Dynamik des Räuber-Beute-Systems:

f[n1_,n2_]:= ϵ1 n1 - γ1 n1 n2
g[n1_,n2_]:= -ϵ2 n2 + γ2 n1 n2
Assum = ϵ1>0&&ϵ2>0&&γ1>0&&γ2>0;

sol=Solve[{f[n1,n2],g[n1,n2]}=={0,0},{n1,n2}]

RBModell[startϵ1_,startϵ2_,startγ1_,startγ2_,startn1_,startn2_,range_]:=Manipulate[
Module[{f,g,n1,n2,plot},
f[n1_,n2_]:= ϵ1 n1 - γ1 n1 n2;
g[n1_,n2_]:= -ϵ2 n2 + γ2 n1 n2;
plot:=StreamPlot[{f[n1,n2],g[n1,n2]},{n1,0,maxn1},{n2,0,maxn2},PlotRange-> range,PerformanceGoal->"Quality",AxesLabel-> {"n1","n2"}];
Show[plot]
],
{{ϵ1,startϵ1,"Wachstumsrate Beute"},0,1},{{ϵ2,startϵ2,"Sterberate Räuber"},0,1},{{γ1,startγ1,"Wahrscheinlichkeit Fresstod"},0,1},{{γ2,startγ2,"Wahrscheinlichkeit Räubertod"},0,1},
{{maxn1,startn1},0,100},{{maxn2,startn2},0,100}
]

RBModell[0.9,0.6,0,0,0.001,0.001,Full]
"2. Fixpunkt (0,0)";

RBModell[0.9,0.6,1,0.5,5,5,{{0,3},{0,3}}]
"Fixpunkt im Zentrum der Spirale bei(\!\(\*FractionBox[\(\(ϵ2\)\(\\ \)\), \(γ2\)]\),((ϵ1)/(γ1)))";

Numerische Lösung der DGL:

parameter={ϵ1-> 0.9,ϵ2-> 0.6,γ1-> 1,γ2-> 0.5}
DGLsol=NDSolve[{n1'[t]==f[n1[t],n2[t]],n2'[t]== g[n1[t],n2[t]],n1[0]==4,n2[0]==1}/.parameter,{n1,n2},{t,0,(10^4)},PrecisionGoal->(10^9)]
"periodische Populationsentwicklung"
ParametricPlot[Evaluate[{n1[t],n2[t]}/.DGLsol],{t,0,100},PlotRange-> {{0,4.5},{0,3}},AxesLabel-> {Beute,Räuber}]
"Oszillationsverhalten"
Plot[Evaluate[{n1[t],n2[t]}/.DGLsol],{t,0,50},PlotLegends-> {Beute,Räuber}]