# The GMS model with threshold extinction

## TL;DR

This paper introduces a modified GMS model where extinction events are governed by random thresholds, providing criteria for system recurrence, transience, and long-term species distribution, revealing a notable symmetry in these conditions.

## Contribution

It presents a novel variation of the GMS model incorporating threshold-based extinctions and establishes fundamental criteria for system behavior and species distribution.

## Key findings

- Criteria for recurrence and transience of the empty system
- Existence of a long-term species distribution
- Conditions for finiteness of species in the distribution

## Abstract

We propose a variation of the GMS model of evolution of species. In this version, as in the GMS model, at each birth, the new species in the system is labeled with a random fitness mark, but in our variation, to each extinction event is associated a random threshold mark and all species with fitness lower than the threshold are removed from the system. We present necessary and suficient criteria for the recurrence and transience of the empty configuration of species; we show the existence of a long time limit distribution of species in the system, and present necessary and suficient criteria for the finiteness of the number of species in that distribution. There is a remarkable symmetry between both sets of criteria.