Bifurcations and hydra effects in Bazykin's predator-prey model

Infection-induced increases to population size during cycles in a discrete-time epidemic model

One-dimensional discrete-time population models, such as those that involve Logistic or Ricker growth, can exhibit periodic and chaotic dynamics. Expanding the system by one dimension to incorporate epidemiological interactions causes an interesting complexity of new behaviors. Here, we examine a discrete-time two-dimensional susceptible-infectious (SI) model with Ricker growth and show that the introduction of infection can not only produce a distinctly different bifurcation structure than that of the underlying disease-free system but also lead to counter-intuitive increases in population size. We use numerical bifurcation analysis to determine the influence of infection on the location and types of bifurcations. In addition, we examine the appearance and extent of a phenomenon known as the 'hydra effect,' i.e., increases in total population size when factors, such as mortality, that act negatively on a population, are increased. Previous work, primarily focused on dynamics at fixed points, showed that the introduction of infection that reduces fecundity to the SI model can lead to a so-called 'infection-induced hydra effect.' Our work shows that even in such a simple two-dimensional SI model, the introduction of infection that alters fecundity or mortality can produce dynamics can lead to the appearance of a hydra effect, particularly when the disease-free population is at a cycle.

Square Root Identities for Harvested Beverton-Holt Models

We introduce the term net-proliferation rate for a class of harvested single species models, where harvest is assumed to reduce the survival probability of individuals. Following the classical maximum sustainable yield calculations, we establish relations between the proliferation and net-proliferation that are economically and sustainably favored. The resulting square root identities are analytically derived for species following the Beverton-Holt recurrence considering three levels of complexity. To discuss the generalization of the results, we compare the square root result to the optimal survival rate of the Pella-Tomlinson model. Furthermore, to test the practical relevance of the square root identities, we fit a stochastic Pella-Tomlinson model to observed Barramundi fishery data from the Southern Gulf of Carpentaria, Australia. The results show that for the estimated model parameters, the equilibrium biomass levels resulting from the MSY harvest and the square root harvest are similar, supporting the claim that the square root harvest can serve as a rule-of-thumb. This application, with its inherited model uncertainty, sparks a risk sensitivity analysis regarding the probability of populations falling below an unsustainable threshold. Characterization of such sensitivity helps in the understanding of both dangers of overfishing and potential remedies.