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Resident-invader dynamics in infinite dimensional systems. (English) Zbl 1388.35119

In the present work, the authors introduce and study infinite-dimensional dynamical systems which models the competition of two phenotypes of the same species in the form: \[ u_t=A(\alpha )u+F(\alpha ,G(\alpha )u+G(\beta )v)u \]
\[ v_t=A(\beta )v+F(\beta ,G(\alpha )u+G(\beta )v)v \]
\[ u|_t=0=u_0\;,\quad v|_{t=0}=v_0\;,\quad u_0,v_0\in X_+ \] with traits and \(\beta\), where \(\alpha ,\beta \in S\subset\mathbb R^1\), \(S\) is an open interval, \(A\) is a linear closed and sectorial operator defined on \(D(A)\), which is assumed to be independent of \(\alpha \in S\). \(D(A)\) is a dense subset of \(X=C(\bar\Omega ,\mathbb R)\), \(X_+\subset X\) is the subset of the nonnegative functions, \(\Omega\) subset \(\mathbb R^n\) is a smooth bounded domain, \(F:S\times X\to X\) and \(G:S\to X\) are smooth functions. This model extend the previous models of S. Geritz and its coauthors. The main motivation for this work is to investigate an important for the evolutionary biology problem concerning the competition of two phenotypes of the same species, appearing in the roles as a resident and an invader for some habitat. As first result, the authors obtain sufficient conditions, under which they prove an infinite dimensional version of the Tube Theorem. The Tube theorem says that if \(u\) and \(v\) are phenotypes of the same species with similar traits \(\alpha \approx \beta\), then the total population of the two will be approximately the total population of a single species, i.e. \(u+v\approx \theta_\alpha\), where \(\theta_\alpha\) is a linearly stable equilibrium. Those conditions are based on invasibility criteria, for instance, evolutionarily stable strategies in the framework of adaptive dynamics. The problem of convergence to equilibrium is solved through adapting the Hadamard graph transform method, which allow to obtain the existence of a one dimensional invariant manifold for the flow within the tube. Note that the adapted Hadamard’s graph transform method does not depend on the monotonicity of the two-species system.

MSC:

35K90 Abstract parabolic equations
35Q92 PDEs in connection with biology, chemistry and other natural sciences
92D25 Population dynamics (general)
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