Structured Model Conserving Biomass for the Size-spectrum Evolution in Aquatic Ecosystems

Mathematical modelling of the evolution of the size-spectrum dynamics in aquatic ecosystems was discovered to be a powerful tool to have a deeper insight into impacts of human- and environmental driven changes on the marine ecosystem. In this article we propose to investigate such dynamics by formulating and investigating a suitable model. The underlying process for these dynamics is given by predation events, causing both growth and death of individuals, while keeping the total biomass within the ecosystem constant. The main governing equation investigated is deterministic and non-local of quadratic type, coming from binary interactions. Predation is assumed to strongly depend on the ratio between a predator and its prey, which is distributed around a preferred feeding preference value. Existence of solutions is shown in dependence of the choice of the feeding preference function as well as the choice of the search exponent, a constant influencing the average volume in water an individual has to search until it finds prey. The equation admits a trivial steady state representing a died out ecosystem, as well as - depending on the parameterregime - steady states with gaps in the size spectrum, giving evidence to the well known cascade effect. The question of stability of these equilibria is considered, showing convergence to the trivial steady state in a certain range of parameters. These analytical observations are underlined by numerical simulations, with additionally exhibiting convergence to the non-trivial equilibrium for specific ranges of parameters

Stopping a reaction-diffusion front

We revisit the problem of pinning a reaction-diffusion front by a defect, in particular by a reaction-free region. Using collective variables for the front and numerical simulations, we compare the behaviors of a bistable and monostable front. A bistable front can be pinned as confirmed by a pinning criterion, the analysis of the time independant problem and simulations. Conversely, a monostable front can never be pinned, it gives rise to a secondary pulse past the defect and we calculate the time this pulse takes to appear. These radically different behaviors of bistable and monostable fronts raise issues for modelers in particular areas of biology, as for example, the study of tumor growth in the presence of different tissues

A phenomenological model of cell-cell adhesion mediated by cadherins

International audienceWe present a phenomenological model intended to describe at the protein population level the formation of cell-cell junctions by the local recruitment of homophilic cadherin adhesion receptors. This modeling may have a much wider implication in biological processes since many adhesion receptors, channel proteins and other membrane-born proteins associate in clusters or oligomers at the cell surface. Mathematically, it consists in a degenerate reaction-diffusion system of two partial differential equations modeling the time-space evolution of two cadherin populations over a surface: the first one represents the diffusing cadherins and the second one concerns the fixed ones. After discussing the stability of the solutions of the model, we perform numerical simulations and show relevant analogies with experimental results. In particular, we show patterns or aggregates formation for a certain set of parameters. Moreover, perturbing the stationary solution, both density populations converge in large times to some saturation level. Finally, an exponential rate of convergence is numerically obtained and is shown to be in agreement, for a suitable set of parameters, with the one obtained in some in vitro experiments