In the Mathematical Biology seminar on 10/12/2011 a very broad generalization of the Lotka-Volltera
equation was considered, and one particular case was seen to have many
biologically and mathematically desirable properties. Specifically, it
was seen to be a transformation of an affine linear process, and as such
it was seen to possess an explicit solution.
This week we consider ways of generalizing this deterministic model
to a stochastic one (deterministic evolution not being a trait one normally
associates with biological systems). By analogy with the Square Root
Model, we seek a biological model that is the square of a stochastic
process having affine linear drift. Conditions on the diffusion term
that will ensure the squared process remains markov are investigated, as
is one possible method of coefficient inference.