Systems of Differential Equations- Two models of interacting species

[thienthientoo]

1. Given Information/Objectives

The following system for "stable competition":

dx/dt=(2-2x-y)x
dy/dt=(2-x-2y)y

Find the equilibrium points for the system.
Using the Jacobian matrix, linearize the system about the equilibrium that has both species present.
Classify this equilibrium.
Plot direction field and orbits.



2. The attempt at a solution


After finding equilibrium points (2/3,2/3), and the Jacobian (provided that the equilibrium points I found were correct) for the previously mentioned points (J[2/3,2/3]=28/9). I don't know where to go from here; how do I know what kind of equilibrium it is??

 

[Unco]

A few points:
1) There are 3 equilibrium (aka fixed) points; you have found one of them.
2) The Jacobian is a 2x2 matrix, with entries functions of x and y; evaluating it at a point is still a 2x2 matrix.
3) The Jacobian evaluated at a point allows you to classify the equilibrium point in the linearised system. For example, if it has two distinct positive eigenvalues, then the equilibrium point is an unstable proper node. The corresponding eigenvectors, one being dominant, allow you to sketch the direct field near the fixed point. I suggest you consult your notes or text to become aware of the other types.