Growth with regulation in fluctuating environments. II. Intrinsic lower bounds to population size
[For part I see the preceding review, Zbl 0553.92016.] A third order deterministic model is proposed for population growth characterized by three equilibrium states: the zero population size, the carrying capacity and a ''threshold level'' after which ultimate extinction is secured. The authors analyze the above model when random fluctuations are introduced in the intrinsic fertility rate which then becomes a stationary normal process \(a+\Lambda (t)\) with \(\Lambda\) (t) being a white noise. A stochastic differential equation of Stratonovich type is thus obtained, which defines a temporally homogeneous diffusion process X(t). By means of a monotone transformation a Wiener process Y(t) is obtained such that its transition p.d.f. allows an easy computation of the transition p.d.f. of X(t); the asymptotic behaviour of that is analyzed. An alternative equation involving a logarithmic nonlinearity is also considered, which still allows three equilibrium states, and, as in the previous case, can be transformed into a stochastic differential equation for a Wiener process. Furthermore, a model which does not always allow a steady state distribution is analyzed; in the case in which, under suitable conditions, it does exist, this distribution is used to obtain the asymptotic moments of the population. Finally, a model with a logarithmic singularity is studied which always admits a steady state distribution, but on the other hand none of the moments always exist.
stochastic differential equation of Stratonovich type
scientific article; zbMATH DE number 3882256