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Discrete stochastic processes, replicator and Fokker-Planck equations of coevolutionary dynamics in finite and infinite populations

Jens Christian Claussen (2008)

Banach Center Publications

Finite-size fluctuations in coevolutionary dynamics arise in models of biological as well as of social and economic systems. This brief tutorial review surveys a systematic approach starting from a stochastic process discrete both in time and state. The limit N → ∞ of an infinite population can be considered explicitly, generally leading to a replicator-type equation in zero order, and to a Fokker-Planck-type equation in first order in 1/√N. Consequences and relations to some previous approaches...

Lévy Processes, Saltatory Foraging, and Superdiffusion

J. F. Burrow, P. D. Baxter, J. W. Pitchford (2008)

Mathematical Modelling of Natural Phenomena

It is well established that resource variability generated by spatial patchiness and turbulence is an important influence on the growth and recruitment of planktonic fish larvae. Empirical data show fractal-like prey distributions, and simulations indicate that scale-invariant foraging strategies may be optimal. Here we show how larval growth and recruitment in a turbulent environment can be formulated as a hitting time problem for a jump-diffusion process. We present two theoretical results. Firstly,...

Macroscopic models of collective motion and self-organization

Pierre Degond, Amic Frouvelle, Jian-Guo Liu, Sebastien Motsch, Laurent Navoret (2012/2013)

Séminaire Laurent Schwartz — EDP et applications

In this paper, we review recent developments on the derivation and properties of macroscopic models of collective motion and self-organization. The starting point is a model of self-propelled particles interacting with its neighbors through alignment. We successively derive a mean-field model and its hydrodynamic limit. The resulting macroscopic model is the Self-Organized Hydrodynamics (SOH). We review the available existence results and known properties of the SOH model and discuss it in view...

Modelling and Mathematical Analysis of the Glass Eel Migration in the Adour River Estuary

M. Odunlami, G. Vallet (2012)

Mathematical Modelling of Natural Phenomena

In this paper we are interested in a mathematical model of migration of grass eels in an estuary. We first revisit a previous model proposed by O. Arino and based on a degenerate convection-diffusion equation of parabolic-hyperbolic type with time-varying subdomains. Then, we propose an adapted mathematical framework for this model, we prove a result of existence of a weak solution and we propose some numerical simulations.

Numerical solution of a new hydrodynamic model of flocking

Kučera, Václav, Živčáková, Andrea (2015)

Programs and Algorithms of Numerical Mathematics

This work is concerned with the numerical solution of a hydrodynamic model of the macroscopic behavior of flocks of birds due to Fornasier et al., 2011. The model consists of the compressible Euler equations with an added nonlocal, nonlinear right-hand side. As noticed by the authors of the model, explicit time schemes are practically useless even on very coarse grids in 1D due to the nonlocal nature of the equations. To this end, we apply a semi-implicit discontinuous Galerkin method to solve the...

On a Caginalp phase-field system with a logarithmic nonlinearity

Charbel Wehbe (2015)

Applications of Mathematics

We consider a phase field system based on the Maxwell Cattaneo heat conduction law, with a logarithmic nonlinearity, associated with Dirichlet boundary conditions. In particular, we prove, in one and two space dimensions, the existence of a solution which is strictly separated from the singularities of the nonlinear term and that the problem possesses a finite-dimensional global attractor in terms of exponential attractors.

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