How to plot a system of recurrence equations

Straightforward method:

With[{NN = 25, a = 2/10, b = 1/10, M0 = 30, L0 = 50},
     ListLinePlot[Transpose[RecurrenceTable[{M[k + 1] == (1 - a) M[k] + a L[k], 
                                             L[k + 1] == (1 - b) L[k] + b M[k],
                                             M[0] == M0, L[0] == L0},
                                            {M, L}, {k, 0, NN}]],
                  DataRange -> {0, NN}, PlotRange -> All]]

Slick method:

With[{NN = 25, a = 2/10, b = 1/10, M0 = 30, L0 = 50},
     ListLinePlot[Transpose[NestList[{{1 - a, a}, {b, 1 - b}}.# &, {M0, L0}, NN]], 
                  DataRange -> {0, NN}, PlotRange -> All]]

Even slicker method:

With[{NN = 25, a = 2/10, b = 1/10, M0 = 30, L0 = 50},
     DiscretePlot[MatrixPower[{{1 - a, a}, {b, 1 - b}}, k, {M0, L0}] // Evaluate,
                  {k, 0, NN}, Filling -> None, Joined -> True,
                  PlotRange -> All]]

All three versions produce the following figure:

Clear["Global`*"]

eqns = {
   M[k + 1] == (1 - a)*M[k] + a*L[k],
   L[k + 1] == (1 - b)*L[k] + b*M[k],
   M[0] == M0, L[0] == L0};

RSolve provides the exact solution to the recurrence equations

sol = RSolve[eqns, {L, M}, k][[1]]

{* {L -> Function[{k}, (a L0 + (1 - a - b)^k b L0 + b M0 - (1 - a - b)^k b M0)/(
   a + b)], M -> 
  Function[{k}, -((-a L0 + a (1 - a - b)^k L0 - a (1 - a - b)^k M0 - b M0)/(
    a + b))]} *}

Verifying,

eqns /. sol // Simplify

{* {True, True, True, True} *}

For your specific parameters,

solEx[k_] = {L[k], M[k]} /. sol /.
   {a -> 2/10, b -> 1/10, M0 -> 30, L0 -> 50} // Simplify

{* {1/3 10^(1 - k) (2 7^k + 13 10^k), 1/3 10^(1 - k) (-4 7^k + 13 10^k)} *}

The functions share a common limit

lim = Limit[solEx[k], k -> Infinity]

(* {130/3, 130/3} *)

Plotting

With[{NN = 25},
 Plot[Evaluate@solEx[k], {k, 0, NN},
  PlotRange -> All,
  PlotLegends -> Placed[{"L", "M"}, {0.5, 0.3}],
  Prolog -> {Gray, Dashed,
    Line[{{0, lim[[1]]}, {NN, lim[[1]]}}]}]]

EDIT: If RSolve is unable to solve:

Clear["Global`*"]

NN = 25; a = 2/10; b = 1/10; M0 = 30; L0 = 50;
rt = RecurrenceTable[{M[k + 1] == (1 - a)*M[k] + a*L[k], 
    L[k + 1] == (1 - b)*L[k] + b*M[k], M[0] == M0, L[0] == L0}, {M, L}, {k, 0,
     NN}];

M[k_Integer] := rt[[k + 1, 1]] /; 0 <= k <= NN

L[k_Integer] := rt[[k + 1, 2]] /; 0 <= k <= NN

DiscretePlot[{L[k], M[k]}, {k, 0, NN},
 PlotRange -> All, Filling -> None,
 PlotLegends -> Placed["Expressions", {0.5, 0.3}]]