## Introduction

Ecological stoichiometry is relatively new as a recognized area within ecology, but the broad implications for population and community interactions were recognized by one of the discipline's modern founders, Alfred Lotka. Stoichiometry deals with quantitative relations among the masses of elements and compounds, so the law of mass conservation is a primary tool. It dictates that masses of substances appearing during a process (products of a chemical reaction, production of new organisms or tissues, excretory wastes, etc.) must be balanced by masses of substances disappearing (reac-tants, dissolved nutrients, food or prey eaten, etc.). Inevitably, there are mass-conservation constraints or mass-balance constraints on the dynamics of such transformations.

Lotka made a concise formal statement of such constraints for ecological communities where n populations with densities N interact according to the governing equations

Each organism in one of these interacting populations will contain a certain mass of element j called the quota, Qij. Supposing that biomass is composed of m elements, which are assumed for the moment not to appear in nonliving forms, conservation of mass leads to the following constraints on population densities:

These constraints reduce the dynamical degrees of freedom among the interacting species from n (the number of species) to n - m. Although this general principle has long been known, it was many decades after Lotka's first statement before ecologists explored the detailed consequences of such mass-conservation constraints for population and community interactions.

The contemporary approach to studying these consequences appends to Lotka's equations another set of equations for the dynamics of free elements, because most nutrient elements are present in the environment in at least one dissolved or gaseous form. The dynamics of free elements typically depend on densities of at least some of the populations, and in turn population dynamics often depend on concentrations of nutrient elements. Thus the set of equations [1] is expanded to two sets:

= fi (N1, N2,... , Nn; R1, R2,... , Rm) for i = 1,..., n dt

"d^ = gj (N1, N2,..., N„; R1, R2,..., Rm) for j = 1,..., m dt [3b]

where Rj is the concentration of nutrient element or resource j free in the environment. The mass-conservation constraints are correspondingly modified to read n

At first sight, such constraints might appear unimportant, since we have now restored the dynamical degrees of freedom to n, the number of interacting populations in the community. Nevertheless, mass-conservation constraints have proven helpful in analyzing model ecological communities represented by equation system [3].

A potential drawback of Lotka's perspective is that mass-conservation constraints of the form of equation [4] most clearly apply to idealized, closed systems. In contrast, natural systems are open, with materials entering and exiting at their boundaries. However, constraints of the form of eqn [4] apply to long-term dynamics for an important class of open systems, those that resemble a laboratory chemostat, a well-mixed culture device into which a medium containing nutrient elements is pumped, and from which all organisms and dissolved nutrients are removed indiscriminately by outflow. Such devices are often used for experiments on nutrients and microbial population dynamics, and caricaturize some of the open systems found in nature. More generally, mass-conservation constraints often emerge in the long-term dynamics of open systems that achieve a steady state.

Table 1 Classification of pairwise population interactions

Signs of interaction coefficients Type of interaction

Table 1 Classification of pairwise population interactions

Signs of interaction coefficients Type of interaction

 aj < 0 and aji < 0 Competition (through resource exploitation, allelopathy, aggression, or interference) a/j > 0 and aji <0 or Predation (including herbivory, aj < 0 and aji >0 parasitism, and disease) a/j > 0 and aji = 0 or Commensalism a,y = 0 and 0j->0 aj < 0 and aji = 0 or Amensalism a/j = 0 and 0j-<0 aj > 0 and aji > 0 Mutualism a/j = 0 and aji = 0 None

mutualistic. Second, eqn [5] and its aj coefficients describe the effects of interactions on population dynamics, but not the mechanisms of those interactions.

Theoretical models incorporating nutrient resource dynamics have been an important source of mechanistic knowledge enriching ecologists' understanding of population interactions. Much progress has been made by coupling equation system [3] with stoichiometric constraints (eqns [4]), for special cases with small numbers of populations and nutrient resources. Although the traditional classification of population interactions has limitations, it provides a convenient way to catalog such efforts.