Stochastic Path Perturbation Approach Applied to Non–Local Non–Linear Equations in Population Dynamics

¹ Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico 87501, USA
² Instituto de Investigaciones Filosóficas, Bulnes 642, 1428 Buenos Aires, Argentina
³ Universidad San Sebastián, Lota 2465, Santiago 7500000, Chile
⁴ Centro Atómico Bariloche, Instituto Balseiro and CONICET, 8400 Bariloche, Argentina

^⋆ Corresponding author. E-mail: caceres@cab.cnea.gov.ar

Abstract

We described the first passage time distribution associated to the stochastic evolution from an unstable uniform state to a patterned one (attractor of the system), when the time evolution is given by an integro-differential equation describing a population model. In order to obtain analytical results we used the Stochastic Path Perturbation Approach introducing a minimum coupling approximation into the nonlinear dynamics, and a stochastic multiscale perturbation expansion. We show that the stochastic multiscale perturbation is a necessary tool to handle other problems like: nonlinear instabilities and multiplicative stochastic partial differential equations. A small noise parameter was introduced to define the random escape of the stochastic field. We carried out Monte Carlo simulations in a non-local Fisher like equation, to show the agreement with our theoretical predictions.