Onno A. van Herwaarden

Resilience and persistence in the context of stochastic population models

The resilience of an ecological system is defined by the velocity of the system as it returns to its equilibrium state after some perturbation. Since the system does not arrive exactly at the equilibrium within a finite time, the definition is based on the time needed to decrease the distance to the equilibrium with some fraction. In this study it is found that for stochastic populations this arbitrarily chosen function disappears because the equilibrium point can be replaced by a small (confidence) domain containing the equilibrium. The size of this domain is a measure for the (local) persistence of the system. This method is fully worked out for the stochastic logistic equation as well as for a prey-predator system.

The Fokker—Planck Equation: One Dimension

Let us consider a stochastic process described by the scalar X(t) with probability density p(t, x) satisfying

Singular Perturbation Analysis of the Differential Equations for the Exit Probability and Exit Time in One Dimension

\frac{{\partial p}}{{\partial t}} = - \frac{\partial }{{\partial x}}(b(x)p) + \frac{{{\varepsilon ^2}}}{2}\frac{{{\partial ^2}}}{{\partial {x^2}}}(a(x)p)

Confidence Domain, Return Time and Control

(3.1a) for 0 0

Extinction in Systems of Interacting Biological Populations

p(0,x) = {p_0}(x),\;\;\;\int\limits_0^1 {{p_0}(x)dx = 1.}

Breakdown of a Chemostat Exposed to Stochastic Noise

(3.1b)