a9 = 20;

\(p[0] = 0.7; p[1] = 0.85; 
\(Do[p[a] = 0.9, {a, 2, a9}]; \)\)m[1] = 0; m[2] = 0.45; Do[m[a] = 0.5, {a, 3, a9}]; Do[l[i, j] = 0, {i, 1, a9}, {j, 1, a9}]; Do[l[1, j] = m[j] p[0], {j, 1, a9}]; Do[l[j + 1, j] = p[j], {j, 1, a9}]; ll = Table[l[i, j], {i, 1, a9}, {j, 1, a9}]; Eigenvalues[ll][[1]]

MatrixForm[ll]

t9 = 30; Log10[u_] := Log[u]/Log[10];

n[0] = Table[100 Random[]*0.99^a, {a, 1, a9}]\(u[0] = 
      ∑\+\(a = 1\)\%a9\( n[0]\)[[a
          ]]; 

    \(Do[n[t] = ll . n[t - 1]; 
      u[t] = ∑\+\(a = 2\)\%a9\( n[t]\)[[a
            ]], {t, 1, t9}]; \)\)g = 
      ListPlot[Table[Log10[u[t]], {t, 1, 30}], PlotJoined -> True, 
        Frame -> True];

t9 = 110; g1 = 
      Plot[(k/(1 + ((k - n0)/(n0 Exp[r t])))) /. {r -> 0.1, 
            k -> 1000, n0 -> 1200}, {t, 0, t9}, 
        DisplayFunction -> Identity]; g2 = 
      Plot[(k/(1 + ((k - n0)/(n0 Exp[r t])))) /. {r -> 0.1, 
            k -> 1000, n0 -> 10}, {t, 0, t9}, 
        DisplayFunction -> Identity]; Show[{g1, g2}, PlotRange -> {{0, t9}, {0, 1200}}, 
      DisplayFunction -> $DisplayFunction];

f[x_] = Exp[r (1 - (x/k))] x; para = {r -> 0.5, k -> 1000};r3=2.2;
g0 = Plot[x, {x, 0, 2000}, DisplayFunction -> Identity];
g1 = 
        Plot[f[x] /. {r -> 0.5, k -> 1000}, {x, 0, 2000}, 
          DisplayFunction -> Identity]; g2 = Plot[f[x] /. {r -> 1.5, k -> 1000}, {x, 0, 2000}, 
        DisplayFunction -> Identity]; g3 = Plot[f[x] /. {r -> r3, k -> 1000}, {x, 0, 2000}, 
        DisplayFunction -> Identity];

x[0] = 1100; Do[x[t] = 
    f[x[t - 1]] /. {r -> r3, k -> 1000}
, {t, 1, 500}]
g4 = ListPlot[Table[{x[Round[t/2 - 
         0.1]], x[Round[t/2 + 0.4]]}, 
     {t, 2, 500}], PlotJoined -> True, 
    DisplayFunction -> Identity];

Show[{g0, g1, g2, g3, g4}, AspectRatio -> 1, 
      PlotRange -> {{0, 1600}, {0, 1600}}, 
      DisplayFunction -> $DisplayFunction];

para={r->r3,k -> 1000};
FindRoot[Evaluate[f[f[x]] /. para] == x, {x, 1500}]eq1 = FindRoot[Evaluate[f[f[x]] /. para] == x, {x, 100}](∂_x) Evaluate[f[f[x]] /. para] /. eq1Plot[{x, f[f[x]] /. para}, {x, 0, 2000}, AspectRatio -> 1];

t9 = 30; x[0] = 900; Do[x[t] = f[x[t - 1]] /. para, {t, 1, t9}]ListPlot[Table[x[t], {t, 1, t9}], PlotJoined -> True, 
      AspectRatio -> 1, PlotRange -> {{0, t9}, {0, 1600}}];

,

Clear[r,k];f[x_] = Exp[r ((1 - x/k))] x; 
para = {r -> 3.15, k -> 1000}; 
eq2 = FindRoot[Evaluate[f[f[f[x]]] /. para] == x, 
   {x, 3000}]
D[f[f[f[x]]], x] /. para /. eq2
Plot[{f[f[f[x]]] /. para, x}, {x, 0, 3000}, 
   AspectRatio -> 1];

x[0] = 2000; t9 = 100; 
Clear[r,k];f[x_] = Exp[r ((1 - x/k))] x; 
Do[x[t] = f[x[t - 1]] /. {r -> 3.1, k -> 10000}, 
  {t, 1, t9}]
ListPlot[Table[Log[x[t]]/Log[10], 
    {t, t9 - 100, t9}], PlotJoined -> True];

=Simplify[f'[x]];

a1 = 1; t9 = 1000; k9 = 100; 
Do[r = N[3 + k/k9]; x[0] = 0.5; 
   Do[x[0] = f[x[0]], {t, 1, 100}]; 
   Do[x[t] = f[x[t - 1]], {t, 1, t9}]; 
   lp[k] = 
    N[Sum[Log[Abs[df[x[t]]]], {t, 1, t9}]/t9]; 
   Print[r, " ", lp[k]], {k, 1, k9}]

;;;;;
;

=g'[x];

t9 = 100; rnd := s0*(Random[] - 0.5); 
  Clear[ll,r]; 
ll[r_] = {{0, r}, {s1, s2}}; 
Do[sa = -0.8*k; para = 
    {s1 -> 0.25, s2 -> 0.8, r0 -> 0.8, 
     s0 -> 1.6}; n[0] = {{10}, {10}}; 
   r[0] = r0 + s0/2 /. para; 
   u[0] = 
    n[0][[1]][[1]] + n[0][[2]][[1]]; 
   Do[r[t] = 
      If[sa > 0, 
       sa*r[t - 1] + 
        (1 - sa)*(r0 + rnd /. para), 
       -sa*(2*r0 - r[t - 1]) + 
        (1 + sa)*(r0 + rnd /. para)]; 
     n[t] = 
      N[ll[r[t]] /. para] . n[t - 1]; 
     u[t] = 
      n[t][[1]][[1]] + n[t][[2]][[1]], 
    {t, 1, t9}]; 
   g[k] = 
    ListPlot[Table[u[t], {t, 0, t9}], 
     PlotJoined -> True, 
     DisplayFunction -> Identity]; 
   Print[u[t9]], {k, 0, 1}]