A preliminary threshold model of parasitism in the Cockle\emph{Cerastoderma edule} using delayed exchange of stability

Thresholds occur in the dynamics of many biological communities. Here we model a persistence type threshold which has been shown experimentally to exist in hyperparasitised flukes in the cockle, a shellfish. Our model consists of a periodically driven slow-fast host-parasite system of equations for a slow flukes population and a fast Unikaryon hyperparasite population. The model exhibits two branches of the critical curve crossing in a transcritical bifurcation scenario. We discuss two thresholds due to immediate and delayed exchange of stability effects; and we derive algebraic relationships for parameters of the periodic solution in the limit of the infinite ratio of the time scales. Flukes parasitise cockles and in turn are hyperparasitised by the microsporidian Unikaryon legeri; the life cycle of flukes includes several life stages and a number of different hosts. That is, the flukes-hyperparasite system in a cockle is part of a larger estuarine ecosystem of species involving parasites, shellfish and birds which prey on shellfish. A population dynamics model which accounts for such multi-species interactions and includes the fluke-hyperparasite model in a cockle as a subsystem is presented. We provide evidence that the threshold effect we observed in the flukes-hyperparasite subsystem remains apparent in the multi-species system. Assuming that flukes damage cockles, and taking into account that the hyperparasite is detrimental to flukes, it is natural to suggest that the hyperparasitism may support the abundance of cockles and, thereby, the persistence of the ecosystem, including shellfish and birds. We confirm the possibility of this scenario in our model by removing the hyperparasite and demonstrating that this may result in a substantial drop in cockle numbers. The result indicates a possible significant role for the microparasite in this estuarine ecosystem