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            <title xml:lang="en">Persistence, extinction and spreading properties of non-cooperative Fisher--KPP systems in space-time periodic media</title>
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                <term xml:lang="en">space-time periodicity</term>
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              <p>This paper is concerned with asymptotic persistence, extinction and spreading properties for non-cooperative Fisher–KPP systems with space-time periodic coefficients. In a preceding paper, a family of generalized principal eigenvalues associated with an appropriate linear problem was studied. Here, a relation with semilinear systems is established. When the maximal generalized principal eigenvalue is negative, all solutions to the Cauchy problem become locally uniformly positive in long-time. In contrast with the scalar case, multiple space-time periodic uniformly positive entire solutions might coexist. When another, possibly smaller, generalized principal eigenvalue is nonnegative, then on the contrary all solutions to the Cauchy problem vanish uniformly and the zero solution is the unique space-time periodic nonnegative entire solution. When the two generalized principal eigenvalues differ and zero is in between, the long-time behavior depends on the decay at infinity of the initial data. Finally, with similar arguments, a Freidlin–Gärtner-type formulafor the asymptotic spreading speed of solutions with compactly supported initial data is established.</p>
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