Modelling the harvest of an insect pathogen

Ensuring a sustainable yield is essential for continued survival of a natural resource, however over-exploitation can easily occur. Therefore, understanding how increasing the harvesting rate affects the yield is vital. Harvesting of infected hosts in a host–pathogen system, for example the fungal pathogen Cordyceps sinensis which is harvested for medicinal use, has not been explored mathematically in the literature. We present a generalized host–pathogen model in which the infected host is harvested. Two strategies are explored; proportional harvesting at a constant rate and in an open-closed setting (a repeating cycle of a period of harvest followed by a period where the resource is left to recover). We present yield-effort curves for both strategies and find that open-closed harvesting affects the traditional yield-effort curve, with the system able to support a greater range of harvesting rates. Furthermore, host–pathogen systems may exhibit more complex population dynamics than single equation/species models, depending on the eigenvalues of the linearised system. In the open-closed setting we find that if there are complex eigenvalues in the absence of harvesting although small changes in the length of open season have little impact on the maximum sustainable yield, it can dramatically change the harvesting rate needed to achieve this. For proportional harvesting in a constant setting our model shows that if there are real eigenvalues in the absence of harvesting, then resilience–harvest relationship agrees with accepted theory, where as yield initially increases so too does the return time (a measure of the long-term resilience). However, when there are complex eigenvalues we see, counter to intuition, that the return time initially decreases whilst still providing increased yield. We also study the transient (short-term) reactivity, which shows that in both cases harvesting can initially decrease the reactivity. These results show that harvesting can in some instances enhance the ability of host–pathogen systems to respond to perturbations in both the short- and long-term.