Lack of symmetry in linear determinacy due to convective effects in reaction-diffusion-convection problems

Main Article Content

Ameera Al-Kiffai
Elaine Crooks

Abstract

This paper is concerned with linear determinacy in monostable reaction- diffusion-convection equations and co-operative systems. We present sufficient conditions for minimal travelling-wave speeds (equivalent to spreading speeds) to equal values obtained from linearisations of the travelling-wave problem about the unstable equilibrium. These conditions involve both reaction and convection terms. We present separate conditions for non-increasing and non-decreasing travelling waves, called `right' and `left' conditions respectively, because of the asymmetry in propagation caused by the convection terms. We also give a necessary condition on the reaction term for the existence of convection terms such that both the right and left conditions can be satisfied simultaneously. Examples show that our sufficient conditions for linear determinacy are not necessary and compare these conditions in the scalar case with alternative conditions observed in Malaguti-Marcelli [15] and Benguria-Depassier-Mendez [3]. We also illustrate, for both an equation and a system, the existence of reaction and (non-trivial) convection terms for which the right and left linear determinacy conditions are simultaneously satisfied. An example is given of an equation which is right but not left linearly determinate.

Article Details

How to Cite
Al-Kiffai, A., & Crooks, E. (2016). Lack of symmetry in linear determinacy due to convective effects in reaction-diffusion-convection problems. Tamkang Journal of Mathematics, 47(1). https://doi.org/10.5556/j.tkjm.47.2016.1891
Section
Special Issue
Author Biographies

Ameera Al-Kiffai

Department ofMathematics, College of Science, Swansea University, Swansea SA2 8PP, U.K. College of Education for Girls, Kufa University, Al-Najaf Al-Ashraf, Iraq.

Elaine Crooks

Department ofMathematics, College of Science, Swansea University, Swansea SA2 8PP, U.K.

References

A. Al-Kiffai, The role of convection on spreading speeds and linear determinacy for reaction-diffusion-convection systems, PhD Thesis, Swansea University, submitted.

R. D. Benguria, M. C. Depassier and V. Mendez, Speed of travelling waves in reaction-diffusion equations, Phys. Rev. E, 3(2001), 109-110.

R. D. Benguria, M. C. Depassier and V. Mendez, Minimal speed of fronts of reaction-convection-diffusion equations, Phys. Rev. E, 69 (2004), 031106, p 7.

C. Castillo-Chavez, B. Li, and H. Wang, Some recent developments on linear determinacy, Math. Biosci. Eng., 10(2013), 1419-1436.

E. C. M. Crooks, Travelling fronts for monostable reaction-diffusion systems with gradient-dependence, Advances in Differential Equations, 8(2003), No.3, 279-314.

R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugenics, 7(1937), 355-369.

B. H. Gilding and R. Kersner, Travelling waves in nonlinear diffusion-convection-reaction}, Progress in Nonlinear Differential Equations and their Applications, 60, Birkhauser Verlag, Basel, (2004).

K. P. Hadeler and F. Rothe, Travelling fronts in nonlinear diffusion equations, J. Math. Biol., 2(1975), 251-263.

E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theor. Biol., 30(1971), 225-234.

A. N. Kolmogorov, I. Petrowskii, N. Piscounov, Etude de lequation de la diffusion avec croissance de la quantitede matiere et son application a un probleme biologique, Moscow Univ. Math. Bull., 1 (1937), 1-25.

B. Li, H. F. Weinberger, M. A. Lewis, Spreading speeds as slowest wave speeds for cooperative systems}, Mathematical Biosciences, 196(2005), 82-98.

X. Liang and X. Zhao, Asymptotic speed of spread and traveling waves for monotone semiflows with applications, Commun. Pure Appl. Math.,60(2007), 1-40.

M. Lucia, C. Muratov and M. Novaga, Linear vs. nonlinear selection for the propagation speed of the solutions of scalar reaction-diffusion equations invading an unstable equilibrium, Comm. Pure Appl. Math., 57(2004), 616-636.

R. Lui, Biological growth and spread modeled by systems of recursions, I Mathematical theory, Mathematical Bioscience, 93(1989), 269-295.

L. Malaguti and C. Marcelli, Travelling wave fronts in reaction-diffusion equation with convection effects and non-regular terms, Math., Nach., 242(2002), 148-164.

J. D. Murray, Mathematical biology, 2nd, corrected edition, Biomathematics Texts, 19(1989).

A. H. Roger and R. J. Charles, Matrix Analysis, Cambridge University Press, 2nd Edition, 2013.

E. Seneta Non-negative Matrices, an Introduction to Theory and Applications, George Allen and Unwin Ltd, London, 1973.

A. I. Volpert and V. A. Volpert, Traveling-wave Solutions of Parabolic Systems, vol:140 of Translations of Mathematical Monographs, American Mathematical Society, Providence, R.I., 1994.

H. Weinberger, On sufficient condition for a linearly determinate spreading speed, Discrete Cont. Dyn. Syst. B, 17(2012), 2267-2280.

H. F. Weinberger, M. A. Lewis, B. Li, Analysis of linear determinacy for spread in cooperative models, J. Math. Biol., 45(2002), 183-218.