Continuous and discrete models in population biology

This dissertation consists of four chapters. Chapter one contains a brief introduction to population biology. Chapter two studies the long term behavior of a discrete time Beverton-Holt competition model where a competition takes place between n different species (traits). It is shown that the population with maximal fitness will outcompete all other populations locally. Global stability results are also provided for the case of two populations and the case of a restrictive condition. Derivation of this model from its continuous version is also provided. In Chapter three, we consider a selection model with safe refuge. It is assumed that the mortality function is density dependent and that individuals with "weak" traits are able to disperse to a safe refuge patch and avoid competition with individuals carrying the strongest trait. It is shown that if any subpopulation with a "weak" trait does not have a safe refuge then it will become extinct. Therefore, for survival of n traits, n-1 safe refuge patches are needed. Global asymptotic stability results and a detailed study of two examples are provided. Chapter four includes a three stage discrete time model. The birth rate is assumed to be constant (continuous breeding) in one part and time-dependent (seasonal breeding) in another. Global asymptotic stability is established for both types. When comparing continuous breeding to seasonal, it is found that continuous breeding with period two is advantageous for all values of birth rates. Numerical simulation suggests that seasonal breeding with period three is advantageous for low birth rates and deleterious for high birth rates.