Mass Balance Equation Versus Logistic Equation in Food Chains

The dynamic behavior of tri-trophic food chains consisting of resources, prey, predator and top-predator is dealt with. We compare a formulation whereby the prey growth is logistic, with a mass balance formulation. In the case of the mass balance formulation both the linear and the hyperbolic functional response are discussed. The consequences of the different formulations on the dynamics of a microbial food chain in chemostat situation are described. Bifurcation diagrams for the nonlinear dynamic systems are given. When the prey grows logistically there is no coexistence of the three species for biologically realistic parameter values for a microbial food chain. The same holds for the mass balance equations with a linear functional response for the prey. For a hyperbolic functional response, however, there is a stable equilibrium for the whole food chain in a rather large region of the parameter space. Furthermore, this model shows more complex dynamic behaviors; besides point attractors, limit cycles and chaotic attractors.