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  • Nonlinear DDS have similiar input as Linear DDS but the computation takes longer to iterate. This produces larger time lags to compute p_n for large values of n.
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  • Nonlinear DDS often don't have a closed form analytic solution so we can't guarantee a shortcut when needing to compute large values of p_n
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  • We can often interpret the nonlinear dds by building it from the difference equation that tells the difference between the current output p_n and the previous output p_(n-1).
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  • Just like we had parameters "a" and "d" for the linear recursion equation, for a nonlinear recursion equation there are also parameters but we might assign different names.
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  • Part of the reason the recusion equation is non-linear is often because these parameters get multiplied by terms that depend on the independent variable which is the previous output p_(n-1).
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  • The Carrying Capacity Logistic Growth Model has two parameters R for Rate of Growth and C for Carrying Capacity​
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  • Start by defining the initial condition and the recursion equation
    • In[]:=
    p[0]=40;​​R=.8;​​Cap=500;​​p[n_]:=p[n-1]+R*(1-(p[n-1]/Cap))*p[n-1]
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  • Test an output
    • In[]:=
    p[2]
    Out[]=
    117.277
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  • Output a table of values for n and p_n. NOTE: Be careful about iterating very far as the nonlinear computation takes more time. You may also choose to save the output with a variable name stored in memory that you can reference without having to recomput the table values every time.
    • In[]:=
    mynonlindds=Table[{n,p[n]},{n,0,13}]
    Out[]=
    {{0,40},{1,69.44},{2,117.277},{3,189.092},{4,283.157},{5,381.398},{6,453.773},{7,487.336},{8,497.21},{9,499.43},{10,499.885},{11,499.977},{12,499.995},{13,499.999}}
    Out[]=
    {{0,40},{1,69.44},{2,117.277},{3,189.092},{4,283.157},{5,381.398},{6,453.773},{7,487.336},{8,497.21},{9,499.43},{10,499.885},{11,499.977},{12,499.995},{13,499.999}}
    In[]:=
    mynonlintab=TableForm[mynonlindds,TableHeadings{None,{"n iterations","p[n] output"}}]
    Out[]//TableForm=
    n iterations
    p[n] output
    0
    40
    1
    69.44
    2
    117.277
    3
    189.092
    4
    283.157
    5
    381.398
    6
    453.773
    7
    487.336
    8
    497.21
    9
    499.43
    10
    499.885
    11
    499.977
    12
    499.995
    13
    499.999
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  • Here we plot the table output values, not the tableform command that displayed it in columns.
    • In[]:=
    ListPlot[mynonlindds,PlotLabel"My Carrying Capacity Non-Linear DDS",PlotLabels"population"]
    Out[]=
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  • We can build a manipulate interactive tool that allows you to adjust certain parameter values like the carrying capacity, growth rate, and initial conditions.