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Two components is too simple: an example of oscillatory Fisher–KPP system with three components

Part of: Parabolic equations and systems Local and nonlocal bifurcation theory

Published online by Cambridge University Press:  24 September 2019

Léo Girardin
Léo Girardin*
Affiliation:
Laboratoire de Mathématiques d'Orsay, Université Paris Sud, CNRS, Université Paris-Saclay, 91405 Orsay Cedex, France ([email protected])
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Abstract

In a recent paper by Cantrell et al. [9], two-component KPP systems with competition of Lotka–Volterra type were analyzed and their long-time behaviour largely settled. In particular, the authors established that any constant positive steady state, if unique, is necessarily globally attractive. In the present paper, we give an explicit and biologically very natural example of oscillatory three-component system. Using elementary techniques or pre-established theorems, we show that it has a unique constant positive steady state with two-dimensional unstable manifold, a stable limit cycle, a predator–prey structure near the steady state, periodic wave trains and point-to-periodic rapid travelling waves. Numerically, we also show the existence of pulsating fronts and propagating terraces.

Keywords

KPP nonlinearityreaction–diffusion systemHopf bifurcationspreading phenomena

MSC classification

Primary: 35K40: Second-order parabolic systems
Secondary: 35K57: Reaction-diffusion equations 37G10: Bifurcations of singular points 92D25: Population dynamics (general)
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 6 , December 2020 , pp. 3097 - 3120
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Copyright
Copyright © 2019 The Royal Society of Edinburgh

