Welcome to Fractal Forums

(* User Parameters *)
len = 100; (* Population-size = list-length *)
count = 1000; (* While loop counter - number of timesteps *)
mRate = .001; (* Mutation rate *)

(*Initialization*)
data = RandomVariate[UniformDistribution[], 
  len]; (* generate a list of random variables *)
mat = {data}; (* put generated data into a second list level - this \
will be the output matrix *)

(*Main Program*)
While[count > 1,
 smallest = 
  Ordering[data][[1]];(* find the element with the smallest value *)
 replace = 
  RandomVariate[UniformDistribution[], 
   3];(* build a list of replace values *)
 data = If[1 < smallest < len,
   ReplacePart[
    data, {smallest - 1 -> replace[[1]], smallest -> replace[[2]], 
     smallest + 1 -> replace[[3]]}], (* 
   replace the correct array positions *)
   If[smallest == 1,
    ReplacePart[
     data, {len -> replace[[1]], 1 -> replace[[2]], 
      2 -> replace[[3]]}], (* special-cases for borders *)
    ReplacePart[
     data, {len - 1 -> replace[[1]], len -> replace[[2]], 
      1 -> replace[[3]]}]]]; (*in order to make it a full circle *)
 
 mat = Append[mat, data]; (* add a line to the history *)
 (* initialize Mutation *)
 i = 1;
 mArray = 
  RandomVariate[UniformDistribution[], 
   len]; (*generate a List of random variables. Again. *)
 
 While[i < len,
   
   If[mArray[[i]] < mRate, 
     data = ReplacePart[data, 
        i -> RandomVariate[UniformDistribution[]]];] (* 
    if the variables are below the mutation rate, 
    replace the data with a random value *)
    i++]
  
  
  count--] (*make sure, the two while loops finish *)

MatrixPlot[mat, ColorFunction -> "TemperatureMap", Frame -> False, 
 PixelConstrained -> 1](*plot with pretty colors*)