Dynamics of predator and modular prey: effects of module consumption on stability of prey-predator system

In traditional models of predator-prey population dynamics, it is usually assumed that consumed prey are immediately removed from the population. However, in plant-herbivore interactions, damaged plants are generally alive after attacks by herbivores. This can result in successive or simultaneous attacks by multiple predators on a single prey item (here, the term prey is expanded to include plants). We constructed a mathematical model with two time scales, taking into account predation processes within a generation, considering post-predation survival and the modularity of prey. We assumed that a prey item can be divided into modules and that it can be fed on by multiple predators or parasitized by multiple parasites at the same time. The model includes two essential factors: the modularity of prey for predators (n) and the detaching/attaching ratio of predators to prey (epsilon). Based on the formulae, we revealed a general property of realistic dynamics in plant-herbivore and host-parasite interactions. The analysis showed that the model could be approximated by models with the type I, type II or Beddington-DeAngelis functional responses by taking appropriate limits to the situations. When modularity is low or the detaching/attaching ratio is high, population dynamics tend to be stabilized. These stabilizing effects may be related to interference competition among predator individuals or increases in free prey modules and free predator individuals. In the limit of high modularity, the ratio of the attached prey modules to the total prey modules becomes negligible and the dynamics tend to be destabilized. However, if quantity and quality of prey modules are negatively correlated, the equilibrium is likely to be stabilized at high modularity as long as it remains feasible. These results suggest that considering post-predation survival and modularity of prey is crucial to understand predator-prey interactions.