Bifurcacions genériques d'atractors en sistemes de reacció i difusió

In this work we write down in some detail the bifurcation theory of stationary states of reaction-diffusion equations. First, we prove, adapting notes of looss on the Navier-Stokes equations, that under some weak hypothesis a reaction-diffusion equation defines a differentiable dynamical systems in the Sobolev space H2 with some boundary conditions . Then it is proven that a rest point where the infinitessimal generator of the linear part of the system has a spectrum in the left hand plane is stable . We prove then that when , depending on a parameter, a simple eigenvalue crosses to the right hand plane, a bifurcation appears (generically). In the last chapter we propose a model for dune formation, which does not have the pretension of being faithful, but which illustrates how the theory given is usefu

Un model de dinàmica de poblacions per a l'anèmia aplàstica

L'anèmia aplàstica és una greu malaltia de la sang que consisteix en grans fluctuacions de la quantitat de glòbuls vermells, i en particular, en disminucions dràstiques del nombre d'aquests, sense que se'n vegin però afectades les propietats individuals. La disminució de la quantitat d'hemoglobina en sang és doncs deguda a una disminució del nombre de glòbuls vermells madurs i no a cap problema intrínsec d'aquests ni a la manca de ferro com en d'altres anèmies comunes

Semilinear formulation of a hyperbolic system of partial differential equations

Acord transformatiu CRUE-CSICIn this paper, we solve the Cauchy problem for a hyperbolic system of first-order PDEs defined on a certain Banach space X. The system has a special semilinear structure because, on the one hand, the evolution law can be expressed as the sum of a linear unbounded operator and a nonlinear Lipschitz function but, on the other hand, the nonlinear perturbation takes values not in X but on a larger space Y which is related to X. In order to deal with this situation we use the theory of dual semigroups. Stability results around steady states are also given when the nonlinear perturbation is Fréchet differentiable. These results are based on two propositions: one relating the local dynamics of the nonlinear semiflow with the linearised semigroup around the equilibrium, and a second relating the dynamical properties of the linearised semigroup with the spectral values of its generator. The later is proven by showing that the Spectral Mapping Theorem always applies to the semigroups one obtains when the semiflow is linearised. Some epidemiological applications involving gut bacteria are commented

Non-local reaction-diffusion equations modelling predator-prey coevolution

In this paper we examine a prey-predator system with a characteristic of the predator subject to mutation. The ultimate equilibrium of the system is found by Maynard-Smith et al. by the so called ESS (Evolutionary Stable Strategy). Using a system of reaction-diffusion equations with non local terms, we conclude the ESS result for the diffusion coefficient tending to zero, without resorting to any optimization criterion

About ghost transients in spatial continuous media

Altres ajuts: acords transformatius de la UABThe impact of space on ecosystem dynamics has been a matter of debate since the dawn of theoretical ecology. Several studies have revealed that space usually involves an increase in transients' times, promoting the so-called supertransients. However, the effect of space and diffusion in transients close to bifurcations has not been thoroughly investigated. In non-spatial deterministic models such as those given by ordinary differential equations transients become extremely long in the vicinity of bifurcations. Specifically, for the saddle-node (s-n) bifurcation the time delay, τ, follows τ∼|μ−μ|; μ and μ being the bifurcation parameter and the bifurcation value, respectively. Such long transients are labeled delayed transitions and are governed by the so-called ghosts. Here, we explore a simple model with intra-specific cooperation (autocatalysis) and habitat loss undergoing a s-n bifurcation using a partial differential equations (PDE) approach. We focus on the effects of diffusion in the ghost extinction transients right after the tipping point found at a critical habitat loss threshold. Our results show that the bifurcation value does not depend on diffusion. Despite transients' length typically increase close to the bifurcation, we have observed that at extreme values of diffusion, both small and large, extinction times remain long and close to the well-mixed results. However, ghosts lose influence at intermediate diffusion rates, leading to a dramatic reduction of transients' length. These results, which strongly depend on the initial size of the population, are shown to remain robust for different initial spatial distributions of cooperators. A simple two-patch metapopulation model gathering the main results obtained from the PDEs approach is also introduced and discussed. Finally, we provide analytical results of the passage times and the scaling for the model under study transformed into a normal form. Our findings are discussed within the framework of ecological transients

On the basic reproduction number in continuously structured populations

In the framework of population dynamics, the basic reproduction number (Formula presented.) is, by definition, the expected number of offspring that an individual has during its lifetime. In constant and time periodic environments, it is calculated as the spectral radius of the so-called next-generation operator. In continuously structured populations defined in a Banach lattice X with concentrated states at birth, one cannot define the next-generation operator in X. In the present paper, we present an approach to compute the basic reproduction number of such models as the limit of the basic reproduction number of a sequence of models for which (Formula presented.) can be computed as the spectral radius of the next-generation operator. We apply these results to some examples: the (classical) size-dependent model, a size-structured cell population model, a size-structured model with diffusion in structure space (under some particular assumptions), and a (physiological) age-structured model with diffusion in structure space

Oscillations in a molecular structured cell population model

We consider a nonlinear cyclin content structured model of a cell population divided into proliferative and quiescent cells. We show, for particular values of the parameters, existence of solutions that do not depend on the cyclin content. We make numerical simulations for the general case obtaining, for some values of the parameters convergence to the steady state but also oscillations of the population for others