Models of individual growth in a random environment: study and application of first passage times

Abstract

We study the first-passage times for models of individual growth of animals in randomly fluctuating environments. In particular, we present results on the mean and variance of the first-passage time by a high threshold value (higher than the initial size). The models considered are stochastic differential equations of the form dY(t)=β(α−Y(t))dt+σdW(t), Y(t0) = y0, where Y(t)= g(X(t)) is a transformed size, g being a strictly increasing C1 function of the actual animal size X(t) at time t, σ measures the effect of random environmental fluctuations on growth, W(t) is the standard Wiener process, and y0 is the transformed size (assumed known) at the initial instant t 0. Results are illustrated using cattle weight data, to which we have applied the Bertalanffy-Richards (g(x) = x^c ) and the Gompertz (g(x) = lnx) stochastic models.