# Dynamic Transition in Symbiotic Evolution Induced by Growth Rate Variation

## TL;DR

This paper uncovers a novel dynamic transition in symbiotic populations where changing growth rates alters the system's behavior and basins of attraction without affecting stationary states.

## Contribution

It demonstrates that growth rate variations can qualitatively change dynamics by modifying basins of attraction, despite stationary states remaining unchanged.

## Key findings

- Growth rates influence the basins of attraction boundaries.
- Dynamic transitions occur without changes in stationary states.
- The effect is illustrated in a two-population symbiotic model.

## Abstract

In a standard bifurcation of a dynamical system, the stationary points (or more generally attractors) change qualitatively when varying a control parameter. Here we describe a novel unusual effect, when the change of a parameter, e.g. a growth rate, does not influence the stationary states, but nevertheless leads to a qualitative change of dynamics. For instance, such a dynamic transition can be between the convergence to a stationary state and a strong increase without stationary states, or between the convergence to one stationary state and that to a different state. This effect is illustrated for a dynamical system describing two symbiotic populations, one of which exhibits a growth rate larger than the other one. We show that, although the stationary states of the dynamical system do not depend on the growth rates, the latter influence the boundary of the basins of attraction. This change of the basins of attraction explains this unusual effect of the quantitative change of dynamics by growth rate variation.