Seminário

[Fabio Prates Machado <fmachado@ime.usp.br>]

 Seminário
 Probabilidade e Processos Estocásticos

 Nesta sexta teremos a visita da (recem) ex-aluna do prof.
 Andre Toom.

 Caliteia Souza - UFPE

 Sexta, 10:00 - sala 247 A (Antonio Gilioli)
  
Conditions for the existence of fixed points in a class of particle processes

We consider a class of particle processes with discrete time and a finite number
of types of particles, which we call Processes Without Collisions or PWC for
short. At every time step any particle may die or transform itself into one or
several particles of any types with certain probabilities, but there are no
collisions, that is every transformation applies to only one particle and its
probability does not depend on the other particles. This excludes predator-prey
and sexual relations, but includes death and asexual reproduction with possible
mutations. Due to this restriction, positions of particles are irrelevant and we
do not mention them.

We prove that if the number of particles tends to infinity, the behavior of the
process becomes deterministic after normalization. In fact, this approximation
is often used without a proper foundation. We also prove that the resulting
dynamic system has a fixed point and that under additional conditions the fixed
point is unique and the system tends to it from any initial condition. Besides,
this approach raises some questions about the manner in which some books on
populational dynamics are written. 

Fabio Prates Machado <fmachado@ime.usp.br>