11/6: Interacting systems: modeling, phase plane, and trajectories

10.6

1) Main Points

Chapter 10.6 continues exploring differential equations but this time with two interacting systems. The book uses the example of the relationship between robins and worms. Independently the differential equations regarding their populations are dw/dt = aw (positive because with no robins they increase) and dr/dt=-br (negative because with no food the robins decrease). In relation to each other though these equation look like: dw/dt=aw - Effects of robins on worms and dr/dt = -br + Effects of worms on robins. With these considered as variables with constants the equations come out to be (dw/dt = aw - cwr) and (dr/dt = -br + kwr). We look at this first assuming the constants a=b=c=k=1. We want to see both graphs over time but first it is easier to plot (w,r) the relationship between the two populations. This graph is called a phase plane and the point is called a phase trajectory. Using the chain rule we know that (dr/dt = dr/dw * dw/dt) so to find dr/dw we solve and get (dr/dw =(-r+wr)/(w-wr)). When w and r both equal 1 we see that their differential equations = 0 which means this is an equilibrium and the populations don't change. If we graph this on a slope field we see that the derivatives at the different points create a closed curve. This means that if you plug in a certain number of worms and robins the derivative will point in a certain direction that if continued will lead to the same starting values of w and r.


2) Challenges


This chapter is a little difficult for me because I am unfamiliar with slope fields. I understand this concept graphically but when it comes to using differential equations I fee like it will be hard for me to connect the relationships between the two variables. Think of a graph as the two variables can be misleading. I am used to using time as the x axis so it is very important with these problem to consider the variables represented.

3) Reflections

This concept has obvious applications for real life. Populations often affect each other and their populations can change periodically overtime.