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Computer Code for A mathematical framework for evo-devo dynamics

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Mauricio González-Forero at University of St Andrews
Mauricio González-Forero
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Computer
code
for:
A
mathematical
framework
for
evo-devo
dynam-
ics
Mauricio
González -Forero*1
1
School
of
Biology,
University
of
St
Andrews,
St
Andrews,
UK
*mgf3@st-andrews.ac.uk
This
file
contains
the
computer
code
used
to
generate
the
figures
given
in
the
main
text.
This
code
was
prepared
in
Mathematica
12.1.1.0
and
it
is
made
available
under
a
Creative
Commons
Attribu-
tion
licence
(CC
BY).
Illustration
of
socio-devo
instability
 Clear["Global`*"]
Nθ=10;
Na =4;
q=0.5;
y[a_] :=0.5;
Table[x[1, θ] = 0.1, {θ, 1, Nθ}];
Table[x[a, 1] = 0.1, {a, 1, Na}];
xsol =Tablex[a+1, θ+1] = x[a, θ+1]+y[a]x[a, θ+1]+q x[a+1, θ]2,
{a, 1, Na -1},{θ, 1, Nθ-1};
PlotStyleFunction[θ_, Nθ_] :=If[θ⩵Nθ,{Black, Thickness[0.01]},
If[θ ⩵ 1, {Black, Dashing[0.05], Thickness[0.01]},{Gray}]]
Export[StringJoin[ToString[NotebookDirectory[]],
StringJoin["Fig.example.social.SDStabilisation"], ".pdf"],
Show[Table[ListLinePlot[Table[x[a, θ],{a, 1, Na}],
PlotStyle →PlotStyleFunction[θ, Nθ], AxesStyle →Large,
PlotRange →{0, 0.5}, PlotStyle →{Black, Thickness[0.01]},
PlotMarkers →{"●", Large}],{θ, 1, Nθ}]]];
 xm[1] = 0.1;
xmm[1] = 0.1;
ym[a_] :=0.6;
(*Obtain SDS resident*)
xb =Table[x[a, Nθ],{a, 1, Na}];
(*Compute mutant in the context of SDS resident*)
Tablexm[a+1] = xm[a] + ym[a]xm[a] + q xb[[a+1]]2,{a, 1, Na -1};
(*Compute mutant in the context of itself*)
Tablexmm[a+1] = xmm[a] + ym[a]xmm[a] + q xm[a+1]2,{a, 1, Na -1};
(*Plot SDS resident, mutant in context of SDS resident,
and mutant in the context of itself*)
Export[StringJoin[ToString[NotebookDirectory[]],
StringJoin["Fig.example.social.SDNon-Eq"], ".pdf"],
ListLinePlot[{Table[xb[[a]],{a, 1, Na}], Table[xm[a],{a, 1, Na}],
Table[xmm[a],{a, 1, Na}]}, PlotStyle →
{{Black, Dashing[0.05], Thickness[0.01]},{Gray},{Black, Thickness[0.01]}},
AxesStyle →Large, PlotMarkers →{"●", Large}]];
Non-social
development
 Clear["Global`*"]
(*Choose parameters*)
NA =4; (*number of ages*)
Pp =0.7; (*survival probability*)
Finalτ=200;
ι[Na_, p_] :=Sumpa-1,{a, 1, Na}
T[Na_, p_] :=Sum[a(1-y[a]) x[a] × l[a, p],{a, 1, Na}]
partialwpartialx[Na_, p_] :=
Table1
T[Na, p]l[a, p] (1-y[a]),{a, 1, Na},{j, 1, 1}
partialwpartialy[Na_, p_] :=Table-1
T[Na, p]l[a, p] × x[a],{a, 1, Na},{j, 1, 1}
partialxpartialy[Na_] :=Table[Table[If[j⩵a+1, x[a], 0],{j, 1, Na}],{a, 1, Na}]
partialxpartialx[Na_] :=
Table[Table[If[j⩵a+1, 1 +y[a], If[j⩵a, 1, 0]],{j, 1, Na}],{a, 1, Na}]
totalxtotalx[Na_] :=
Table[Table[If[j>a, Product[partialxpartialx[Na][[k, k +1]],{k, a, j -1}],
If[j⩵a, 1, 0]],{j, 1, Na}],{a, 1, Na}]
totalxtotaly[Na_] :=partialxpartialy[Na].totalxtotalx[Na];
totalwtotalx[Na_, p_] :=totalxtotalx[Na].partialwpartialx[Na, p]
totalwtotaly[Na_, p_] :=
partialwpartialy[Na, p] + totalxtotaly[Na].partialwpartialx[Na, p]
2
���
Example.nb
l[a_, p_] :=pa-1
Hy[Na_,τ_] :=Table
Ifa⩵j, ysol[τ][[a, 1]] 1-ysol[τ][[a, 1]], 0,{a, 1, Na},{j, 1, Na}
(*Substitute solutions in total selection gradient*)
Totalwtotalx[Na_,τ_, p_] :=
totalwtotalx[Na, p] /. Table[y[a]→ysol[τ][[a, 1]],{a, 1, Na}] /.
