a=2;b=0.5;c=1.9;d=0.4;
Clear[x,y]
sol1=NDSolve[{x'[t]==a x[t]-b x[t] y[t],y'[t]==-c y[t]+d x[t]y[t],y[0]==4,x[0]==4},{x,y},{t,0,20}];

sol2=NDSolve[{x'[t]==a x[t]-b x[t] y[t],y'[t]==-c y[t]+d x[t]y[t],y[0]==1,x[0]==2},{x,y},{t,0,20}];

sol3=NDSolve[{x'[t]==a x[t]-b x[t] y[t],y'[t]==-c y[t]+d x[t]y[t],y[0]==2,x[0]==3},{x,y},{t,0,20}];

sol4=NDSolve[{x'[t]==a x[t]-b x[t] y[t],y'[t]==-c y[t]+d x[t]y[t],y[0]==1,x[0]==3},{x,y},{t,0,20}];
Plot[Evaluate[{x[t],y[t]}/.sol1],{t,0,20},PlotLegends->{Beute,Räuber}]
ParametricPlot[{Evaluate[{x[t],y[t]}/.sol1],Evaluate[{x[t],y[t]}/.sol2],Evaluate[{x[t],y[t]}/.sol3],Evaluate[{x[t],y[t]}/.sol4]},{t,0,10},PlotRange-> {{0,15},{0,15}}]

a=2;b=0.5;c=1.9;d=0.4;
Clear[x,y]
Manipulate[sol1=NDSolve[{x'[t]==a x[t]-b x[t] y[t]-Δa x[t],y'[t]==-c y[t]+d x[t]y[t]-Δc y[t],y[0]==3,x[0]==4},{x,y},{t,0,100}];
Plot[{Evaluate[{x[t],y[t]}/.sol1],(c+Δc)/d,(a-Δa)/b},{t,0,20},PlotLegends->{Beute,Räuber},AxesLabel->{Zeit, Population}]
ParametricPlot[{Evaluate[{x[t],y[t]}/.sol1]},{t,0,10},PlotRange-> {{0,15},{0,15}},AxesLabel->{Beute,Jäger}],{Δa,0,3(a/4)},{Δc,0,3(c/4)}]

Clear[x,y]
Manipulate[a=2;b=0.5;c=1.9;d=0.4;e=0.6;f=1;
sol1=NDSolve[{
x'[t]==a x[t](1-x[t]/m)-b x[t] y[t],y'[t]==-c y[t]+d x[t]y[t],
y[0]==3,x[0]==4},
{x,y},{t,0,5000}];
Plot[{Evaluate[{x[t],y[t]}/.sol1]},{t,0,20},PlotLegends->{Beute,Räuber},AxesLabel->{Zeit, Population}]
ParametricPlot[{Evaluate[{x[t],y[t]}/.sol1]},{t,0,500},PlotRange-> {{0,6.5},{0,5.4}}],{{m,250,"Maximalkapazität"},1,300}]

Clear[x,y,z,q]
Manipulate[sol1=NDSolve[{
x'[t]==-a1 x[t] y[t], 
y'[t]==-c1 y[t]+a1 x[t]y[t],
z'[t]==c1 y[t],
y[0]==y0,x[0]==x0,z[0]==0},
{x,y,z,q},{t,0,1000}];
Plot[{Evaluate[{x[t],y[t],z[t],q[t]}/.sol1]},{t,0,30},PlotLegends->{"Gesund","Krank","Immun","Tote"},AxesLabel->{Zeit, Population},PlotRange->{0,1}]
,{{x0,0.9,"Anfangswert gesunder Menschen" },0,1},{{y0,0.1,"Anfangswert kranker Menschen"},0,1},{{a1,0,"Ansteckungsrate"},0,1},{{c1,0,"Immun/Todesrate"},0,1}]

Clear[x,y,z,q]
Manipulate[sol1=NDSolve[{
x'[t]==-a1 x[t] y[t]+b1 y[t], 
y'[t]==-c1 y[t]+a1 x[t]y[t]-s1 y[t]-b1 y[t],
z'[t]==c1 y[t],
q'[t]==s1 y[t],
y[0]==y0,x[0]==x0,z[0]==z0,q[0]==0},
{x,y,z,q},{t,0,1000}];
Plot[{Evaluate[{x[t],y[t],z[t],q[t]}/.sol1]},{t,0,20},PlotLegends->{"Gesund","Krank","Immun","Tote"},AxesLabel->{Zeit, Population},PlotRange->{0,1}]
,{{x0,0.9,"Anfangswert gesunder Menschen" },0.1,10},{{y0,0.1,"Anfangswert kranker Menschen"},0.1,10},{{z0,0,"Anfangswert von keine Ansteckung mehr möglich"},0,10},{{l0,0,"Anfangswert von Toten"},0,10},{{a1,0,"Ansteckungsrate"},0,1},{{c1,0,"Immun"},0,1},{{b1,0,"Wieder anfällig"},0,1},{{s1,0,"Sterberate"},0,1}]