Abstract:
There has been a surge of interest in developing and analysing models of interacting species in ecosystems,
with specific interest in investigating the existence of limit cycles in systems describing the dynamics of
these species. The original Lotka–Volterra model does not possess any limit cycles. In recent years this
model has been modified to take disturbances into consideration and allow populations to return to their
original numbers. By introducing logistic growth and a Holling Type II functional response to the traditional
Lotka–Volterra-type models, it has been proven analytically that a unique, stable limit cycle exists. These
proofs make use of Dulac functions, Liénard equations and invariant regions, relying on theory developed by
Poincaré, Poincaré-Bendixson, Dulac and Liénard, and are generally perceived as difficult. Computer algebra
systems are ideally suited to apply numerical methods to confirm or refute the analytical findings with
respect to the existence of limit cycles in non-linear systems. In this paper a class of predator–prey models
of a Gause type is used as the vehicle to illustrate the use of a simple, yet novel numerical algorithm. This
algorithm confirms graphically the existence of at least one limit cycle that has analytically been proven to
exist. Furthermore, adapted versions of the proposed algorithm may be applied to dynamic systems where
it is difficult, if not impossible, to prove analytically the existence of limit cycles.
