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

  • Elena Trofimchuk (a1), Manuel Pinto (a2) and Sergei Trofimchuk (a3)


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.



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1Alfaro, M. and Coville, J.. Rapid traveling waves in the nonlocal Fisher equation connect two unstable states. Appl. Math. Lett. 25 (2012), 20952099.
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), 103122.
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), 221227.
4Berestycki, H. and Nirenberg, L.. On the method of moving planes and the sliding method. Bol. Soc. Brasil. Mat. (N.S.) 22 (1991), 137.
5Berestycki, H. and Nirenberg, L.. Travelling fronts in cylinders. Ann. Inst. H. Poincaré Anal. Non Linéaire 9 (1992), 497572.
6Berestycki, H., Nadin, G., Perthame, B. and Ryzhik, L.. The non-local Fisher-KPP equation: travelling waves and steady states. Nonlinearity 22 (2009), 28132844.
7Coville, J., Dávila, J. and Martínez, S.. Nonlocal anisotropic dispersal with monostable nonlinearity. J. Differ. Equ. 244 (2008), 30803118.
8Ducrot, A. and Nadin, G.. Asymptotic behaviour of traveling waves for the delayed Fisher-KPP equation. J. Differ. Equ. 256 (2014), 31153140.
9Fang, J. and Zhao, X.-Q.. Monotone wavefronts of the nonlocal Fisher-KPP equation. Nonlinearity 24 (2011), 30433054.
10Faria, T., Huang, W. and Wu, J.. Traveling waves for delayed reaction-diffusion equations with non-local response. Proc. R. Soc. A 462 (2006), 229261.
11Feng, W. and Lu, X.. On diffusive population models with toxicants and time delays. J. Math. Anal. Appl. 233 (1999), 373386.
12Gomez, A. and Trofimchuk, S.. Monotone traveling wavefronts of the KPP-Fisher delayed equation. J. Differ. Equ. 250 (2011), 17671787.
13Gopalsamy, K., Kulenovic, M. S. C. and Ladas, G.. Time lags in a ‘food-limited’ population model. Appl. Anal. 31 (1988), 225237.
14Gourley, S. A.. Travelling front solutions of a nonlocal Fisher equation. J. Math. Biol. 41 (2000), 272284.
15Gourley, S. A.. Wave front solutions of a diffusive delay model for populations of Daphnia Magna. Comput. Math. Appl. 42 (2001), 14211430.
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), 7589.
17Hale, J. K. and Lin, X.-B.. Heteroclinic orbits for retarded functional differential equations. J. Differ. Equ. 65 (1986), 175202.
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), 12031236.
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),
20Ivanov, A., Gomez, C. and Trofimchuk, S.. On the existence of non-monotone non-oscillating wavefronts. J. Math. Anal. Appl. 419 (2014), 606616.
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), 553557.
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), 3341.
23Ou, C. and Wu, J.. Traveling wavefronts in a delayed food-limited population model. SIAM J. Math. Anal. 39 (2007), 103125.
24Pinto, M., Robledo, G., Liz, E., Tkachenko, V. and Trofimchuk, S.. Wright type delay differential equations with negative Schwarzian. DCDS-A 9 (2003), 309321.
25Smith, F. E.. Population dynamics in Daphnia Magna. Ecology 44 (1963), 651663.
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), 9911005.
27Solar, A. and Trofimchuk, S.. A simple approach to the wave uniqueness problem. J. Differ. Equ. 266 (2019), 66476660.
28Trofimchuk, E., Alvarado, P. and Trofimchuk, S.. On the geometry of wave solutions of a delayed reaction-diffusion equation. J. Differ. Equ. 246 (2009), 14221444.
29Trofimchuk, E., Pinto, M. and Trofimchuk, S.. Monotone waves for non-monotone and non-local monostable reaction-diffusion equations. J. Differ. Equ. 261 (2016), 2031236.
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), 185232.
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), 699712.
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), 305320.
33Widder, D. V.. The Laplace transform. Princeton Math. Ser., vol. 6 (Princeton, NJ: Princeton University Press, 1941).
34Wu, J. and Zou, X.. Traveling wave fronts of reaction-diffusion systems with delay. J. Dynam. Diff. Eqns. 13 (2001), 651687.


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