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Register a new userA Markov jump process modelling animal group size statistics
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We translate a coagulation-framentation model, describing the dynamics of animal group size distributions, into a model for the population distribution and associate the blue{nonlinear} evolution equation with a Markov jump process of a type introduced in classic work of H.~McKean. In particular this formalizes a model suggested by H.-S. Niwa [J.~Theo.~Biol.~224 (2003)] with simple coagulation and fragmentation rates. Based on the jump process, we develop a numerical scheme that allows us to approximate the equilibrium for the Niwa model, validated by comparison to analytical results by Degond et al. [J.~Nonlinear Sci.~27 (2017)], and study the population and size distributions for more complicated rates. Furthermore, the simulations are used to describe statistical properties of the underlying jump process. We additionally discuss the relation of the jump process to models expressed in stochastic differential equations and demonstrate that such a connection is justified in the case of nearest-neighbour interactions, as opposed to global interactions as in the Niwa model.
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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.
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