Short time analysis of deterministic ODE solutions and the expected value of the corresponding birth-death process
Date
2009
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Abstract
There is a standard way to construct a discrete birth-death probability model for an evolution system, in which an ODE model of the system is used to define the probabilities governing the evolution of the stochastic model. Given the significant differences in the dynamical behavior of ODE solutions which are inherently smooth, and stochastic models which are subject to random variation, the question naturally arises about the connection between the two models. In particular, we investigate the validity of using a continuum model to define the evolution of a stochastic model.
We show a consistent way to define the probabilities for the stochastic model if the ODE has the form y = f(y) = yg(y). The deterministic model can then be compared to the expected value of the discrete probability model. For an ODE of this form describing population dynamics, we can describe each individual of the population as a categorical random variable. In this formulation, the probability for a birth or death to occur in a population of size y over a time interval of length Δt is given by yg(y)Δt + o(Δt), depending on the sign of g (y).
We derive local and global bounds for the difference between the expected value of the discrete probability model and the solution of the ODE. Locally, the two models behave similarly. Global bounds, however, imply the difference between these two models may be at most exponential in nature. Such a large bound must account for the possibility that the associated probability model may jump across a steady state of the ODE and exhibit divergent behavior while the ODE remains stable.
We explore our results for a number of models. In particular, we provide examples that show that there can be fundamental differences in the dynamical behavior of the stochastic and ODE model solutions, even when they are close over any given step. Our results represent a different approach from another view, which derives some ODE that governs the expected value of the stochastic system. That ODE and the original ODE are not the same in general.
We show a consistent way to define the probabilities for the stochastic model if the ODE has the form y = f(y) = yg(y). The deterministic model can then be compared to the expected value of the discrete probability model. For an ODE of this form describing population dynamics, we can describe each individual of the population as a categorical random variable. In this formulation, the probability for a birth or death to occur in a population of size y over a time interval of length Δt is given by yg(y)Δt + o(Δt), depending on the sign of g (y).
We derive local and global bounds for the difference between the expected value of the discrete probability model and the solution of the ODE. Locally, the two models behave similarly. Global bounds, however, imply the difference between these two models may be at most exponential in nature. Such a large bound must account for the possibility that the associated probability model may jump across a steady state of the ODE and exhibit divergent behavior while the ODE remains stable.
We explore our results for a number of models. In particular, we provide examples that show that there can be fundamental differences in the dynamical behavior of the stochastic and ODE model solutions, even when they are close over any given step. Our results represent a different approach from another view, which derives some ODE that governs the expected value of the stochastic system. That ODE and the original ODE are not the same in general.
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Subject
birth-death process
conditional expectation
discrete probability
local bound
ecology
mathematics
statistics