Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-29T15:22:29.211Z Has data issue: false hasContentIssue false
  • English
  • Français

Existence and uniqueness of monotone wavefronts in a nonlocal resource-limited model

Part of: Functional-differential and differential-difference equations Parabolic equations and systems

Published online by Cambridge University Press:  07 May 2019

Elena Trofimchuk ,
Manuel Pinto  and
Sergei Trofimchuk
Elena Trofimchuk
Affiliation:
Department of Differential Equations, Igor Sikorsky Kyiv Polytechnic Institute, Kyiv, Ukraine ([email protected])
Manuel Pinto
Affiliation:
Facultad de Ciencias, Universidad de Chile, Santiago, Chile ([email protected])
Sergei Trofimchuk
Affiliation:
Instituto de Matemática y Fisica, Universidad de Talca, Talca, Chile ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We are revisiting the topic of travelling fronts for the food-limited (FL) model with spatio-temporal nonlocal reaction. These solutions are crucial for understanding the whole model dynamics. Firstly, we prove the existence of monotone wavefronts. In difference with all previous results formulated in terms of ‘sufficiently small parameters’, our existence theorem indicates a reasonably broad and explicit range of the model key parameters allowing the existence of monotone waves. Secondly, numerical simulations realized on the base of our analysis show appearance of non-oscillating and non-monotone travelling fronts in the FL model. These waves were never observed before. Finally, invoking a new approach developed recently by Solar et al., we prove the uniqueness (for a fixed propagation speed, up to translation) of each monotone front.

Keywords

food-limited modelmonotone wavenonlocal interactionexistenceuniqueness

MSC classification

Primary: 34K60: Qualitative investigation and simulation of models
Secondary: 35K57: Reaction-diffusion equations 92D25: Population dynamics (general)
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 5 , October 2020 , pp. 2462 - 2483
Check for updates
Copyright
Copyright © Royal Society of Edinburgh 2019

