Hydra effects in stable food chain models

In this paper, we explore the occurrence of the hydra effect in food chains, a popular research theme in the current decade. The hydra effect, one of the paradoxical results in theoretical and applied ecology refers to the fact where increasing mortality rate on a population enhances its own stock. The main focus is to propose a dynamical system model of food chain showing a stable steady state and estimate the variation of stock of targeted species with increasing mortality. In our model, the per capita growth rate of any predator trophic level does not depend upon its density. The prey-predator model incorporating such a feature for predator growth is referred to as 'pure predator system' (see Sieber and Hilker (2012), J. Math. Biol. (2012) 64: 341-360, Journal of Mathematical Biology). Keeping the above feature in mind, we study a Rosenweig-MacArthur food chain model with logistic prey growth and Holling type II functional responses. It is shown that hydra effect at stable state appears on (a) prey in a four-trophic system, (b) first predator in a five-trophic system, and (c) prey and second predator in a six-trophic system. Xiao and Cao (2009) (Mathematical and Computer Modelling 50 (2009) 360-379) established that limit cycle may be observed due to harvesting in a system with the ratio-dependent prey-predator system (example of a non "non-pure predator system"). Therefore, if harvesting causes instability on some range of mortality rate, the hydra effect cannot occur at a stable state. Some results show that the unique stable steady state in our model remains stable under harvesting of either trophic level. As a whole, our investigations have some contribution in understanding population interactions, fishery management and biological pest control tactic.