Abstract
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.
Highlights
The scientific literature on natural resource exploitation is displaying a steady growth and a number of new research directions have been explored
In this paper we examined three Lotka-Volterra tritrophic chains: a linear chain Sl, a trophic chain with omnivory So and a trophic chain with omnivory and source partition Sp
We identified the equilibrium solutions of each system and analyzed the related stability properties
Summary
The scientific literature on natural resource exploitation is displaying a steady growth and a number of new research directions have been explored. In this paper we consider how optimal resource management policies depend on special structures of the trophic chain and investigate infinite time horizon optimal control problems with the objective of characterizing management strategies that sustain the human population’s biomass and promote welfare. Since we are assuming that all the model parameters (including the controls) are constant over time, to avoid trivial dynamics starting from (ξ10, y0, z0, ξ20) > 0 we have to rule out the condition θyξ Dyξ K1 − qy < 0, which would imply limt→∞ y(t) = 0 in view of the second equation in (3) This would imply limt→∞ z(t) = 0 in the case of Sl. if θzξ Dzξ K1 − (qz + C) < 0 in So and θzξ Dzξ K2 − (qz + C) < 0 in Sp, limt→∞ z(t) = 0 in these cases. We provide a list of the non-coexistence equilibrium states of systems Sl, So and Sp:
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