Diffusion with stochastic resetting of interacting particles emerging from a model of population genetics

Abstract

In this paper, we put forward a connection between diffusion with resetting and a certain extension of Ohta-Kimura model, inspired on what was carried out in Da Silva and Fragoso (2018 J. Phys. A: Math. Theor. 51 505002). The contribution is twofold: (1) we derive a new extension of Ohta-Kimura model, dubbed here new extended version of Ohta-Kimura ladder model (NOKM) which bears a strong liaison with the so-called jump-type Fleming-Viot process. The novelty here, when we compare with the classical Ohta-Kimura model, is that we now have an operator which allows simultaneous interaction among many individuals. It has to do with a generalized branching mechanism i.e. m individual types extinguish and one individual type splits into m copies. The system of evolution equations arising from NOKM can be seen as a system of n-dimensional Kolmogorov forward equations (or Fokker-Planck equations). The analysis requires an amenable armory of concepts and mathematical technique to analyze some relevant issues such as correlation, indistinguishability of individuals and stationarity; (2) nudged by the ideas brought to bear in Da Silva and Fragoso (2018 J. Phys. A: Math. Theor. 51 505002), we advance in this agenda here by making an initial incursion on the connection between diffusion with resetting and the NOKM. The connection which relies on the similarities between the models allows, in some cases, that relevant results obtained for one model can be translated to the other model framework by taking advantage of the technique used to derive a result in one of the models. Through the development of the population genetic model and its reinterpretation in terms of diffusion with stochastic resetting, we show an invariance property of the correlation between two interacting particles that reset at a time-inhomogeneous resetting rate. Pushing forward the ideas we obtain the stationary state of a new model for an n-particle system under an anisotropic diffusion with resetting. Although the results using this approach are of recent vintage, we believe that this avenue of research seems to be very encouraging.