Modelling of predator-prey trophic interactions. Part I: two trophic levels

A class of lumped parameter models to describe the local dynamics in a controlled environment of a two-trophic chain is considered. The class is characterized by a trophic function (functional response of predator to the abundance of prey) depending on the ratio of prey biomass x and a linear function of predator biomass y: f(qx/[(1-rho)k+rho y]), where q is the efficiency of the predation process, k is a reference biomass, and rho (0 <= rho <= 1) specifies the predation model. The trophic function is defined only by some properties determining its shape. A stability analysis of the models has been performed by taking the parameters q and rho as bifurcation parameters: the regions in the (rho,q) plane of existence and stability of nonnegative equilibrium states and limit cycles are determined. This analysis shows that the behaviour of the models is qualitatively similar for 0 <= rho < 1 (in particular the null state is always a saddle point), while the value rho=1 gives rise to some kind of structural instability of the system (in particular the null state becomes an attractor for sufficiently high predation efficiency).