Jules Guilberteau

The well-posedness of a non-local advection-selection-mutation problem deriving from adaptive dynamics models is shown for a wide family of initial data. A particle method is then developed, in order to approximate the solution of such problem by a regularised sum of weighted Dirac masses whose characteristics solve a suitably defined ODE system. The convergence of the particle method over any finite interval is shown and an explicit rate of convergence is given. Furthermore, we investigate the asymptotic-preserving properties of the method in large times, providing sufficient conditions for it to hold true as well as examples and counter-examples. Finally, we illustrate the method in two cases taken from the literature.

In this paper, we study the stability of an Ordinary Differential Equation (ODE) usually referred to as Cyclic Feedback Loop, which typically models a biological network of d molecules where each molecule regulates its successor in a cycle. Regulations, which can be either positive or negative, are modelled by increasing or decreasing functions. We make a complete analysis of this model for a wide range of functions (including affine and Hill functions) by determining the parameters for which bistability and oscillatory behaviours arise. These results encompass previous theoretical studies of gene regulatory networks, which are particular cases of this model.

In this paper, we study the stability of an Ordinary Differential Equation (ODE) usually referred to as Cyclic Feedback Loop, which typically models a biological network of molecules where each molecule regulates its successor in a cycle. Regulations, which can be either positive or negative, are modelled by increasing or decreasing functions. We make a complete analysis of this model for a wide range of functions (including affine and Hill functions) by determining the parameters for which bistability and oscillatory behaviours arise. These results encompass previous theoretical studies of gene regulatory networks, which are particular cases of this model.

We study the long-time behaviour of the advection-selection equation ∂tn(t, x) + ∇ • (f (x)n(t, x)) = (r(x) − ρ(t)) n(t, x), ρ(t) = \int_{R^d}{n(t, x)dx} t ≥ 0, x ∈ R^d, with an initial condition n(0, •) = n^0. In the field of adaptive dynamics, this equation typically describes the evolution of a phenotype-structured population over time. In this case, n(t, x) represents the density of the population characterised by a phenotypic trait x, the advection term '∇ • (f (x)n(t, x))' a cell differentiation phenomenon driving the individuals toward specific regions, and the selection term '(r(x) − ρ(t)) n(t, x)' the growth of the population, which is of logistic type through the total population size ρ(t) = \int_{R^d}{n(t, x)}dx. In the one-dimensional case x ∈ R, we prove that the solution to this equation can either converge to a weighted Dirac mass or to a function in L^1. Depending on the parameters n^0 , f and r, we determine which of these two regimes of convergence occurs, and we specify the weight and the point where the Dirac mass is supported, or the expression of the L^1-function which is reached.