Review of a predator-prey model with two limit cycles

Abstract

It is well-known that the Lotka–Volterra predator-prey model has a family of periodic orbits, but does not possess limit cycles and therefore the model is said to be structurally unstable. The Lotka–Volterra model is a special case of a much larger group namely the quadratic population models and it can be shown that none of them can produce limit cycles. The surprising finding is that by combining two quadratic models a quadratic population model with two limit cycles is uncovered. Although the model looks simple at first glance it provides a rich source of dynamics and deserves attention. In this paper, we revisit a model that has its origin in the work of Dubois and Closset. A set of two quadratic population models interact as piecewise defined differential equations. The model has been discussed by Ren Yongtai and Han Li, cryptically written and showing some linguistic and typographical errors, but providing an excellent vehicle for developing skills in mathematical modelling, differential equations and technology for the young researcher. We explore the model in clearer detail and supplement the theory with rich graphical illustration. The paper has the purpose of providing an example of how a young researcher, such as a postgraduate student in biomathematics, can expand on an existing model by making use of current technology.