Stochastic Dynamical Theta-Logistic Population Growth Model

Abstract

Population growth models are abstract representation of the real world objects, systems or processes to illustrate the theoretical concepts that these days are increasingly being used in more applied situations such as predicting future outcomes or simulation experiments. In mathematical literature, many population models have been considered, from deterministic and stochastic population models where the population size is represented by a discrete random variable, to very complex continuous stochastic models. A nonrandom case, ignores natural variation and produces a single value result, whereas a stochastic model incorporates some natural variations in to model to state unpredictable situations such as weather or random fluctuations in resources and itgenerates a mean or most probable result. Nowadays, the well-known model like logistic isplaying a major role in modern ecological theory. In paper [1], the Laguerre-type derivatives and the Laguerre-type exponentials are introduced and then by using Laguerre-type exponentials the LExponential and L-Logistic population growth models are derived, and output of these models is given for world population growth in the period 1955-2005. The paper [2], develops a stochastic logistic population growth model with immigration and multiple births. The differential equation for the low-order cumulant functions (i.e., mean, variance, and skewness) of the single birth model is reviewed, and the corresponding equations for the multiple birth model are derived. Accurate approximate solutions for the cumulant functions are obtained using moment closure methods for two families of model parameterizations, one for badger and the other for fox population growth. For both model families, the equilibrium size distribution may be approximated well using the normal approximation, and even more accurately using the saddle point approximation and it is shown that in comparison with the corresponding single birth model, the