Howard University, Washington DC, USA
Arizona State University, Tempe, USA
National Center of Health, Bethesda, USA

fsberezo@hotmail.com
ikareva@asu.edu
karev@ncbi.nlm.nih.gov
 

Abstract. We analyze a “resource-consumer” type system, where consumers can not only deplete the common renewable resource but are also capable of contributing to its restoration. Within the frameworks of this model, consumers are defined as clones which are characterized by their intrinsic (parametric) rate of interaction with the resource, from producers up to over-consumers. The exploitive depletion of a shared resource is known as the ”tragedy of the commons”. We first study the dynamics of parametrically homogeneous system, identifying a series of basic and transitional regimes with respect to parameter variations. In order to analyze the parametrically heterogeneous population and its evolution over time we apply the Reduction theorem, reducing an otherwise infinitely dimensional system of ODEs to finite dimensionality. We show that collapse of the heterogeneous system is still possible with the same set of initial conditions but the heterogeneous population survives longer than a homogeneous one. We also analyze the possibility of preventing the tragedy of the commons through external regulation, i.e., by inflicting punishment upon over-consumers and/or encouraging under-consumers.  We observe that the most effective mode of regulation requires both punishment and reward, which need to be adjusted depending on the initial distribution of clones in the population, and “non-linear” punishment of over-consumers is required to prevent the tragedy of the commons.

Key words: resource- consumer system, population heterogeneity, renewable resource, the system collapse.