Abstract
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.
 
Highlights
Predator-prey interactions have attracted the great attention of both ecologists and mathematical biologists, because of their vast existence in nature and because of their diversified forms and rich consequences in the real world
A recent experimental study [25] in fields observed a phenomenon of trophic cascade in a food chain population system consisting of three species, i.e., meso-carnivore on top, its prey in the middle and the prey’s prey in the bottom, caused by the fear of virtual large carnivores which is implemented by playback of the large carnivores
In order to incorporate the benefit term into the model, we have to add the population of the large carnivores into the interplay, making (1.5) a 4-D system
Summary
Predator-prey interactions have attracted the great attention of both ecologists and mathematical biologists, because of their vast existence in nature and because of their diversified forms and rich consequences in the real world. The population of the large carnivores does not appear in the system because in the field study [25], only their voices are played, and they only have fear (indirect) effect on the meso-carnivores represented by B(α), where the net birth (production) rate B(α) depends on a parameter α standing for the meso-carnivore’s anti-predation response level. The function a34(α) denotes the encounter rate between N3 and N4 which is affected by the protective behaviours of N3 species characterized by its dependence on the anti-predation response level α By their biological meanings of B(α, N4) and a34(α), they are assumed to satisfy the following conditions: B(α, N4) is decreasing in α and N4, B(0, N4) = B(α, 0) = B3 > 0, lim B(α, N4) = lim B(α, N4) = 0, α→∞. We discuss some possible future projects along this direction of anti-predation response in predator-prey interactions
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