Numerical approximation of a coagulation-fragmentation model for animal group size statistics

We study numerically a coagulation-fragmentation model derived by Niwa [17] and further elaborated by Degond et al. [5]. In [5] a unique equi- librium distribution of group sizes is shown to exist in both cases of continuous and discrete group size distributions. We provide a numerical investigation of these equilibria using three different methods to approximate the equilibrium: a recursive algorithm based on the work of Ma et. al. [12], a Newton method and the resolution of the time-dependent problem. All three schemes are val- idated by showing that they approximate the predicted small and large size asymptotic behaviour of the equilibrium accurately. The recursive algorithm is used to investigate the transition from discrete to continuous size distributions and the time evolution scheme is exploited to show uniform convergence to equilibrium in time and to determine convergence rates

A Mixed Discrete-Continuous Fragmentation Model

Motivated by the occurrence of "shattering" mass-loss observed in purely continuous fragmentation models, this work concerns the development and the mathematical analysis of a new class of hybrid discrete--continuous fragmentation models. Once established, the model, which takes the form of an integro-differential equation coupled with a system of ordinary differential equations, is subjected to a rigorous mathematical analysis, using the theory and methods of operator semigroups and their generators. Most notably, by applying the theory relating to the Kato--Voigt perturbation theorem, honest substochastic semigroups and operator matrices, the existence of a unique, differentiable solution to the model is established. This solution is also shown to preserve nonnegativity and conserve mass