Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-28T12:23:03.307Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic and transient behaviour for a nonlocal problem arising in population genetics

Part of: Miscellaneous topics - Partial differential equations Partial differential equations Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  18 September 2018

J.-B. BURIE ,
R. DJIDJOU-DEMASSE  and
A. DUCROT
J.-B. BURIE*
Affiliation:
University of Bordeaux, IMB, UMR 5251, F-33400 Talence, France emails: jean-baptiste.burie@u-bordeaux.fr; arnaud.ducrot@u-bordeaux.fr CNRS, IMB, UMR 5251, F-33400 Talence, France
R. DJIDJOU-DEMASSE
Affiliation:
UMR 1065, INRA, Villenave d’Ornon F-33882, France email: ddramses@gmail.com
A. DUCROT
Affiliation:
University of Bordeaux, IMB, UMR 5251, F-33400 Talence, France emails: jean-baptiste.burie@u-bordeaux.fr; arnaud.ducrot@u-bordeaux.fr CNRS, IMB, UMR 5251, F-33400 Talence, France
Get access Rights & Permissions [Opens in a new window]

Abstract

This work is devoted to the study of an integro-differential system of equations modelling the genetic adaptation of a pathogen by taking into account both mutation and selection processes. First, we study the asymptotic behaviour of the system and prove that it eventually converges to a stationary state. Next, we more closely investigate the behaviour of the system in the presence of multiple EAs. Under suitable assumptions and based on a small mutation variance asymptotic, we describe the existence of a long transient regime during which the pathogen population remains far from its asymptotic behaviour and highly concentrated around some phenotypic value that is different from the one described by its asymptotic behaviour. In that setting, the time needed for the system to reach its large time configuration is very long and multiple evolutionary attractors may act as a barrier of evolution that can be very long to bypass.

Keywords

Nonlocal equationasymptotic behaviourtransient behaviourpopulation genetics

MSC classification

Primary: 35B40: Asymptotic behavior of solutions
Secondary: 35R09: Integro-partial differential equations 35Q92: PDEs in connection with biology and other natural sciences 92D10: Genetics
Type
Papers
Information
European Journal of Applied Mathematics , Volume 31 , Issue 1 , February 2020 , pp. 84 - 110
Copyright
© Cambridge University Press 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alikakos, N. D. & Fusco, G. (1998) Slow dynamics for the Cahn–Hilliard equation in higher space dimensions: the motion of bubbles. Arch. Ration. Mech. Anal. 141(1), 1–61.CrossRefGoogle Scholar
Bonnefon, O., Coville, J. & Legendre, G. (2017) Concentration phenomenon in some non-local equation. Discrete Cont. Dyn.-B 22(3), 763–781.Google Scholar
Calsina, À. & Cuadrado, S. (2005) Stationary solutions of a selection mutation model: the pure mutation case. Math. Models Methods Appl. Sci. 15(07), 1091–1117.CrossRefGoogle Scholar
Calsina, À., Cuadrado, S., Desvillettes, L. & Raoul, G. (2013) Asymptotics of steady states of a selection-mutation equation for small mutation rate. Proc. Math. Roy. Soc. Edinb. 143, 1123–1146.CrossRefGoogle Scholar
Carr, J. & Pego, R. L. (1989) Metastable patterns in solutions of ∂tu = ε2uxx − f(u). Commun. Pure Appl. Math. 42(5), 523–576.CrossRefGoogle Scholar
Cuadrado, S. (2009) Equilibria of a predator prey model of phenotype evolution. J. Math. Anal. Appl. 354, 286–294.CrossRefGoogle Scholar
Day, T. & Gandon, S. (2007) Applying population-genetic models in theoretical evolutionary epidemiology. Ecol. Lett. 10, 876–888.CrossRefGoogle ScholarPubMed
Diekmann, O., Jabin, P. E., Mischler, S. & Perthame, B. (2005) The dynamics of adaptation: an illuminating example and a Hamilton–Jacobi approach. Theor. Popul. Biol. 67, 257–271.CrossRefGoogle Scholar
Djidjou-Demasse, R., Ducrot, A. & Fabre, F. (2017) Steady state concentration for a phenotypic structured problem modeling the evolutionary epidemiology of spore producing pathogens. Math. Models Methods Appl. Sci. 27(02), 385–426.CrossRefGoogle Scholar
Ducrot, A. (2016) Spatial propagation for a two component reaction-diffusion system arising in population dynamics. J. Differ. Equations 260, 8316–8357.CrossRefGoogle Scholar
Fusco, G. & Hale, J. K. (1989) Slow-motion manifolds, dormant instability, and singular perturbations. J. Dyn. Differ. Equations 1(1), 75–94.CrossRefGoogle Scholar
Geritz, S. A., Metz, J. A., Kisdi, É. & Meszéna, G. (1997) Dynamics of adaptation and evolutionary branching. Phys. Rev. Lett. 78(10), 2024–2027.CrossRefGoogle Scholar
Iacono, G. L., van den Bosch, F. & Paveley, N. (2012) The evolution of plant pathogens in response to host resistance: factors affecting the gain from deployment of qualitative and quantitative resistance. J. Theor. Biol. 304, 152–163.CrossRefGoogle ScholarPubMed
Laforgue, J. G. L. & O’Malley, R. E. Jr. (1994) On the motion of viscous shocks and the super-sensitivity of their steady-state limits. Methods Appl. Anal. 1(4), 465–487.Google Scholar
Lannou, C. (2012) Variation and selection of quantitative traits in plant pathogens. Annu. Rev. Phytopathol. 50, 319–338.CrossRefGoogle ScholarPubMed
Mascia, C. & Strani, M. (2013) Metastability for nonlinear parabolic equations with application to scalar viscous conservation laws. SIAM J. Math. Anal. 45(5), 3084–3113.CrossRefGoogle Scholar
Metz, J. A. J., Geritz, S. A. H., Meszéna, G., Jacobs, F. J. A. & van Heerwaarden, J. S. (1996) Adaptive dynamics, a geometrical study of the consequences of nearly faithful reproduction. In: van Strien, S. J. & Verduyn Lunel, S. M. (editors), Stochastic and Spatial Structures of Dynamical Systems, North-Holland, Amsterdam, pp. 183–231.Google Scholar
Pego, R. L. (1989) Front migration in the nonlinear Cahn–Hilliard equation. Proc. Roy. Soc. London Ser. A 422(1863), 26–278.CrossRefGoogle Scholar
Reyna, L. G. & Ward, M. J. (1995) On the exponentially slow motion of a viscous shock. Commun. Pure Appl. Math. 48(2), 79–120.CrossRefGoogle Scholar
Sun, X. & Ward, M. J. (1999) Metastability for a generalized Burgers equation with applications to propagating flame-fronts. Eur. J. Appl. Math. 10(1), 27–53.CrossRefGoogle Scholar
van Nimwegen, E. & Crutchfield, J. P. (2000) Metastable evolutionary dynamics: crossing fitness barriers or escaping via neutral paths? Bull. Math. Biol. 62, 799–848.CrossRefGoogle ScholarPubMed
Wilke, C. O., Wang, J. L., Ofria, C., Lenski, R. E. & Adami, C. (2001). Evolution of digital organisms at high mutation rates leads to survival of the flattest. Nature 412, 331–333.CrossRefGoogle ScholarPubMed
Zhan, J., Thrall, P. H., Papaïx, J., Xie, L. & Burdon, J. J. (2015) Playing on a pathogen’s weakness: using evolution to guide sustainable plant disease control strategies. Annu. Rev. Phytopathol. 53, 19–43.CrossRefGoogle ScholarPubMed