Populationsdynamik

1  Erste Vorgehensweise

Remove["Global`*"]

Bemerkung zur Populationsmatrix B

B = {{3.5, -25}, {0.05, 0.5}};
B // MatrixForm

B = {{7/2, -25}, {5/100, 5/10}};
B // MatrixForm

Eigenvalues[B]

Eigenvectors[B]

a) Negative Resultate

x[1]={1,1}; B={{3.5,-25},{0.05,0.5}};
x[n_]:= B.x[n-1];
z[k_]:=Prepend[Append[x[k],x[k][[1]]/x[k][[2]]],k];
Prepend[Table[z[k],{k,1,20}],{"Nr.","x","y","Quot. x/y"}]//N//MatrixForm

x[1]={5,1}; B={{3.5,-25},{0.05,0.5}};
x[n_]:= B.x[n-1];
z[k_]:=Prepend[Append[x[k],x[k][[1]]/x[k][[2]]],k];
Prepend[Table[z[k],{k,1,20}],{"Nr.","x","y","Quot. x/y"}]//N//MatrixForm

b) Stabilität

x[1]={10,1}; B={{3.5,-25},{0.05,0.5}};
x[n_]:= B.x[n-1];
z[k_]:=Prepend[Append[x[k],x[k][[1]]/x[k][[2]]],k];
Prepend[Table[z[k],{k,1,20}],{"Nr.","x","y","Quot. x/y"}]//N//MatrixForm

c) Zunahme, positive Resultate, Quotient konvergiert von unten her

x[1]={25,1}; B={{3.5,-25},{0.05,0.5}};
x[n_]:= B.x[n-1];
z[k_]:=Prepend[Append[x[k],x[k][[1]]/x[k][[2]]],k];
Prepend[Table[z[k],{k,1,20}],{"Nr.","x","y","Quot. x/y"}]//N//MatrixForm

d) Proportionale Zunahme

x[1]={50,1}; B={{3.5,-25},{0.05,0.5}};
x[n_]:= B.x[n-1];
z[k_]:=Prepend[Append[x[k],x[k][[1]]/x[k][[2]]],k];
Prepend[Table[z[k],{k,1,20}],{"Nr.","x","y","Quot. x/y"}]//N//MatrixForm

e) Zunahme, positive Resultate, Quotient konvergiert von oben her

x[1]={70,1}; B={{3.5,-25},{0.05,0.5}};
x[n_]:= B.x[n-1];
z[k_]:=Prepend[Append[x[k],x[k][[1]]/x[k][[2]]],k];
Prepend[Table[z[k],{k,1,20}],{"Nr.","x","y","Quot. x/y"}]//N//MatrixForm

x[1]={100,1}; B={{3.5,-25},{0.05,0.5}};
x[n_]:= B.x[n-1];
z[k_]:=Prepend[Append[x[k],x[k][[1]]/x[k][[2]]],k];
Prepend[Table[z[k],{k,1,20}],{"Nr.","x","y","Quot. x/y"}]//N//MatrixForm

2  Andere Vorgehensweise

Remove[x]

x[1] = 25; y[1] = 1;
x[
          n_] := (B . {x[n - 1], 
                y[n - 1]})[[1]];
y[n_] := (B . {x[n - 1], y[n - 1]})[[2]];

{x[1], y[1]}

{x[2], y[2]} // N

Table[{x[k], y[k]}, {k, 1, 15}] // 
        N//MatrixForm