Spreading speeds for multidimensional reaction–diffusion systems of the prey–predator type

By Arnaud Ducrot, Thomas Giletti, Hiroshi Matano. Calculus of Variations and Partial Differential Equations, 2019

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We investigate spreading properties of solutions of a large class of two-component reaction–diffusion systems, including prey–predator systems as a special case. By spreading properties we mean the long time behaviour of solution fronts that start from localized (i.e. compactly supported) initial data. Though there are results in the literature on the existence of travelling waves for such systems, very little has been known—at least theoretically—about the spreading phenomena exhibited by solutions with compactly supported initial data. The main difficulty comes from the fact that the comparison principle does not hold for such systems. Furthermore, the techniques that are known for travelling waves such as fixed point theorems and phase portrait analysis do not apply to spreading fronts. In this paper, we first prove that spreading occurs with definite spreading speeds. Intriguingly, two separate fronts of different speeds may appear in one solution—one for the prey and the other for the predator—in some situations.

Subfields: Computer Networks and Communications, Genetics, Public Health, Environmental and Occupational Health

View Paper: https://www.semanticscholar.org/paper/8f514a628184c1ef2437a7a5da7eb5cc52069afb

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Persistence of species in a predator-prey system with climate change and either nonlocal or local dispersal (2021, methodology): “Since such a result is also needed for our argument, we state it below and refer to [15] for the details.”