On hierarchical competition through reduction of individual growth.

We consider a population organised hierarchically with respect to size in such a way that the growth rate of each individual depends only on the presence of larger individuals. As a concrete example one might think of a forest, in which the incidence of light on a tree (and hence how fast it grows) is affected by shading by taller trees. The classic formulation of a model for such a size-structured population employs a first order quasi-linear partial differential equation equipped with a non-local boundary condition. However, the model can also be formulated as a delay equation, more specifically a scalar renewal equation, for the population birth rate. After discussing the well-posedness of the delay formulation, we analyse how many stationary birth rates the equation can have in terms of the functional parameters of the model. In particular we show that, under reasonable and rather general assumptions, only one stationary birth rate can exist besides the trivial one (associated to the state in which there are no individuals and the population birth rate is zero). We give conditions for this non-trivial stationary birth rate to exist and analyse its stability using the principle of linearised stability for delay equations. Finally, we relate the results to the alternative, partial differential equation formulation of the model.

On the Reproduction Number of a Gut Microbiota Model.

A spatially structured linear model of the growth of intestinal bacteria is analysed from two generational viewpoints. Firstly, the basic reproduction number associated with the bacterial population, i.e. the expected number of daughter cells per bacterium, is given explicitly in terms of biological parameters. Secondly, an alternative quantity is introduced based on the number of bacteria produced within the intestine by one bacterium originally in the external media. The latter depends on the parameters in a simpler way and provides more biological insight than the standard reproduction number, allowing the design of experimental procedures. Both quantities coincide and are equal to one at the extinction threshold, below which the bacterial population becomes extinct. Optimal values of both reproduction numbers are derived assuming parameter trade-offs.

Asymptotic profile in selection-mutation equations: Gauss versus Cauchy distributions.

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).

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.

A general structured model for a sequential hermaphrodite population.

This paper introduces and analyzes a model of sequential hermaphroditism in the framework of continuously structured population models with sexual reproduction. The model is general in the sense that the birth, transition (from one sex to the other) and death processes of the population are given by arbitrary functions according to a biological meaningful hypotheses. The system is reduced to a single equation introducing the intrinsic sex-ratio subspace. The steady states are analyzed and illustrated for several cases. In particular, neglecting the competition for resources we have explicitly found a unique non-trivial equilibrium which is unstable.

Asymptotic stability of equilibria of selection-mutation equations.

We study local stability of equilibria of selection-mutation equations when mutations are either very small in size or occur with very low probability. The main mathematical tools are the linearized stability principle and the fact that, when the environment (the nonlinearity) is finite dimensional, the linearized operator at the steady state turns out to be a degenerate perturbation of a known operator with spectral bound equal to 0. An example is considered where the results on stability are applied.

Small mutation rate and evolutionarily stable strategies in infinite dimensional adaptive dynamics.

An integrodifferential equations model for the distribution of individuals with respect to the age at maturity is considered. Mutation is modeled by an integral operator. Results concerning the behaviour of the steady states and their relation to evolutionarily stable strategies when the mutation rate is small are given. The same results are obtained for a (rather) general class of models that include the one mentioned before.

Hopf bifurcation in a structured population model for the sexual phase of monogonont rotifers.

We are studying a population of monogonont rotifers in the context of non-linear age-dependent models. In the sexual phase of their reproductive cycle we consider the population structured by age, and composed of three subclasses: virgin mictic females, mated mictic females, and haploid males. The model system has a unique stationary population density which is stable as long as a parameter, related to male-female encounter rate, remains below a critical value. When the parameter increases beyond this critical value, the stationary solution becomes unstable and a stable limit cycle (isolated periodic orbit) appears. The occurrence of this supercritical Hopf bifurcation is shown analytically.