The Equilibrium and Stability of Simple Marine Biological Systems. I. Primary Nutrient Consumers

In this paper I examine the general interactions among species of primary nutrient users competing for these nutrients in a homogeneous biological system. No specific assumptions are made concerning the functional dependence of the assimilation rates and population growth rates on the concentrations of nutrients involved, and conditions on these functions are sought for the existence of equilibrium states and the stability of the system about the equilibrium states. It is shown that necessary conditions for equilibrium are (1) that in the absence of predation, the number of removable nutrients must be greater than or equal to the number of species of primary consumer, and (2) that for each species involved in the system, there must be at least one domain of values of the removable nutrients in which this population alone can grow, all others dying out. The stability of an equilibrium state is studied in general and two examples are given in more detail, one involving a single limiting nutrient and one species, the other with two limiting nutrients and one species of primary consumer. It is shown that if the population density is limited by toxic agents, the equilibrium is generally unstable; instability can also be provoked by sufficiently strong cross-linkage in which the assimilation rate of one nutrient depends strongly on the concentration of another. Specific criteria for stability are presented; they generally depend on the forms of the population growth rates and assimilation rates as functions of the concentrations involved. It is shown further that in a stable system with more than one species of primary nutrient consumer, if the rate of supply of certain (but not all) of the consumable nutrients is either increased or decreased sufficiently, then the number of species necessarily decreases, at least one of them becoming extinct.