Stochastic prey-predator relationships: A random differential equation approach

A stochastic model describing two interacting populations is considered. The model involves a random differential equation of the form dX/dt=A(t)X+Y(t) where the random matrixA and vectorY represent the interactions and growth rates respectively andX is a (random) vector the components of which are the logarithms of the population's sizes. An expression for the solution of the above equation is obtained whence its statistical properties can be determined. Alternatively, a method based on Liouville's theorem is used to obtain the probability distribution of the solution. Application of both methods to simple cases indicates that the random solution is asymptotically stable in the mean even when the solution to the associated deterministic equation is not, viz. in the absence of self interactions.