Table[x[a]→xsol[τ, a, 1],{a, 1, Na}]
Totalwtotaly[Na_,τ_, p_] :=
totalwtotaly[Na, p] /. Table[y[a]→ysol[τ][[a, 1]],{a, 1, Na}] /.
Table[x[a]→xsol[τ, a, 1],{a, 1, Na}]
(*Initial conditions*)
ysol[1] = Table12, {a, 1, 4},{j, 1, 1};
Runs[Na_, T_, p_] :=Tablexsol[τ, 1, 1] = 1;
Tablexsol[τ, a +1, 1] = 1+ysol[τ][[a, 1]]xsol[τ, a, 1],{a, 1, Na -1};
ysol[τ+1] = ysol[τ] + ι[Na, p]×Hy[Na, τ].Totalwtotaly[Na, τ, p];,{τ, 1, T}
(*Plots of runs*)
PlotStyleFunction[τ_] :=If[τ ⩵ Finalτ,{Black, Thickness[0.01]},
If[τ ⩵ 1, {Black, Dashing[0.05], Thickness[0.01]},{Gray}]]
PlotRuns[Na_, T_, p_] := {Runs[Na, T, p];,
Export[StringJoin[ToString[NotebookDirectory[]],
StringJoin["Fig.example.control.p=", ToString[p]], ".pdf"],
Show[Table[ListLinePlot[Flatten[ysol[τ]], PlotRange →{-0.1, 1.1},
PlotStyle →PlotStyleFunction[τ], AxesStyle →Large,
PlotMarkers →{"●", Large}],{τ,{1, 2, 5, 10, 20, Finalτ}}]]],
Export[StringJoin[ToString[NotebookDirectory[]],
StringJoin["Fig.example.state.p=", ToString[p]], ".pdf"],
Show[Table[ListLinePlot[Table[xsol[τ, a, 1],{a, 1, Na}],
PlotRange →{0, 5}, PlotStyle →PlotStyleFunction[τ], AxesStyle →Large,
PlotMarkers →{"●", Large}],{τ,{1, 2, 5, 10, 20, Finalτ}}]]]}
(*Do runs and plots*)
PlotRuns[NA, Finalτ, Pp];
(*Hz plots*)
(*To fix color scale in matrix plot*)
cf =Blend[{{0., RGBColor[0.260487, 0.356, 0.891569]},
{0.166667, RGBColor[0.230198, 0.499962, 0.848188]},
{0.333333, RGBColor[0.392401, 0.658762, 0.797589]},
{0.499999, RGBColor[0.964837, 0.982332, 0.98988]},
{0.5, RGBColor[1, 1, 1]},{0.500001, RGBColor[0.95735, 0.957281, 0.896269]},
{0.666667, RGBColor[0.913252, 0.790646, 0.462837]},
{0.833333, RGBColor[0.860243, 0.558831, 0.00695811]},
Example.nb
���
3
{1., RGBColor[1., 0.42, 0.]}},#1]&;
(*cfScaled=cf@Rescale[#,{-1,1},{0,1}]&;*)
cfScaled =cf@Rescale[#,{-4, 4},{0, 1}] &;
(*Compute Hx*)
Totalxtotaly[Na_,τ_] :=
totalxtotaly[Na] /. Table[y[a]→ysol[τ][[a, 1]],{a, 1, Na}] /.
