Multiple coexistence equilibria in a two parasitoid-one host model

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

We consider Bazykin's model to address harvesting induced stability exchanges through bifurcation analysis. We examine the existence of hydra effects and analyze the stock pattern under predator harvesting. Prey harvesting cannot produce hydra effects in our model, whereas predator harvesting may cause multiple hydra effects. Our study reveals that type II response function and mutual interference among predators jointly induce multiple hydra effects and bistability. Bifurcations such as single Hopf-bifurcation, multiple Hopf-bifurcations and multiple saddle-node bifurcations appear for increasing harvesting rate on the predators. However, over-exploitation of the predators cannot generate any such bifurcation in our study. In simulations, the maximum sustainable yield (MSY) exists at a globally stable state. When predator is culled under increasing effort, basin of attraction of the equilibrium corresponding to the higher predator stock gets expanded, which alternatively is in favor of stock benefit for predators. The ecological theory developed in this study might be useful to understand conservation policy and fishery management.

Host coexistence in a model for two host-one parasitoid interactions

Building from a continuous-time host-parasitoid model introduced by Murdoch et al. (Am Nat 129:263-282, 1987), we study the dynamics of a 2 host-parasitoid model assuming, for the sake of simplicity, that larval stages have a fixed duration. If each host is subjected to density-dependent mortality in its larval stage, we obtain explicit conditions for the existence of an equilibrium where the two host species coexist with the parasitoid. However, if host demography is density-independent, equilibrium coexistence is impossible. If at least one of the 1 host-parasitoid systems has an oscillatory dynamics (which happens under some parameter values), we found, through numerical bifurcation, that coexistence is favoured. Coexistence between the two hosts may occur along a periodic solution even without density-dependence. Models of this type may be relevant for the use of parasitoids as biocontrol agents of insect pests.