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

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
2Ashwin, P. B., Bartuccelli, M. V., Bridges, T. J. and Gourley, S. A.. Travelling fronts for the KPP equation with spatio-temporal delay. Z. Angew. Math. Phys. 53 (2002), 103–122.CrossRefGoogle Scholar
3Benguria, R. D. and Depassier, M. C.. Variational characterization of the speed of propagation of fronts for the nonlinear diffusion equation. Comm. Math. Phys. 175 (1996), 221–227.CrossRefGoogle Scholar
4Berestycki, H. and Nirenberg, L.. On the method of moving planes and the sliding method. Bol. Soc. Brasil. Mat. (N.S.) 22 (1991), 1–37.CrossRefGoogle Scholar
5Berestycki, H. and Nirenberg, L.. Travelling fronts in cylinders. Ann. Inst. H. Poincaré Anal. Non Linéaire 9 (1992), 497–572.CrossRefGoogle Scholar
6Berestycki, H., Nadin, G., Perthame, B. and Ryzhik, L.. The non-local Fisher-KPP equation: travelling waves and steady states. Nonlinearity 22 (2009), 2813–2844.CrossRefGoogle Scholar
7Coville, J., Dávila, J. and Martínez, S.. Nonlocal anisotropic dispersal with monostable nonlinearity. J. Differ. Equ. 244 (2008), 3080–3118.CrossRefGoogle Scholar
8Ducrot, A. and Nadin, G.. Asymptotic behaviour of traveling waves for the delayed Fisher-KPP equation. J. Differ. Equ. 256 (2014), 3115–3140.CrossRefGoogle Scholar
9Fang, J. and Zhao, X.-Q.. Monotone wavefronts of the nonlocal Fisher-KPP equation. Nonlinearity 24 (2011), 3043–3054.CrossRefGoogle Scholar
10Faria, T., Huang, W. and Wu, J.. Traveling waves for delayed reaction-diffusion equations with non-local response. Proc. R. Soc. A 462 (2006), 229–261.CrossRefGoogle Scholar
11Feng, W. and Lu, X.. On diffusive population models with toxicants and time delays. J. Math. Anal. Appl. 233 (1999), 373–386.CrossRefGoogle Scholar
12Gomez, A. and Trofimchuk, S.. Monotone traveling wavefronts of the KPP-Fisher delayed equation. J. Differ. Equ. 250 (2011), 1767–1787.CrossRefGoogle Scholar
13Gopalsamy, K., Kulenovic, M. S. C. and Ladas, G.. Time lags in a ‘food-limited’ population model. Appl. Anal. 31 (1988), 225–237.CrossRefGoogle Scholar
14Gourley, S. A.. Travelling front solutions of a nonlocal Fisher equation. J. Math. Biol. 41 (2000), 272–284.CrossRefGoogle ScholarPubMed
15Gourley, S. A.. Wave front solutions of a diffusive delay model for populations of Daphnia Magna. Comput. Math. Appl. 42 (2001), 1421–1430.CrossRefGoogle Scholar
16Gourley, S. A. and Chaplain, M. A. J.. Traveling fronts in a food-limited population model with time delay. Proc. R. Soc. Edinb. Sect. A 132 (2002), 75–89.CrossRefGoogle Scholar
17Hale, J. K. and Lin, X.-B.. Heteroclinic orbits for retarded functional differential equations. J. Differ. Equ. 65 (1986), 175–202.CrossRefGoogle Scholar
18Hasik, K., Kopfová, J., Nábělková, P. and Trofimchuk, S.. Traveling waves in the nonlocal KPP-Fisher equation: different roles of the right and the left interactions. J. Differ. Equ. 261 (2016), 1203–1236.Google Scholar
19Hernández, E. and Trofimchuk, S.. Nonstandard quasi-monotonicity: an application to the wave existence in a neutral KPP-Fisher equation. J. Dyn. Diff. Equat. (2019), https://doi.org/10.1007/s10884-019-09748-z.Google Scholar
20Ivanov, A., Gomez, C. and Trofimchuk, S.. On the existence of non-monotone non-oscillating wavefronts. J. Math. Anal. Appl. 419 (2014), 606–616.CrossRefGoogle Scholar
21Nadin, G., Perthame, B. and Tang, M.. Can a traveling wave connect two unstable states? The case of the nonlocal Fisher equation. C. R. Acad. Sci. Paris, Ser. I 349 (2011), 553–557.CrossRefGoogle Scholar
22Nadin, G., Rossi, L., Ryzhik, L. and Perthame, B.. Wave-like solutions for nonlocal reaction-diffusion equations: a toy model. Math. Model. Nat. Phenom. 8 (2013), 33–41.CrossRefGoogle Scholar
23Ou, C. and Wu, J.. Traveling wavefronts in a delayed food-limited population model. SIAM J. Math. Anal. 39 (2007), 103–125.CrossRefGoogle Scholar
24Pinto, M., Robledo, G., Liz, E., Tkachenko, V. and Trofimchuk, S.. Wright type delay differential equations with negative Schwarzian. DCDS-A 9 (2003), 309–321.Google Scholar
25Smith, F. E.. Population dynamics in Daphnia Magna. Ecology 44 (1963), 651–663.CrossRefGoogle Scholar
26So, J. W.-H. and Yu, J. S.. On the uniform stability for a ‘food limited’ population model with time delay. Proc. R. Soc. Edinb. A 125 (1995), 991–1005.CrossRefGoogle Scholar
27Solar, A. and Trofimchuk, S.. A simple approach to the wave uniqueness problem. J. Differ. Equ. 266 (2019), 6647–6660.CrossRefGoogle Scholar
28Trofimchuk, E., Alvarado, P. and Trofimchuk, S.. On the geometry of wave solutions of a delayed reaction-diffusion equation. J. Differ. Equ. 246 (2009), 1422–1444.CrossRefGoogle Scholar
29Trofimchuk, E., Pinto, M. and Trofimchuk, S.. Monotone waves for non-monotone and non-local monostable reaction-diffusion equations. J. Differ. Equ. 261 (2016), 203–1236.CrossRefGoogle Scholar
30Wang, Z. C., Li, W. T. and Ruan, S.. Travelling wave fronts in reaction-diffusion systems with spatio-temporal delays. J. Differ. Equ. 222 (2006), 185–232.CrossRefGoogle Scholar
31Wang, Z. C. and Li, W. T.. Monotone travelling fronts of a food-limited population model with nonlocal delay. Nonlinear Anal.: Real World Appl. 8 (2007), 699–712.CrossRefGoogle Scholar
32Wei, J., Tian, L., Zhou, J., Zhen, Z. and Xu, J.. Existence and asymptotic behavior of traveling wave fronts for a food-limited population model with spatio-temporal delay. Japan J. Indust. Appl. Math. 34 (2017), 305–320.CrossRefGoogle Scholar
33Widder, D. V.. The Laplace transform. Princeton Math. Ser., vol. 6 (Princeton, NJ: Princeton University Press, 1941).Google Scholar
34Wu, J. and Zou, X.. Traveling wave fronts of reaction-diffusion systems with delay. J. Dynam. Diff. Eqns. 13 (2001), 651–687.CrossRefGoogle Scholar