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IMA 2004 Summer Program:

Birth and Death Processes and Mass Action Laws in Biology and Chemistry: AN Interacting Particle Systems Approach

Birth and Death Processes and Mass Action Laws in Biology and Chemistry: AN Interacting Particle Systems Approach

**Shay Gueron **

Technion - Israel Institute of Technology

Haifa, 32000, Israel

shay@math.technion.ac.il

Birth-death processes describe the stochastic evolution in time of a random variable whose value increases or decreases by one in a single event. Many models, for example in chemistry and in population biology, can be viewed as (coupled) birth-death processes, although this interpretation is not often made explicitly.

A typical modeling approach of a stochastic process is implemented by writing down deterministic differential equations for the time evolution of the process.

Deterministic models hope to approximate the expectation of the stochastic process when the population is sufficiently large, but this is not always the case. To illustrate, we discuss the example of the coagulation-fragmentation process where the deterministic model is not necessarily the limit of the finite state space process.

To compare the dynamics of stochastic and deterministic models
we use birth-death processes on a finite state space with *N*
states. We focus on asymptotically linear birth-death processes,
mass action laws, and their applications to population biology.
Asymptotic expansions at equilibrium, of the expectations of
these processes will derived, and compared with their deterministic
counterparts. As *N * , we show that both asymptotic expansions agree, although only
up to the first two terms.