## Underlying Mathematical Structure

For expository purposes, assume for the time being that the number of h-states is finite. In that case, the expected numbers of individuals in different h-states at time t for a given realized environment E can be written as a column vector N(t). If it is moreover assumed that time runs in steps, the definitions of state and environment introduced above make it possible to write

while for continuous time

The i, j-th component of the matrix A equals the probability of a transition from h-state j to h-state i, multiplied with the corresponding survival probability, plus the average number of offspring with h-state i produced by an individual in h-state j. The matrix B is built up from per capita rates. The off-diagonal components equal the transition rates between the corresponding h-states plus the h-state-dependent average rates of offspring production differentiated according to their birth h-state. The diagonal components equal minus the overall rates of state transitions from the h-states, minus the h-state-dependent death rates, plus the average rates of giving birth to offspring with the parental h-state.

Equations  and  apply to populations of all sizes, large as well as small, but with different interpretations. Small populations do not influence the environment, but are subject to demographic chance fluctuations. Hence N in eqns  and  refers only to the expected and not to the realized population state. For large populations, the components of N may be interpreted as realized population densities in a large spatial area or volume. The step from expected numbers to realized densities is based on law-of-large-number considerations. Moreover, for large populations the assumption that E is given independent of N takes on the status of a thought experiment and not that of an idealized representation of reality. In reality the feedback loop is closed, that is, E is determined at least in part by N. (As example let E(t) = F(O(t)) with the population output O given by O(t) = HN(t) , with H a matrix consisting of weight factors telling the extent to which individuals in different h-states impinge on various aspects of the environment and F determined by a fast dynamics of the environmental state plus the map from environmental states to the environmental conditions experienced by the focal individuals.)

The situation for infinitely many h-states is not different in principle, except that to deal with it more powerful abstract machinery has to be invoked. This is still an active research area. In the general case, even the appropriate law-of-large-number theorems have not been proven. However, all evidence indicates that under biologically reasonable restrictions the picture will be similar to that for finitely many h-states. Below, little distinction will be made between finite and infinite h-state spaces. However, the reader should bear in mind that the most general statements have in the strict mathematical sense only the status of conjectures. 