| The population growth of a single species or the population interaction of two or more species can be described by mathematical models. This study deals with prey-predator interactions that are modeled by systems of differential equations, with one equation for each species. Two distinct types of models are considered: deterministic models represented by systems of ordinary differential equations and stochastic models represented by systems of stochastic differential equations.; The general deterministic model studied is a Gause-type model with explicit predator self-limiting term. Criteria are given for the global stability of a positive equilibrium. The Michaelis-Menten model with explicit predator self-limitation is a special case of the Gause-type model. Specific conditions on the model parameters of the Michaelis-Menten model guaranteeing global stability of a positive equilibrium are determined.; A stochastic Gause-type model is also studied. For a special case of the stochastic model, the existence of an invariant distribution is verified. The stochastic Michaelis-Menten model, which is an example of this special case, is shown to have an invariant distribution under certain conditions on the model parameters which are consistent with those conditions which guarantee global stability for the deterministic analog.; Also, for the stochastic Michaelis-Menten model, a condition on the model parameters is given which implies the solution is transient. Criteria are also presented for which the general stochastic Gause-type model is transient. Finally, some exit probabilities are investigated for the stochastic Michaelis-Menten model. |
