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Include Delay Differential Equations Directly in Dynamic Simulations
This shows a dynamic simulation of a differential delay logistic equation y(t)=a y(t) (1-y(t-1)) compared to an ordinary differential logistic equation y(t)=a y(t) (1-y(t)).
In[1]:=

Click for copyable input
Manipulate[

 Module[{T = 50, sol, sold},

  sol = First[

    y /. NDSolve[{y'[t] == a y[t] (1 - y[t]), y[0] == 0.1}, 

      y, {t, 0, T}]]; 

  sold = First[

    y /. NDSolve[{y'[t] == a y[t] (1 - y[t - 1]), 

       y[t /; t <= 0] == 0.1}, y, {t, 0, T}]];

   ParametricPlot[{{sol[t], sol[t - 1]}, {sold[t], sold[t - 1]}}, {t, 

    1, T}, PlotRange -> {{0, 3}, {0, 3}}, 

   PlotStyle -> {Gray, Blue}]], {{a, 1}, 0, 2}]
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