Àngel Calsina

A model of physiologically structured population dynamics with a nonlinear individual growth rate

In this article we consider a size structured population model with a nonlinear growth rate depending on the individuals size and on the total population. Our purpose is to take into account the competition for a resource (as it can be light or nutrients in a forest) in the growth of the individuals and study the influence of this nonlinear growth in the population dynamics. We study the existence and uniqueness of solutions for the model equations, and also prove the existence of a (compact) global attractor for the trajectories of the dynamical system defined by the solutions of the model. Finally, we obtain sufficient conditions for the convergence to a stationary size distribution when the total population tends to a constant value, and consider some simple examples that allow us to know something about their global dynamics.

Equations for biological evolution

In this paper we consider mathematical models inspired by the mechanisms of biological evolution. We take populations which are subject to interaction and mutation. In the cases we consider, the interaction is through competition or through a prey-predator relationship. The models consider the specific characteristics as taking values in real intervals and the equations are of the integro—differential type. In the case of competition, thanks to the fact that some of the equations have solutions which are quite explicit, we succeed in proving the existence of attracting stationary solutions. In the case of prey and predator, using techniques of dynamical systems in infinite-dimensional spaces, we succeed in showing the existence of a global attractor, which in some instances reduces to a point. Our analysis takes into account having δ distributions, corresponding to all individuals having the same characteristics, as possible populations.

Stability and instability of equilibria of an equation of size structured population dynamics

In this paper we consider a quasilinear equation with a nonlinear boundary condition modelling the dynamics of a biological population structured by size. We suppose vital rates depending on the total population. This hypothesis introduces some nonlinearities on the equation and on the boundary condition. We study the existence and uniqueness of solution of the initial value problem and the existence of stationary solutions. After we calculate the spectrum of the linearization at an equilibrium and we study its (local) stability.

BASIC THEORY FOR A CLASS OF MODELS OF HIERARCHICALLY STRUCTURED POPULATION DYNAMICS WITH DISTRIBUTED STATES IN THE RECRUITMENT

In this paper we present a proof of existence and uniqueness of solution for a class of PDE models of size structured populations with distributed state-at-birth and having nonlinearities defined by an infinite-dimensional environment. The latter means that each member of the population experiences an environment according to a sort of average of the population size depending on the individual size, rank or any other variable structuring the population. The proof of the local existence and uniqueness of solution as well as the continuous dependence on initial continuous is based on the general theory of quasi-linear evolution equations in nonreflexive Banach spaces, while the global existence proof is based on the integration of the local solution along characteristic curves.

Steady states in a structured epidemic model with Wentzell boundary condition

We introduce a non-linear structured population model with diffusion in the state space. Individuals are structured with respect to a continuous variable which represents a pathogen load. The class of uninfected individuals constitutes a special compartment that carries mass; hence the model is equipped with generalized Wentzell (or dynamic) boundary conditions. Our model is intended to describe the spread of infection of a vertically transmitted disease, for e.g., Wolbachia in a mosquito population. Therefore, the (infinite dimensional) non-linearity arises in the recruitment term. First, we establish global existence of solutions and the principle of linearised stability for our model. Then, in our main result, we formulate simple conditions which guarantee the existence of non-trivial steady states of the model. Our method utilises an operator theoretic framework combined with a fixed-point approach. Finally in the last section, we establish a sufficient condition for the local asymptotic stability of the positive steady state.

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.

OPTIMAL LATENT PERIOD IN A BACTERIOPHAGE POPULATION MODEL STRUCTURED BY INFECTION-AGE

We study the lysis timing of a bacteriophage population by means of a continuously infection-age-structured population dynamics model. The features of the model are the infection process of bacteria, the death process, and the lysis process which means the replication of bacteriophage viruses inside bacteria and the destruction of them. The time till lysis (or latent period) is assumed to have an arbitrary distribution. We have carried out an optimization procedure, and we have found that the latent period corresponding to maximal fitness (i.e. maximal growth rate of the bacteriophage population) is of fixed length. We also study the dependence of the optimal latent period on the amount of susceptible bacteria and the number of virions released by a single infection. Finally, the evolutionarily stable strategy of the latent period is also determined as a fixed period taking into account that super-infections are not considered.

Evolution of age-dependent sex-reversal under adaptive dynamics.

We investigate the evolution of the age (or size) at sex-reversal in a model of sequential hermaphroditism, by means of the function-valued adaptive dynamics. The trait is the probability law of the age at sex-reversal considered as a random variable. Our analysis starts with the ecological model which was first introduced and analyzed by Calsina and Ripoll (Math Biosci 208(2), 393–418, 2007). The structure of the population is extended to a genotype class and a new model for an invading/mutant population is introduced. The invasion fitness functional is derived from the ecological setting, and it turns out to be controlled by a formula of Shaw–Mohler type. The problem of finding evolutionarily stable strategies is solved by means of infinite-dimensional linear optimization. We have found that these strategies correspond to sex-reversal at a single particular age (or size) even if the set of feasible strategies is considerably broader and allows for a probabilistic sex-reversal. Several examples, including in addition the population-dynamical stability, are illustrated. For a special case, we can show that an unbeatable size at sex-reversal must be larger than 69.3% of the expected size at death.