Fitting a logistic curve to population size data
A stochastic logistic process N(,t) is viewed as a homogeneous birth and death process with population birth and death rates (lamda)(,n) = (a(,1)-b(,1)n)n and (mu)(,n) = (a(,2)+b(,2)n)n, respectively, where a(,1), a(,2) > 0, b(,1), b(,2) (GREATERTHEQ) 0, a(,1)(NOT=)a(,2), b(,1)+b(,2) > 0, and n is the population size, a realization of N(,t);The mean value and the covariance function of the stochastic logistic growth were estimated using computer simulations. The stochastic mean at time t was found to be extremely close to the deterministic mean, K/(1+Ce('-rt)), where r = a(,1)-a(,2) is the intrinsic individual growth rate, K = r/(b(,1)+b(,2)) is the carrying capacity, and C = K/N(,0) - 1, N(,0) being the initial population size (fixed). The covariance function was found to be dependent not only on time, order of the covariance, N(,0), r, and K, but also on a(,1), a(,2), b(,1) and b(,2);An approximating function for the covariance was developed using search techniques on the estimated covariance from the simulations. The variance and the covariance of the first order were computed theoretically from the transition probabilities for one case, and they were found to agree with the corresponding values of the approximating function;The approximating function found for the covariance function was used in an attempt to apply a quasi-maximum likelihood criterion in the estimation of the parameters of the logistic process. This and other methods of fitting the logistic curve to population size data were discussed, and they were compared in a Monte Carlo study. Linearization methods of fitting were found to be biased and some of them presented serious problems of estimation;Finally, the problem of goodness of fit was addressed, and the distribution of a (chi)('2) type statistic for goodness of fit was studied.