# A mathematical analysis of population models

#### Abstract

Just as calculus played an important role in the development of astronomy, mechanics and physics, mathematics is helping people understand the living world. The research in biomathematics which has become an independent subject since 1970's, is extremely active. One of the main topics of biomathematics is investigating population dynamics and the dynamics of population interactions. Actually, the study of population phenomena or, growth phenomena, is a dominant problem in many scientific disciplines. In this dissertation we will focus on problems involving population dynamics in several settings. The first is a model of a single population, based on the logistic equation, introduced by Cui and Lawson. We also examine an age-dependent population model in the form of a partial differential equation, and establish the stability theory by using properties of the semigroup of linear operators. This method is simpler than that used by Marcati (1982) who obtained the same result for a special case of our model by using the usual analytic methods. Next, a general predator-prey model which includes the Lotka-Volterra model, Gause model, generalized Gause model, Hsu model, Kuang-Freedman model, etc. is presented. Five theorems on the local and global stability are obtained concerning this model. Since the paper of May (1972), determining conditions that guarantee uniqueness of limit cycles in predator-prey models has become an outstanding problem. We also establish theorems concerning that problem for our general predator-prey system. Several applications and examples are given. The first deals with a simple model simulating an immune response which is based on the Cui-Lawson model. In addition, the Rosenzweig-MacArthur model, Hsu-Hubbell-Waltman model and Kazarinoff-Van Driessche model are discussed in reference to problems involving limit cycles. For all these models and two other examples we prove the existence and uniqueness of limit cycles.

#### Recommended Citation

Huang, Xun-Cheng, "A mathematical analysis of population models" (1988). *Dissertations (1962 - 2010) Access via Proquest Digital Dissertations*. AAI8904257.

https://epublications.marquette.edu/dissertations/AAI8904257