Stability and optimal control for some classes of tritrophic systems

The objective of this paper is to study an optimal resource management problem for some classes of tritrophic systems composed by autotrophic resources (plants), bottom level consumers (herbivores) and top level consumers (humans). The first class of systems we discuss are linear chains, in which biomass flows from plants to herbivores, and from herbivores to humans. In the second class of systems humans are omnivorous and hence compete with herbivores for plant resources. Finally, in the third class of systems humans are omnivorous, but the plant resources are partitioned so that humans and herbivores do not complete for the same ones. The three trophic chains are expressed as Lotka-Volterra models, which seems to be a suitable choice in contexts where there is a shortage of food for the consumers. Our model parameters are taken from the literature on agro-pastoral systems in Sub-Saharan Africa.