Population Extinction Probabilities and Methods of Estimation for Population Stochastic Differential Equation Models

Abstract

The basic population growth models in a randomly fluctuating environment are the stochastic differential equations dlnN/dt=r (Maithusian), dlnN/dt=r(1-N/K) (K > o; logistic), and dlnN/dt=r (lnK -lnN) (lnK > o; Gompertz), where N = N(t) is the population size at time t and r = r(t) = ro + σ ɛ(t) (σ > o) is a random process with ɛ(t) “standard” white noise. A reference to “colored” noise is also made. In applications to real populations we need parameter estimates based on the usually available discrete observations of a single realization. This paper gives moment and ML estimators. It also gives estimates of the probability of the population dropping below an extinction threshold within a given time. These results can be applied to fisheries and environmental impact assessment.