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Generalized traveling waves for time-dependent reaction–diffusion systems

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Abstract

Traveling wave solutions in general time-dependent (including time-periodic) reaction–diffusion equations and systems of equations have attracted great attention in the last two decades. The aim of this paper is to study the propagation phenomenon in a general time-heterogeneous environment. More specifically, we investigate generalized traveling wave solutions for a two-component time-dependent non-cooperative reaction–diffusion system which has applications in epidemiology and ecology. Sufficient conditions on the existence and nonexistence of generalized traveling wave solutions are established. In the susceptible-infectious epidemic model setting, generalized traveling waves describe the spatio-temporal invasion of a disease into a totally susceptible population. In the context of predator–prey systems, the generalized traveling waves describe the spatial invasion of predators introduced into a new environment where the prey population is at its carrying capacity.

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Acknowledgements

We are very grateful to two anonymous referees for their careful reading and helpful comments.

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Correspondence to Shigui Ruan.

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Communicated by Y. Giga.

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Research was partially supported by National Science Foundation (DMS-1853622)

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Ambrosio, B., Ducrot, A. & Ruan, S. Generalized traveling waves for time-dependent reaction–diffusion systems. Math. Ann. 381, 1–27 (2021). https://doi.org/10.1007/s00208-020-01998-3

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  • DOI: https://doi.org/10.1007/s00208-020-01998-3

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