Equilibrium Stole Some Sleep from Me

I am troubled somewhat by something I learned in class last semester. It has to do with the wolf/rabbit populations. The book implies that there is always a balance between rabbits and wolves according to this handy little vector field. We learned no matter where you start on the phase trajectory you will always end up on the path. There are equilibrium points along that trajectory. This is all well and good, but so many factors can change anything. Fortunately the book says there are many equilibrium points. I guess that means there isn’t a general equilibrium, it is just at that place in that time, with that weather and external forces there is an equilibrium point. How is that an equilibrium though, isn’t it just a point? If there are 500 wolves and 10 rabbits, hell, we’ll just call that equilibrium point. If there are 1000 rabbits and 50 wolves, well hell, there’s another equilibrium point. What are the criteria for an equilibrium point? Could it just be within a function? Does the book say there are many equilibrium points to avoid any problems that might be caused by a minor change in any one of the environmental factors? Can’t one small change at one early point send the whole function into madness? I have to give Frank credit. He did mention that. Yes, I looked back at some of my notes last night found this in red, “In a chaotic system a change of just one rabbit can change the entire equation into spirals and fractals.” (He didn’t want to go into it, and I think I can see why). Can we just call any one of those chaos points an equilibrium point? Are there equilibrium points when we have a chaotic function? What the hell, why not call every point an equilibrium point? Is Stewart (textbook author) just giving us a way to solve a problem in easy terms so in the future we learn that that was just a way to introduce a more difficult concept? Math texts like to do that. Or is it just that I hadn’t slept enough and read too much before I ‘tried’ to go to sleep. It could be that I just remember it wrong. Fortunately in math you get to learn things more than once.

I know later on in my college endeavor this answer might be more obvious and I may laugh at the fact that I was even up last night thinking about this. I could be way off base.

Those damn rabbits.