Sílvia Cuadrado

Measure Solutions for Some Models in Population Dynamics

We give a direct proof of well-posedness of solutions to general selection-mutation and structured population models with measures as initial data. This is motivated by the fact that some stationary states of these models are measures and not L1 functions, so the measures are a more natural space to study their dynamics. Our techniques are based on distances between measures appearing in optimal transport and common arguments involving Picard iterations. These tools provide a simplification of previous approaches and are applicable or adaptable to a wide variety of models in population dynamics.

STATIONARY SOLUTIONS OF A SELECTION MUTATION MODEL: THE PURE MUTATION CASE ∗

A selection mutation equations model for the distribution of individuals with respect to the age at maturity is considered. In this model we assume that a mutation, perhaps very small, occurs in every reproduction where the noncompactness of the domain of the structuring variable and the two-dimensionality of the environment are the main features. Existence of stationary solutions is proved using the theory of positive semigroups and the infinite-dimensional version in Banach lattices of the Perron Frobenius theorem. The behavior of these stationary solutions when the mutation is small is studied.

Asymptotics of steady states of a selection–mutation equation for small mutation rate

A.C. and S. C. were partly supported by Grant nos MTM2008-06349-C03-03, 2009-SGR-345 and MTM2011-27739-C04-02. L. D. and G. R. were partly supported by Project CBDif-Fr ANR-08-BLAN-0333-01. G. R. was partly supported by Award no. KUK-I1-007-43 of Peter A. Markowich, made by the King Abdullah University of Science and Technology (KAUST). Finally, all authors were partly supported by the bilateral PICASSO project POLYCELL, Grant no. 22978WA.

A model for the adaptive dynamics of the maturation age

A density-dependent time-continuous model with two groups of age is considered both from the ecological and from the evolutionary point of view. In the ecological context, the maturation age is dealt with as a parameter and the existence of a non-trivial equilibrium attracting every non-zero solution is proved under some hypotheses including that the birth rate is an increasing function of the maturation age. In the evolutionary framework, conditions for the existence and uniqueness and for the property of being convergence-stable of the evolutionarily stable value of the maturation age are given.

Asymptotic profile in selection–mutation equations: Gauss versus Cauchy distributions

Abstract In this paper, we study the asymptotic (large time) behaviour of a selection–mutation–competition model for a population structured with respect to a phenotypic trait when the rate of mutation is very small. We assume that the reproduction is asexual, and that the mutations can be described by a linear integral operator. We are interested in the interplay between the time variable t and the rate ε of mutations. We show that depending on α > 0 , the limit ε → 0 with t = ε − α can lead to population number densities which are either Gaussian-like (when α is small) or Cauchy-like (when α is large).