How to emit at an exponential rate?

I’m trying to make a scene where bacteria multiply through by using an emission particle system. In order to keep it true to the exponential rate at which bacteria multiply, I was hoping to set the start and end point of the particle system at 1 and use keyframes for the particle count, at which point I would make the animation of those keyframes exponential in the graph editor. However, the number of particles can’t be animated, and a reverse idea I had by animating the end point of the emission to become exponentially smaller (while keeping the particle count and starting point constant) didn’t work either for the same reason. Is there any way to change the rate at which particles are emitted to be exponential, or is there an alternate way to achieve the effect I’m looking for?

i havent tried this, but i think you can use the children render value. if you press I on it, the value can be animated.

keep in mind that most values in blender can be animated by just hovering over the slide and pressing I. all though i noticed that most values in a particle system is not animatable.