Journal
DIFFERENTIAL EQUATIONS AND DYNAMICAL SYSTEMS
Volume 31, Issue 3, Pages 547-580Publisher
SPRINGER INDIA
DOI: 10.1007/s12591-020-00552-6
Keywords
Impulsive event; Partial differential equation; Recursion equation; Monotone cooperative system; Spreading speed; Traveling wave; Savanna; Pulse fire
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This paper addresses two main issues of monotone impulsive reaction-diffusion equations: the existence of traveling waves and the computation of spreading speeds. It applies the methodology to a planar system of impulsive reaction-diffusion equations that models tree-grass interactions in fire-prone savannas. Numerical simulations are provided to support the theoretical results.
Many systems in life sciences have been modeled by reaction-diffusion equations. However, under some circumstances, these biological systems may experience instantaneous and periodic perturbations (e.g. harvest, birth, release, fire events, etc) such that an appropriate formalism like impulsive reaction-diffusion equations is necessary to analyze them. While several works tackled the issue of traveling waves for monotone reaction-diffusion equations and the computation of spreading speeds, very little has been done in the case of monotone impulsive reaction-diffusion equations. Based on vector-valued recursion equations theory, we aim to present in this paper results that address two main issues of monotone impulsive reaction-diffusion equations. Our first result deals with the existence of traveling waves for monotone systems of impulsive reaction-diffusion equations. Our second result tackles the computation of spreading speeds for monotone systems of impulsive reaction-diffusion equations. We apply our methodology to a planar system of impulsive reaction-diffusion equations that models tree-grass interactions in fire-prone savannas. Numerical simulations, including numerical approximations of spreading speeds, are finally provided in order to illustrate our theoretical results and support the discussion.
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