Main facts about eigenvalues and eigenvectors: 1. By definition, a square matrix P has eigenvalue lambda with associated eigenvector v if P v = lambda v 2. If v is an eigenvector associated with eigenvalue lambda, then c v is also an eigenvector associated with lambda for any non-zero constant c 3. For many matrices that arise in a biological context, there is a single dominant eigenvalue (eigenvalue with the largest magnitude), and after a long period of time the population structure (or state structure) associated with this model is the same as that of the eigenvector associated with this dominant eigenvalue. This is true for the Leslie matrix models. For all models (like the Markov chains and the compartment models with transfer matrices), the dominant eigenvalue will be 1 and the eigenvector associate with this (that is the vector v so that P v = v) gives the long-term structure of the system. 4. For the Leslie matrix model, he long-term growth rate of the population is the dominant eigenvalue, lambda. This means that eventually, if the vector of the population at time t is v(t) then v(t+1) = lambda v(t) for t large enough. This is geometric growth. 5. A matrix P of size nxn will generally have n eigenvalues and n associated eigenvectors, but there are a variety of exceptions to this, and the eigenvalues, which may be found as the roots of an n-th degree polynomial, may be complex numbers. Compartment models: Key idea: Break a system down into pieces (called compartments) and follow the flow of some material between the pieces. In this course, the compartments will generally be given to you, but a key problem in modeling biological systems is deciding what the appropriate compartments should be, based upon our limited knowledge and limited ability to observe and measure the system. Examples: (1) A drug infused into the body, followed in the blood stream, the liver, the kidneys, etc. (2) A contaminant in an aquatice system - e.g. PCB distributed between the water and sediments in a stream (3) An element (C, N, or mass) distributed amongst the components of a food web, say phytoplankton, zooplankton, fish, etc. Main assumption: Linear donor control - this means that the amount of material transfered from one compartment to another in one time period is a proportion of the amount present in the compartment it is coming from.