Geometric criteria for the non-existence of cycles in predator–prey systems with group defense

In this paper, we study the existence of cycles in a predator–prey system in which the prey species is equipped with the group defense capability. Some geometric criteria are developed, relating the location of the two positive equilibria on the prey isocline and the non-existence of cycles. We show that under a general geometric condition, if both positive equilibria lie on a downslope or both lie on an upslope of the prey isocline, cycles do not exist.