Abstract
This paper is concerned with non-cooperative parabolic reaction–diffusion systems which share structural similarities with the scalar Fisher–KPP equation. These similarities make it possible to prove, among other results, an extinction and persistence dichotomy and, when persistence occurs, the existence of a positive steady state, the existence of traveling waves with a half-line of possible speeds and a positive minimal speed and the equality between this minimal speed and the spreading speed for the Cauchy problem. Non-cooperative KPP systems can model various phenomena where the following three mechanisms occur: local diffusion in space, linear cooperation and superlinear competition.
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Recommended by Dr Tasso J Kaper
Footnotes
- 1
This convention being superseded by the previous one when the dimension is specifically equal to 1.
- 2
Same exception.
- 3
Regarding functions, some authors use
to denote what is here denoted . Thus the use of these two functional notations will be as sparse as possible and we will prefer the less ambiguous expressions 'nonnegative nonzero' and 'positive'. - 4
Let us emphasize once and for all that the vector field
is not to be confused with the real number c. The former is named after 'competition' whereas the latter is traditionally named after 'celerity'. - 5
They both prove the same type of results but we will refer hereafter only to the latter because the former does not cover, as stated, the one-dimensional space case.
