STOCHASTIC PDE MODEL FOR SPATIAL POPULATION GROWTH IN RANDOM ENVIRONMENTS.

The paper is concerned with a class of stochastic reaction-diffusion equations arising from a spatial population growth model in random environments. Under some sufficient conditions, Theorem 3.1 shows that the equation has a unique positive global solution in space H¹(D). Then it is proven in Theorem 4.1 that the solution, as the population size, is ultimately bounded in the mean L²-norm as the time tends to infinity. An almost-sure upper bound is also obtained for the long run time-average of the exponential rate of the population growth in L²-norm together with the L^p-moment of the population size with p ≥ 2: It is also shown in Theorem 4.3 that there is a unique invariant measure that leads to a stationary population distribution. For illustration, an example is given. [ABSTRACT FROM AUTHOR]