Abstract
In this work, we consider a continuous–time branching process with interaction where the birth and death rates are non linear functions of the population size. We prove that after a proper renormalization our model converges to a generalized continuous state branching process solution of the SDE \[\begin{aligned} Z_t^x=& x + \int _{0}^{t} f(Z_r^x) dr + \sqrt{2c} \int _{0}^{t} \int _{0}^{Z_{r}^x }W(dr,du) + \int _{0}^{t}\int _{0}^{1}\int _{0}^{Z_{r^-}^x}z \overline{M} (ds, dz, du) &+ \int _{0}^{t}\int _{1}^{\infty }\int _{0}^{Z_{r^-}^x}z M(ds, dz, du), \end{aligned} \] where $W$ is a space-time white noise on $(0,\infty )^2$ and $\overline{M} (ds, dz, du)= M(ds, dz, du)- ds \mu (dz) du$, with $M$ being a Poisson random measure on $(0,\infty )^3$ independent of $W,$ with mean measure $ds\mu (dz)du$, where $(1\wedge z^2)\mu (dz)$ is a finite measure on $(0, \infty )$.
Highlights
Consider a population evolving in continuous time with m ancestors at time t = 0, in which to each individual is attached a random vector describing her lifetime and her number of offsprings
We assume that those random vectors are independent and identically distributed
Each individual lives for an exponential time with parameter ν(Z+), and is replaced by a random number of children according to the probability ν(Z+)−1ν
Summary
Consider a population evolving in continuous time with m ancestors at time t = 0, in which to each individual is attached a random vector describing her lifetime and her number of offsprings. We assume that those random vectors are independent and identically distributed. Given a function f : R+ → R, which satisfies assumption (H2) below, whenever the total size of the population is k, the total additional birth rate due to interactions is k i=1. We prove that, when properly renormalized, the above continuous time branching process with interaction converges to the solution of the SDE.
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