Traveling wave fronts in a single species model with cannibalism and strongly nonlocal effect

Xijun Deng; Aiyong Chen; Xijun Deng; Aiyong Chen

doi:10.3934/math.20241298

AIMS Mathematics

2024, Volume 9, Issue 10: 26688-26701. doi: 10.3934/math.20241298

Previous Article Next Article

Research article

Traveling wave fronts in a single species model with cannibalism and strongly nonlocal effect

Xijun Deng ^{1,2
,
,},
Aiyong Chen ³

1.
School of Mathematics, Physics and Optoelectronic Engineering, Hubei University of Automotive Technology, Shiyan, 442002, China
2.
Department of Mathematics and Physics, Hunan University of Arts and Science, Changde, 415000, China
3.
Department of Mathematics, Hunan First Normal University, Changsha, 410205, China

Received: 08 November 2023 Revised: 12 March 2024 Accepted: 13 March 2024 Published: 13 September 2024
MSC : 34D15, 34E15, 35C07, 35Q53

In this paper we studied traveling front solutions of a single species model with cannibalism and nonlocal effect. For a particular class of kernels, the existence of traveling front solutions connecting the extinction state with the positive equilibrium was established for the strongly nonlocal effect case. Our approach was to reformulate it as a singular perturbed problem, and then tackle this problem by using dynamical systems techniques, in particular, geometric singular perturbation theory and Fenichel's invariant manifold theory.
- traveling wave front,
- strongly nonlocal effect,
- cannibalism,
- geometric singular perturbation theory,
- invariant manifold theory
Citation: Xijun Deng, Aiyong Chen. Traveling wave fronts in a single species model with cannibalism and strongly nonlocal effect[J]. AIMS Mathematics, 2024, 9(10): 26688-26701. doi: 10.3934/math.20241298

Related Papers:

Abstract

In this paper we studied traveling front solutions of a single species model with cannibalism and nonlocal effect. For a particular class of kernels, the existence of traveling front solutions connecting the extinction state with the positive equilibrium was established for the strongly nonlocal effect case. Our approach was to reformulate it as a singular perturbed problem, and then tackle this problem by using dynamical systems techniques, in particular, geometric singular perturbation theory and Fenichel's invariant manifold theory.