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References

1Alfaro, M. and Coville, J.. Rapid traveling waves in the nonlocal fisher equation connect two unstable states. Appl. Math. Lett. 25 (2012), 2095–2099.CrossRefGoogle Scholar
2Alfaro, M., Coville, J. and Raoul, G.. Travelling waves in a nonlocal reaction-diffusion equation as a model for a population structured by a space variable and a phenotypic trait. Commun. Partial Differ. Equ. 38 (2013), 2126–2154.CrossRefGoogle Scholar
3Arnold, A., Desvillettes, L. and Prévost, C.. Existence of nontrivial steady states for populations structured with respect to space and a continuous trait. Commun. Pure Appl. Anal. 11 (2012), 83–96.CrossRefGoogle Scholar
4Bénichou, O., Calvez, V., Meunier, N. and Voituriez, R.. Front acceleration by dynamic selection in fisher population waves. Phys. Rev. E 86 (2012), 041908.CrossRefGoogle ScholarPubMed
5Berestycki, H., Nadin, G., Perthame, B. and Ryzhik, L.. The non-local Fisher–KPP equation: travelling waves and steady states. Nonlinearity 22 (2009), 2813.CrossRefGoogle Scholar
6Bouin, E. and Calvez, V.. Travelling waves for the cane toads equation with bounded traits. Nonlinearity 27 (2014), 2233–2253.Google Scholar
7Bouin, E., Calvez, V., Meunier, N., Mirrahimi, S., Perthame, B., Raoul, G. and Voituriez, R.. Invasion fronts with variable motility: phenotype selection, spatial sorting and wave acceleration. Comptes Rendus Mathematique 350 (2012), 761–766.CrossRefGoogle Scholar
8Bouin, E., Henderson, C. and Ryzhik, L.. Super-linear spreading in local and non-local cane toads equations. ArXiv e-prints, (2015).Google Scholar
9Cantrell, R. S., Cosner, C. and Yu, X.. Dynamics of populations with individual variation in dispersal on bounded domains. J. Biol. Dyn. 12 (2018), 288–317.CrossRefGoogle ScholarPubMed
10Ducrot, A., Giletti, T. and Matano, H.. Existence and convergence to a propagating terrace in one-dimensional reaction-diffusion equations. Trans. Amer. Math. Soc. 366 (2014), 5541–5566.CrossRefGoogle Scholar
11Eaton, J. W., Bateman, D., Hauberg, S. and Wehbring, R.. GNU Octave version 5.1.0 manual: a high-level interactive language for numerical computations, (2019).Google Scholar
12Elliott, E. C. and Cornell, S. J.. Dispersal polymorphism and the speed of biological invasions. PLOS ONE 7 (2012), 1–10, 07.CrossRefGoogle ScholarPubMed
13Faye, G. and Holzer, M.. Modulated traveling fronts for a nonlocal fisher-kpp equation: a dynamical systems approach. J. Differ. Equ. 258 (2015), 2257–2289.Google Scholar
14Fife, P. C.. Mathematical aspects of reacting and diffusing systems, volume 28 of Lecture Notes in Biomathematics (Berlin-New York: Springer-Verlag, 1979).CrossRefGoogle Scholar
15Fraile, J. M. and Sabina, J.. Kinetic conditions for the existence of wave fronts in reaction-diffusion systems. Proc. Roy. Soc. Edinburgh Sect. A 103 (1986), 161–177.CrossRefGoogle Scholar
16Fraile, J. M. and Sabina, J. C.. General conditions for the existence of a ‘critical point-periodic wave front’ connection for reaction-diffusion systems. Nonlinear Anal. 13 (1989), 767–786.CrossRefGoogle Scholar
17Girardin, L.. Non-cooperative Fisher–KPP systems: Asymptotic behavior of traveling waves. Math. Models Methods Appl. Sci. 28 (2018), 1067–1104.Google Scholar
18Girardin, L.. Non-cooperative Fisher–KPP systems: traveling waves and long-time behavior. Nonlinearity 31 (2018), 108.CrossRefGoogle Scholar
19Griette, Q. and Raoul, G.. Existence and qualitative properties of travelling waves for an epidemiological model with mutations. J. Differ. Equ. 260 (2016), 7115–7151.Google Scholar
20Horn, R. A. and Johnson, C. R.. Topics in Matrix Analysis (Cambridge: Cambridge University Press, 1991).CrossRefGoogle Scholar
21Kopell, N. and Howard, L. N.. Plane wave solutions to reaction-diffusion equations. Studies in Appl. Mat. 52 (1973), 291–328.CrossRefGoogle Scholar
22Kuznetsov, Y. A.. Elements of Applied Bifurcation Theory, volume 112 of Applied Mathematical Sciences, 3rd edn (New York: Springer-Verlag, 2004).CrossRefGoogle Scholar
23Maginu, K.. Stability of spatially homogeneous periodic solutions of reaction-diffusion equations. J. Differ. Equ. 31 (1979), 130–138.CrossRefGoogle Scholar
24Maginu, K.. Stability of periodic travelling wave solutions with large spatial periods in reaction-diffusion systems. J. Differ. Equ. 39 (1981), 73–99.CrossRefGoogle Scholar
25May, R. M. and Leonard, W. J.. Nonlinear aspects of competition between three species. SIAM J. Appl. Math. 29 (1975), 243–253.Special issue on mathematics and the social and biological sciences.CrossRefGoogle Scholar
26Morris, A., Börger, L. and Crooks, E. C. M.. Individual variability in dispersal and invasion speed. ArXiv e-prints, dec 2016.Google Scholar
27Murray, J. D.. Mathematical Biology. II, volume 18 of Interdisciplinary Applied Mathematics, 3rd edn (New York: Springer-Verlag, 2003), Spatial models and biomedical applications.CrossRefGoogle Scholar
28Nadin, G.. Traveling fronts in space-time periodic media. J. Math. Pures Appl. (9) 92 (2009), 232–262.Google Scholar
29Nadin, G., Perthame, B. and Tang, M.. Can a traveling wave connect two unstable states?: The case of the nonlocal Fisher equation. C. R. Math. Acad. Sci. Paris 349 (2011), 553–557.Google Scholar
30Petrovskii, S., Kawasaki, K., Takasu, F. and Shigesada, N.. Diffusive waves, dynamical stabilization and spatio-temporal chaos in a community of three competitive species. Japan J. Indust. Appl. Math. 18 (2001), 459–481.Recent topics in mathematics moving toward science and engineering.CrossRefGoogle Scholar
31Prévost, C.. Applications of partial differential equations and their numerical simulations of population dynamics. PhD thesis, PhD Thesis, University of Orleans, (2004).Google Scholar
32Sherratt, J. A.. Invading wave fronts and their oscillatory wakes are linked by a modulated travelling phase resetting wave. Phys. D 117 (1998), 145–166.CrossRefGoogle Scholar
33Smith, M. J. and Sherratt, J. A.. The effects of unequal diffusion coefficients on periodic travelling waves in oscillatory reaction-diffusion systems. Phys. D 236 (2007), 90–103.CrossRefGoogle Scholar
34Uno, T. and Odani, K.. On a Lotka–Volterra model which can be projected to a sphere. In Proceedings of the Second World Congress of Nonlinear Analysts, Part 3 (Athens, 1996), volume 30, pp. 1405–1410, (1997).Google Scholar
35Zeeman, M.-L.. Hopf bifurcations in competitive three-dimensional Lotka–Volterra systems. Dynam. Stability Systems 8 (1993), 189–217.CrossRefGoogle Scholar