POPULATION-GROWTH WITH RANDOMLY DISTRIBUTED JUMPS

The growth of populations with continuous deterministic and random jum p components is treated. Three special models in which random jumps oc cur at the time of events of a Poisson process and admit formal explic it solutions are considered: A) Logistic growth with random disasters having exponentially distributed amplitudes; B) Logistic growth with r andom disasters causing the removal of a uniformly distributed fractio n of the population size; and C) Exponential decay with sudden increas es (bonanzas) in the population and with each increase being an expone ntially distributed fraction of the current population. Asymptotic and numerical methods are employed to determine the mean extinction time for the population, qualitatively and quantitatively. For Model A, thi s time becomes exponentially large as the carrying capacity becomes mu ch larger than the mean disaster size. Implications for colonizing spe cies for Model A are discussed. For Models B and C, the practical noti on of a small, but positive, effective extinction level is chosen, and in these cases the expected extinction time rises rapidly with popula tion size, yet at less than an exponentially large order.