References

[1]	H. Zhang, L. Li, Traveling wave fronts of a single species model with cannibalism and nonlocal effect, Chaos Solitons Fractals, 108 (2018), 148–153. https://doi.org/10.1016/j.chaos.2018.01.038 doi: 10.1016/j.chaos.2018.01.038
[2]	G. Sun, G. Zhang, Z. Jin, L. Li, Predator cannibalism can give rise to regular spatial pattern in a predator-prey system, Nonlinear Dyn., 58 (2009), 75–84. https://doi.org/10.1007/s11071-008-9462-z doi: 10.1007/s11071-008-9462-z
[3]	A. Basher, E. Quansah, S. Bhowmick, R. Parshad, Prey cannibalism alters the dynamics of Holling-Tanner-type predator-prey models, Nonlinear Dyn., 85 (2016), 2549–2567. https://doi.org/10.1007/s11071-016-2844-8 doi: 10.1007/s11071-016-2844-8
[4]	Z. C. Wang, W. T. Li, S. Ruan, Traveling wave fronts in reaction-diffusion systems with spatio-temporal delays, J. Differ. Equ., 222 (2006), 185–232. https://doi.org/10.1016/j.jde.2005.08.010 doi: 10.1016/j.jde.2005.08.010
[5]	K. Q. Lan, J. H. Wu, Travelling wavefronts of scalar reaction-diffusion equations with and without delays, Nonlinear Anal. Real World Appl., 4 (2003), 173–188. https://doi.org/10.1016/S1468-1218(02)00020-2 doi: 10.1016/S1468-1218(02)00020-2
[6]	S. A. Gourley, Traveling front solutions of a nonlocal fisher equation, J. Math. Biol., 41 (2000), 272–284. https://doi.org/10.1007/s002850000047 doi: 10.1007/s002850000047
[7]	Q. X. Ye, Z. Y. Li, M. X. Wang, Y. P. Wu, Introduction of reaction-diffusion equation, (Chinese), 2 Eds., Beijing: Science Press, 2011.
[8]	N. Fenichel, Persistence and smoothness of invariant manifolds and flows, Indiana Univ. Math. J., 21 (1971), 193–226. https://doi.org/10.1512/iumj.1972.21.21017 doi: 10.1512/iumj.1972.21.21017
[9]	N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differ. Equ., 31 (1979), 53–98. https://doi.org/10.1016/0022-0396(79)90152-9 doi: 10.1016/0022-0396(79)90152-9
[10]	C. K. R. T. Jones, Geometric singular perturbation theory, In: Lecture notes in mathematics, Heidelberg: Springer, 1609 (1995). https://doi.org/10.1007/BFb0095239
[11]	P. N. Davis, P. van Heijster, R. Marangell, M. R. Rodrigo, Traveling wave solutions in a model for tumor invasion with the acid-mediation hypothesis, J. Dyn. Differ. Equ., 31 (1979), 53–98. https://doi.org/10.1007/s10884-021-10003-7 doi: 10.1007/s10884-021-10003-7
[12]	J. D. Murray, Mathematical biology, In: Interdisciplinary applied mathematics, New York: Springer, 17 (2002). https://doi.org/10.1007/b98868
[13]	S. A. Gourley, Travelling fronts in the diffusive Nicholson's blowflies equation with distributed delays, Math. Comput. Model., 32 (2000), 843–853. https://doi.org/10.1016/S0895-7177(00)00175-8 doi: 10.1016/S0895-7177(00)00175-8
[14]	S. A. Gourley, M. A. J. Chaplain, Travelling fronts in a food-limited population model with time delay, Proc. Roy. Soc. Edinburgh Sect. A, 132 (2002), 75–89. https://doi.org/10.1017/S0308210500001530 doi: 10.1017/S0308210500001530
[15]	P. D. Miller, Nonmonotone wave in a three species reaction-diffusion model, Methods Appl. Anal., 4 (1997), 261–282. http://dx.doi.org/10.4310/MAA.1997.v4.n3.a3 doi: 10.4310/MAA.1997.v4.n3.a3
[16]	T. Tsujikawa, Singular limit analysis of planar equilibrium solutions to a chemotaxis model equation with growth, Methods Appl. Anal., 3 (1996), 401–431. http://dx.doi.org/10.4310/MAA.1996.v3.n4.a1 doi: 10.4310/MAA.1996.v3.n4.a1
[17]	M. Taniguchi, Bifurcation from flat-layered solutions to reaction-diffusion systems in two space dimensions, J. Math. Sci. Univ. Tokyo., 1 (1994), 339–367.
[18]	Y. Nishiura, H. Fujii, Stability of singularly perturbed solutions to systems of reaction-diffusion equations, SIAM J. Math. Anal., 18 (1987), 1726–1770. http://dx.doi.org/10.1137/0518124 doi: 10.1137/0518124
[19]	H. Ikeda, Y. Nishiura, H. Suzuki, Stability of traveling waves and a relation between the Evans function and the SLEP equation, J. Reine Angew. Math., 475 (1996), 1–37. https://doi.org/10.1515/crll.1996.475.1 doi: 10.1515/crll.1996.475.1
[20]	C. K. R. T. Jones, Stability of the travelling wave solution of the FitzHugh-Nagumo system, Trans. Amer. Math. Soc., 286 (1984), 431–469. https://doi.org/10.2307/1999806 doi: 10.2307/1999806
[21]	B. Sandstede, Stability of multiple-pulse solutions, Trans. Amer. Math. Soc., 350 (1998), 429–472. https://doi.org/10.1090/S0002-9947-98-01673-0 doi: 10.1090/S0002-9947-98-01673-0

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)