Identifying stability conditions and Hopf bifurcations in a consumer resource model using a consumption threshold

The existence, or not, of cyclic dynamics is one of the pivotal aspects of ecological populations. This work considers a consumer resource model found in ecology that can describe both cyclic and non-cyclic dynamics depending on parameter conditions. A threshold consumption number C0 is introduced, similar to the basic reproduction R0 in epidemiological models. It is shown that consumer survival requires C0 > 1 and that a Hopf bifurcation occurs at C0=C0*, where C0* is defined here and is greater than 1. This result is discussed with an example and extensions to other more complicated models.