Table[x[a]→xsol[τ, a, 1],{a, 1, Na}]
Hx[Na_,τ_] :=Transpose[Totalxtotaly[Na, τ]].Hy[Na, τ].Totalxtotaly[Na, τ]
(*Compute Hz*)
Totalztotaly[Na_,τ_] :=
ArrayFlatten[{{Totalxtotaly[Na, τ], IdentityMatrix[Na]}}]
Hz[Na_,τ_] :=Transpose[Totalztotaly[Na, τ]].Hy[Na, τ].Totalztotaly[Na, τ]
(*Plots of Hz*)
HzPlot[Na_, p_] :=Table[Export[
StringJoin[ToString[NotebookDirectory[]], StringJoin["Fig.example.Hz.p=",
ToString[p], ".tau=", ToString[τ]], ".pdf"], MatrixPlot[Hz[Na, τ],
FrameTicks →{{{{1, 1},{2, 2},{3, 3},{4, 4},{5, 1},{6, 2},{7, 3},{8, 4}},
{{1, 1},{2, 2},{3, 3},{4, 4},{5, 1},{6, 2},{7, 3},{8, 4}}},
{{{1, 1},{2, 2},{3, 3},{4, 4},{5, 1},{6, 2},{7, 3},{8, 4}},
{{1, 1},{2, 2},{3, 3},{4, 4},{5, 1},{6, 2},{7, 3},{8, 4}}}},
FrameStyle →Large, PlotLegends →BarLegend[{Automatic, {-4, 4}},
LabelStyle →Large], ColorFunction →cfScaled,
ColorFunctionScaling →False]],{τ,{1, 2, 5, 10, 20, Finalτ}}]
(*Do plots of Hz*)
HzPlot[NA, Pp];
(*Plots of the total selection gradients*)
Totalwtotalz[Na_,τ_, p_] :=
ArrayFlatten[{{Totalwtotalx[Na, τ, p]},{Totalwtotaly[Na, τ, p]}}]
dwdyxPlot[Na_, p_] := {Export[StringJoin[ToString[NotebookDirectory[]],
StringJoin["Fig.example.dwdy.p=", ToString[p]], ".pdf"],
Show[Table[ListLinePlot[Flatten[Totalwtotaly[Na, τ, p]],
PlotStyle →PlotStyleFunction[τ], AxesStyle →Large,
PlotRange →{-0.4, 0.1}, PlotStyle →{Black, Thickness[0.01]},
PlotMarkers →{"●", Large}],{τ,{1, 2, 5, 10, 20, Finalτ}}]]],
Export[StringJoin[ToString[NotebookDirectory[]],
StringJoin["Fig.example.dwdx.p=", ToString[p]], ".pdf"],
Show[Table[ListLinePlot[Flatten[Totalwtotalx[Na, τ, p]],
PlotStyle →PlotStyleFunction[τ], AxesStyle →Large,
PlotRange →{0, 0.5}, PlotStyle →{Black, Thickness[0.01]},
PlotMarkers →{"●", Large}],{τ,{1, 2, 5, 10, 20, Finalτ}}]]]}
4
���
Example.nb
dwdyxPlot[NA, Pp];
(*Plots of the partial selection gradients*)
Partialwpartialx[Na_,τ_, p_] :=
partialwpartialx[Na, p] /. Table[y[a]→ysol[τ][[a, 1]],{a, 1, Na}] /.
Table[x[a]→xsol[τ, a, 1],{a, 1, Na}]
Partialwpartialy[Na_,τ_, p_] :=
partialwpartialy[Na, p] /. Table[y[a]→ysol[τ][[a, 1]],{a, 1, Na}] /.
Table[x[a]→xsol[τ, a, 1],{a, 1, Na}]
PartialwpartialyxPlot[Na_, p_] :=
{Export[StringJoin[ToString[NotebookDirectory[]],
StringJoin["Fig.example.partialwpartialy.p=", ToString[p]], ".pdf"],
Show[Table[ListLinePlot[Flatten[Partialwpartialy[Na, τ, p]],
PlotStyle →PlotStyleFunction[τ], AxesStyle →Large,
PlotRange →{-0.4, 0.1}, PlotStyle →{Black, Thickness[0.01]},
PlotMarkers →{"●", Large}],{τ,{1, 2, 5, 10, 20, Finalτ}}]]],
Export[StringJoin[ToString[NotebookDirectory[]],
StringJoin["Fig.example.partialwpartialx.p=", ToString[p]], ".pdf"],
Show[Table[ListLinePlot[Flatten[Partialwpartialx[Na, τ, p]],
PlotStyle →PlotStyleFunction[τ], AxesStyle →Large,
PlotRange →{0, 0.5}, PlotStyle →{Black, Thickness[0.01]},
PlotMarkers →{"●", Large}],{τ,{1, 2, 5, 10, 20, Finalτ}}]]]}
PartialwpartialyxPlot[NA, Pp];
Social
development
 Clear["Global`*"]
(*Choose parameters*)
NA =4; (*number of ages*)
Pp =0.7; (*survival probability*)
Qp =0.5; (*rate of social interaction*)
Finalτ=200;
ι[Na_, p_] :=Sumpa-1,{a, 1, Na}
