cmath16.nb - Ordinary Differential Equations 
Systems of ODEs - Numerical Solution

Clear["Global`*"]

definition and general solution

deq1=x'[t]==x[t](1-x[t]-y[t])
deq2=y'[t]==(1/10)y[t](1-x[t])
sol=DSolve[{deq1,deq2},{x,y},t]

Numerical Solutions

Partial Solution 1

x0=0.1; y0=0.2; tmin=0; tmax=30;
nsol=NDSolve[{deq1,deq2,x[0]==x0, y[0]==y0},{x,y},{t,tmin,tmax}]

solution in user functions

xt=x[t]/.nsol[[1]] 
yt=y[t]/.nsol[[1]] 
Plot[xt,{t,0,tmax}, PlotRange->All, AxesLabel->{t,x}]
Plot[yt,{t,0,tmax}, PlotRange->All,AxesLabel->{t,y}]
ParametricPlot[{xt,yt},{t,0,tmax}, PlotRange->All,FrameLabel->{x,y}, Frame->True]

Partial Solution 2

x0=1.5; y0=0.1; tmin=0; tmax=50;
nsol=NDSolve[{deq1,deq2,x[0]==x0, y[0]==y0},{x,y},{t,tmin,tmax}]

solution in user functions

xt=x[t]/.nsol[[1]] 
yt=y[t]/.nsol[[1]] 
Plot[xt,{t,0,tmax}, PlotRange->All, AxesLabel->{t,x}]
Plot[yt,{t,0,tmax}, PlotRange->All,AxesLabel->{t,y}]
ParametricPlot[{xt,yt},{t,0,tmax}, PlotRange->All,FrameLabel->{x,y}, Frame->True, AspectRatio->1]