Extinction and quasi-stationarity in the Verhulst logistic model.
We formulate and analyse a stochastic version of the Verhulst deterministic model for density-dependent growth of a single population. Three parameter regions with qualitatively different behaviours are identified. Explicit approximations of the quasi-stationary distribution and of the expected time to extinction are presented in each of these regions. The quasi-stationary distribution is approximately normal, and the time to extinction is long, in one of these regions. Another region has a short time to extinction and a quasi-stationary distribution that is approximately truncated geometric. A third region is a transition region between these two. Here the time to extinction is moderately long and the quasi-stationary distribution has a more complicated behaviour. Numerical illustrations are given.
Nåsell
I
I
Department of Mathematics, The Royal Institute of Technology, Stockholm, S-100 44, Sweden. ingemar@math.kht.se
D000818:Animals; D005075:Biological Evolution; D001723:Birth Rate; D016015:Logistic Models; D009026:Mortality; D011158:Population Growth; D013269:Stochastic Processes; D013997:Time Factors