Asymptotic Dynamics in Populations Structured by Sensitivity to Global Warming and Habitat Shrinking
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Asymptotic Dynamics in Populations Structured by Sensitivity to Global Warming and Habitat Shrinking Tommaso Lorenzi · Alexander Lorz · Giorgio Restori
Received: 14 September 2012 / Accepted: 23 September 2013 © Springer Science+Business Media Dordrecht 2013
Abstract How to recast effects of habitat shrinking and global warming on evolutionary dynamics into continuous mutation/selection models? Bearing this question in mind, we consider differential equations for structured populations, which include mutations, proliferation and competition for resources. Since mutations are assumed to be small, a parameter ε is introduced to model the average size of phenotypic changes. A well-posedness result is proposed and the asymptotic behavior of the density of individuals is studied in the limit ε → 0. In particular, we prove the weak convergence of the density to a sum of Dirac masses and characterize the related concentration points. Moreover, we provide numerical simulations illustrating the theorems and showing an interesting sample of solutions depending on parameters and initial data. Keywords Structured populations · Asymptotic analysis · Concentration phenomena · Integrodifferential equations
1 Introduction Recent scientific literature shows the spreading of mathematical models for the evolution of species under the effects of external selective pressures [8, 14, 15, 21, 24, 25, 31, 33, 34,
T.L. is supported by the FIRB project, RBID08PP3J. A.L. is supported by a postdoc grant from the Fondation Sciences Mathématiques de Paris. T. Lorenzi Department of Mathematics, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy e-mail: [email protected]
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A. Lorz ( ) UPMC Univ Paris 06, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, INRIA-Rocquencourt, EPI BANG, 4, pl. Jussieu, 75252 Paris cedex 05, France e-mail: [email protected] G. Restori Sustainability Department, Pirelli & C. S.p.A., Via Pirelli 25, 20124 Milano, Italy e-mail: [email protected]
T. Lorenzi et al.
36]. In this paper, we consider a population structured by two continuous variables x and y representing the sensitivity to, respectively, habitat shrinking and global warming. The density of individuals with a sensitivity level u := (x, y) at time t is modeled by the real function f (t, x, y) ≥ 0 that satisfies ⎧ ∂ ⎪ ⎪ ⎨ f (t, u) = M[f ](t, u) + P t, u, (t) f (t, u) ∂t (1.1) ⎪ ⎪ ⎩ (t) = f (t, u)du, U
where: • t ∈ R+ , x ∈ X := [aX , bX ]k ⊂ Rk , y ∈ Y := [aY , bY ]l ⊂ Rl , with −∞ < aX , bX , aY , bY < ∞, integers k, l ≥ 1 and u ∈ U := X × Y ; • M[f ](t, u) describes the effects of renewal and mutations from parent to offspring; • P (t, u, (t)) models the per capita net growth rate of the population at time t and it is assumed to depend on the total size of the population at the same time instant, which is identified by (t). It is worth noting that we allow dimension k to be greater than one to account for the fact that sensitivity to habitat shrinking can result from the simultaneous expression of seve
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