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

• DOI: 10.3934/nhm.2017009
• Publication Date: 2017-05-23T19:33:36Z
• Authors: Pierre Degond; ; Maximilian Engel
• Keywords: Mathematics, Interdisciplinary Applications; POPULATION BALANCE-EQUATIONS; Science & Technology; 330; SCHEME; Applied Mathematics; Populations and Evolution (q-bio.PE); 92D50, 92C31, 82B40, 82B44; FOS: Physical sciences; Mathematical Physics (math-ph); AGGREGATION; 510; FISH; FOS: Biological sciences; Physical Sciences; DISTRIBUTIONS; Interdisciplinary Applications; [MATH]Mathematics [math]; Quantitative Biology - Populations and Evolution; LAW; Mathematics; Mathematical Physics

We study numerically a coagulation-fragmentation model derived by Niwa and further elaborated by Degond et al., where a unique equilibrium 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., a Newton method and the resolution of the time-dependent problem. All three schemes are validated 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.