T[Na_, p_, q_] :=Suma(1-y[a]) x[a] × l[a, p]1+q
1-q y[a],{a, 1, Na}
partialwpartialx[Na_, p_, q_] :=
Table1
T[Na, p, q]l[a, p] (1-y[a]),{a, 1, Na},{j, 1, 1}
partialwpartialy[Na_, p_, q_] :=
Table-1
T[Na, p, q]l[a, p] × x[a]1+q
1-q y[a],{a, 1, Na},{j, 1, 1}
Example.nb
���
5
partialxpartialy[Na_, q_] :=
TableTableIfj⩵a+1, x[a]1+q
1-q y[a], 0,{j, 1, Na},{a, 1, Na}
partialxpartialx[Na_] :=
Table[Table[If[j⩵a+1, 1 +y[a], If[j⩵a, 1, 0]],{j, 1, Na}],{a, 1, Na}]
totalxtotalx[Na_] :=
Table[Table[If[j>a, Product[partialxpartialx[Na][[k, k +1]],{k, a, j -1}],
If[j⩵a, 1, 0]],{j, 1, Na}],{a, 1, Na}]
totalxtotaly[Na_, q_] :=partialxpartialy[Na, q].totalxtotalx[Na];
totalwtotalx[Na_, p_, q_] :=totalxtotalx[Na].partialwpartialx[Na, p, q]
totalwtotaly[Na_, p_, q_] :=
partialwpartialy[Na, p, q] + totalxtotaly[Na, q].partialwpartialx[Na, p, q]
l[a_, p_] :=pa-1
Hy[Na_,τ_] :=Table
Ifa⩵j, ysol[τ][[a, 1]] 1-ysol[τ][[a, 1]], 0,{a, 1, Na},{j, 1, Na}
(*Substitute solutions in total selection gradient*)
Totalwtotalx[Na_,τ_, p_, q_] :=
totalwtotalx[Na, p, q] /. Table[y[a]→ysol[τ][[a, 1]],{a, 1, Na}] /.
Table[x[a]→xsol[τ, a, 1],{a, 1, Na}]
Totalwtotaly[Na_,τ_, p_, q_] :=
totalwtotaly[Na, p, q] /. Table[y[a]→ysol[τ][[a, 1]],{a, 1, Na}] /.
Table[x[a]→xsol[τ, a, 1],{a, 1, Na}]
(*Initial conditions*)
ysol[1] = Table12, {a, 1, 4},{j, 1, 1};
Runs[Na_, T_, p_, q_] :=Tablexsol[τ, 1, 1] = 1;
Tablexsol[τ, a +1, 1] = 1+ysol[τ][[a, 1]]
1-q ysol[τ][[a, 1]] xsol[τ, a, 1],{a, 1, Na -1};
ysol[τ+1] = ysol[τ] + ι[Na, p] × Hy[Na, τ].Totalwtotaly[Na, τ, p, q];,
{τ, 1, T}
(*Plots of runs*)
PlotStyleFunction[τ_, T_] :=If[τ ⩵ T, {Black, Thickness[0.01]},
If[τ ⩵ 1, {Black, Dashing[0.05], Thickness[0.01]},{Gray}]]
PlotRuns[Na_, T_, p_, q_] := {Runs[Na, T, p, q];,
Export[StringJoin[ToString[NotebookDirectory[]], StringJoin[
"Fig.example.social.control.p=", ToString[p], ".q=", ToString[q]],
".pdf"], Show[Table[ListLinePlot[Flatten[ysol[τ]], PlotRange →{-0.1, 1.1},
PlotStyle →PlotStyleFunction[τ, T], AxesStyle →Large,
PlotMarkers →{"●", Large}],{τ,{1, 2, 5, 10, 20, Finalτ}}]]],
Export[StringJoin[ToString[NotebookDirectory[]], StringJoin[
"Fig.example.social.state.p=", ToString[p], ".q=", ToString[q]], ".pdf"],
Show[Table[ListLinePlot[Table[xsol[τ, a, 1],{a, 1, Na}], PlotRange →{0, 20},
PlotStyle →PlotStyleFunction[τ, T], AxesStyle →Large,
6
���
Example.nb
PlotMarkers →{"●", Large}],{τ,{1, 2, 5, 10, 20, Finalτ}}]]]}
(*Do runs and plots*)
PlotRuns[NA, Finalτ, Pp, Qp];
(*Lz plots*)
(*To fix color scale in matrix plot*)
cf =Blend[{{0., RGBColor[0.260487, 0.356, 0.891569]},
{0.166667, RGBColor[0.230198, 0.499962, 0.848188]},
{0.333333, RGBColor[0.392401, 0.658762, 0.797589]},
{0.499999, RGBColor[0.964837, 0.982332, 0.98988]},
{0.5, RGBColor[1, 1, 1]},{0.500001, RGBColor[0.95735, 0.957281, 0.896269]},
{0.666667, RGBColor[0.913252, 0.790646, 0.462837]},
{0.833333, RGBColor[0.860243, 0.558831, 0.00695811]},
{1., RGBColor[1., 0.42, 0.]}},#1]&;
(*cfScaled=cf@Rescale[#,{-1,1},{0,1}]&;*)
cfScaled =cf@Rescale[#,{-50, 50},{0, 1}] &;
(*Compute Hx*)
Totalxtotaly[Na_,τ_, q_] :=
totalxtotaly[Na, q] /. Table[y[a]→ysol[τ][[a, 1]],{a, 1, Na}] /.
