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A Markov jump process modelling animal group size statistics. (English) Zbl 1450.60045

Summary: We translate a coagulation-fragmentation model, describing the dynamics of animal group size distributions, into a model for the population distribution and associate the 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. Theor. Biol. 224, No. 4, 451–457 (2003; Zbl 1441.92049)] 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 P. Degond et al. [J. Nonlinear Sci. 27, No. 2, 379–424 (2017; Zbl 1382.92260)], 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.

MSC:

60J74 Jump processes on discrete state spaces
65C30 Numerical solutions to stochastic differential and integral equations
65C35 Stochastic particle methods
45J05 Integro-ordinary differential equations
70F45 The dynamics of infinite particle systems
92D50 Animal behavior
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