Model for Competition and Simulation

We begin with a mathematical model for competition and show some basic simulations of this system of differential equation.

> dex := diff(x(t),t) = 0.2*x(t)*(1 - x(t)/8000 - 1.1*y(t)/8000);
dey := diff(y(t),t) = 0.15*y(t)*(1 - 0.7*x(t)/6000 - y(t)/6000);

> with(DEtools): with(plots):

Warning, the name changecoords has been redefined

Maple can be used to show the direction field and typical solutions for this model.

> Plot1 := DEplot([dex,dey],[x(t),y(t)], t=0..100, x=0..8000, y=0..8000,[[x(0)=200,y(0)=200], [x(0)=200,y(0)=7000], [x(0)=7000,y(0)=200], [x(0)=7000,y(0)=7000]], arrows=MEDIUM, title=`Competition Model`, linecolor=t/50, color=CYAN):
Plot1;

Maple's dsolve can numerically solve this system of equations. It produces a procedure that can be used for graphing the solutions.

> ff := dsolve({dex,dey,x(0)=200,y(0)=200},
{x(t),y(t)}, type=numeric, output=listprocedure);

> fx := subs(ff,x(t)): fy := subs(ff,y(t)):

A sequence of points can be created for plotting. We will use both 2D and 3D plot routines.

> xpts := [seq([n,fx(n)],n=0..100)]:
ypts := [seq([n,fy(n)],n=0..100)]:
xy3pts := [seq([n,fx(n),fy(n)],n=0..100)]:

> Xgraph := plot(xpts, t=0..100, title = `Competition Model`, color = BLUE):
Ygraph := plot(ypts, color = RED):
display(Xgraph,Ygraph);

The 3D graph shows the trajectory of the differential equation.

> spacecurve(xy3pts, axes=NORMAL, orientation=[-70,30]);