Application of stochastic differential equations and stochastic delay differential equations in population dynamics

Abstract

Modelling population dynamics in random environment is a way of studying the fluctuations of population size that was affected by the stochasticity of the outside factors like weather. Marion et al. (2002) showed the ability of this environmental Brownian noise (if we assume weather effect follows Brownian noise) in suppressing population explosion, using model that is restricted by its hypothesis. Consequently, this restriction excludes model like power law logistic model. On the other hand, power logistic model has emerged as a good model for variety of natural phenomenas. Thus the exclusion of this model from Marion et al. (2002) model really define a gap in literature that need further work in order to adapt a new hypothesis. Thus this thesis expands their work in three different perspectives that is adapting new assumption to include power law logistic model, improving the existing estimate by means of addition and changing the way the noise is perturbed especially noise that is independent of population size, thus establishing a new way of proving nonnegative global solution and how to detect population extinction. Finally constant delay is also included into the model with some modification from the original model which the followed by the study of its nonnegative global solution, boundedness in s-th moment, asymptotic behaviour and also the intensed noise that may lead to population extinction.