A general dynamical model of one consumer-one resource interactions
AbstractThe dynamical interaction between a consumer population of x individuals and a consumable resource of abundance r that obeys the differential equations 
$$\begin{gathered} \dot r = \sigma (r) - x\kappa (r,x), \hfill \\ \dot x = xy(r,x) \hfill \\ \end{gathered} $$
 is sufficiently general to include as special cases all well-known, continuous, one consumer-one resource models proposed to date. More importantly, its functions can be represented explicitly as compositions of primitive functions that describe how resource is produced and how consumer individuals acquire and use resource and interact with each other. Derivation of the model from these principles highlights the wealth of biological information it contains. Global asymptotic analysis, which generalizes previous analyses of special cases, establishes conditions under which all solutions tend toward an equilibrium point and other conditions under which all solutions tend toward a limit cycle. Dissection of special cases exposes four stabilizing forces and one destabilizing force inherent in the model. The fact that the former are both widespread and readily apparent in nature while the latter is uncommon suggests that the components of natural consumer-resource interactions that appear in this model tend to drive most such systems toward an equilibrium condition. The model's limitations as a description of nature are discussed.