The local dynamics of a tri-trophic system

A lumped parameter model to describe the local dynamics of a tri-trophic chain (resource, herbivore, carnivore) is presented. The time evolution of the system is determined by ordinary differential equations and written in terms of normalized biomass of the three levels of the chain. In these equations we have introduced bio-ecological parameters (specific rates, conversion factors) and two parameters p and q which measure the efficiency of the interaction processes (e.g. predation processes). A stability and persistence analysis of the solutions to these equations has been performed by assuming that: (i) the bio-ecological parameters are given (in other work we suggest procedures to estimate them from individual data and demographic models); (ii) the function determining the growth of the first level and the functional responses of the second and third levels to the abundances of the first and second levels, respectively, are characterized only by a shape of functional response and not by analytical expressions; and (iii) by considering the behavioural parameters p and q as bifurcation parameters. The regions in the (p, q) plane of existence and stability of the non-negative steady states, and those of persistence and limit cycles of the system are determined. Results of numerical simulations are shown.