The Stochastic Beverton-Holt Equation and the

In the Beverton Holt difference equation of population biology with intrinsic growth parameter above its critical value, any initial non-zero population will approach an asymptotically stable fixed point, the carrying capacity of the environment. When this carrying capacity is allowed to vary periodically it is known that there is a globally asymptotically stable periodic solution and the average of the state variable along this solution is strictly less than the average of the carrying capacities, i.e. the varying environment has a deleterious effect on the state average. In this work we consider the case of a randomly varying environment and show that there is a unique invariant density to which all other density distributions on the state variable converge. Further, for every initial non-zero state variable and almost all random sequences of carrying capacities, the averages of the state variable along an orbit and the carrying capacities exist and the former is strictly less than the latter.