On mechanisms of trophic cascade caused by anti-predation response in food chain systems

Motivated by a recent field study [Nat. Commun. 7(2016), 10698] on the impact of fear of large carnivores on the populations in a cascading ecosystem of food chain type with the large carnivores as the top predator, in this paper we propose two model systems in the form of ordinary differential equations to mechanistically explore the cascade of such a fear effect. The models are of the Lotka-Volterra type, one is three imensional and the other four dimensional. The 3-D model only considers the cost of the anti-predation response reflected in the decrease of the production, while the 4-D model considers also the benefit of the response in reducing the predation rate, in addition to the cost by reducing the production. We perform a thorough analysis on the dynamics of the two models. The results reveal that the 3-D model and 4-model demonstrate opposite patterns for trophic cascade in terms of the dependence of population sizes for each species at the co-existence equilibrium on the anti-predation response level parameter, and such a difference is attributed to whether or not there is a benefit for the anti-predation response by the meso-carnivore species.