator prey reaction diffusion system with nonlocal effects”, J. Math. Biol. 34 (1996) 297–333.CrossRefGoogle Scholar
[13]Gourley, S. A. and Chaplain, M. A. J., “Travelling fronts in a food-limited population model with time delay”, Proc. Roy. Soc. Edinburgh Sect. A 132 (2002) 75–89.CrossRefGoogle Scholar
[14]Gourley, S. A. and So, J. W.-H., “Extinction and wavefront propagation in a reaction-diffusion model of a structured population with distributed maturation delay”, Proc. Roy. Soc. Edinburgh Sect. A 133 (2003) 527–548.CrossRefGoogle Scholar
[15]Jones, C. K. R. T., “Geometric singular perturbation theory”, in Dynamical Systems, Lecture Notes in Mathematics 1609, (Springer, New York, 1995) 44–120.CrossRefGoogle Scholar
[16]Lin, G. and Yuan, R., “Periodic solution for a predator-prey system with distributed delay”, Math. Comput. Modelling 42 (2005) 959–966.CrossRefGoogle Scholar
[17]Murray, J. D., Mathematical Biology (Springer, Berlin, 1989).CrossRefGoogle Scholar
[18]Ruan, S., “Delay differential equations in single species dynamics”, in Delay Differential Equations with Applications (eds. Ait Dads, E., Arino, O. and Hbid, M.), (Springer, Berlin, 2005) 31–70.Google Scholar
[19]Ruan, S. and Xiao, D., “Stability of steady states and existence of travelling waves in a vector disease model”, Proc. Roy. Soc. Edinburgh Sect. A 134 (2004) 991–1011.CrossRefGoogle Scholar
[20]Wu, J. and Zou, X., “Travelling wave fronts of reaction-diffusion systems with delay”, J. Dynam. Differential Equations 13 (2001) 651–687.CrossRefGoogle Scholar