Find the Equilibria and Sketch Nullclines

Standard analysis of differential equations begins with finding the equilibria. Equilibria are found by solving the right hand sides of the system of differential equations equal to zero. Geometrically, this occurs when the nullclines intersect. Nullclines are lines (more generally curves) when each of the right hand sides of differential equation are equal to zero.

First we write the right hand sides of the differential equations, then solve them simultaneously equal to zero.

> f1 := (X,Y) -> 0.2*X*(1 - X/8000 - 1.1*Y/8000);
f2 := (X,Y) -> 0.15*Y*(1 - 0.7*X/6000 - Y/6000);

> xyeq := solve({f1(X,Y)=0,f2(X,Y)=0},{X,Y});

Maple finds all 4 equilibria very easily.

Below we graph the nullclines with BLUE lines for dx/dt = 0 and RED lines for dy/dt = 0.

> Y1 := solve(f1(X,Y) = 0,Y); Y2 := solve(f2(X,Y)/Y = 0,Y);

> Null1 := plot([20,y,y=0..8000],color = BLUE):

> Null2 := plot(Y1, X = 0..8000, color = BLUE):

> Null3 := plot([x,30,x=0..9000],color = RED):

> Null4 := plot(Y2, X = 0..8570, color = RED):

> display({Null1, Null2, Null3, Null4});

Below we show the overlay of the Phaseportrait with the Nullcines.

> display({Null1, Null2, Null3, Null4,Plot1});

>