Table[x[a]→xsol[τ, a, 1],{a, 1, Na}]
Hx[Na_,τ_, q_] :=Transpose[Totalxtotaly[Na, τ, q]].
Hy[Na, τ].Totalxtotaly[Na, τ, q]
(*Compute Hz*)
Totalztotaly[Na_,τ_, q_] :=
ArrayFlatten[{{Totalxtotaly[Na, τ, q], IdentityMatrix[Na]}}]
Hz[Na_,τ_, q_] :=Transpose[Totalztotaly[Na, τ, q]].
Hy[Na, τ].Totalztotaly[Na, τ, q]
(*Compute Lz*)
partialxpartialxbar[Na_, q_] :=
Table[Table[If[j⩵a, y[a]q, 0],{j, 1, Na}],{a, 1, Na}]
Partialxpartialxbar[Na_,τ_, q_] :=
partialxpartialxbar[Na, q] /. Table[y[a]→ysol[τ][[a, 1]],{a, 1, Na}] /.
Table[x[a]→xsol[τ, a, 1],{a, 1, Na}]
totalxtotalxbar[Na_, q_] :=partialxpartialxbar[Na, q].totalxtotalx[Na]
Totalxtotalxbar[Na_,τ_, q_] :=
totalxtotalxbar[Na, q] /. Table[y[a]→ysol[τ][[a, 1]],{a, 1, Na}] /.
Table[x[a]→xsol[τ, a, 1],{a, 1, Na}]
sxsxbarTranspose[Na_,τ_, q_] :=
Inverse[IdentityMatrix[Na]-Transpose[Totalxtotalxbar[Na, τ, q]]]
sxsyTranspose[Na_,τ_, q_] :=
sxsxbarTranspose[Na, τ, q].Transpose[Totalxtotaly[Na, τ, q]]
szsyTranspose[Na_,τ_, q_] :=
Example.nb
���
7
ArrayFlatten[{{sxsyTranspose[Na, τ, q]},{IdentityMatrix[Na]}}]
Lz[Na_,τ_, q_] :=szsyTranspose[Na, τ, q].Hy[Na, τ].Totalztotaly[Na, τ, q]
(*Plots of Lz*)
LzPlot[Na_, p_, q_] :=Table[Export[StringJoin[ToString[NotebookDirectory[]],
StringJoin["Fig.example.social.Lz.p=", ToString[p], ".tau=", ToString[τ]],
".pdf"], MatrixPlot[Lz[Na, τ, q],
FrameTicks →{{{{1, 1},{2, 2},{3, 3},{4, 4},{5, 1},{6, 2},{7, 3},{8, 4}},
{{1, 1},{2, 2},{3, 3},{4, 4},{5, 1},{6, 2},{7, 3},{8, 4}}},
{{{1, 1},{2, 2},{3, 3},{4, 4},{5, 1},{6, 2},{7, 3},{8, 4}},
{{1, 1},{2, 2},{3, 3},{4, 4},{5, 1},{6, 2},{7, 3},{8, 4}}}},
FrameStyle →Large, PlotLegends →BarLegend[{Automatic, {-50, 50}},
LabelStyle →Large], ColorFunction →cfScaled,
ColorFunctionScaling →False]],{τ,{1, 2, 5, 10, 20, Finalτ}}]
(*Do plots of Lz*)
LzPlot[NA, Pp, Qp];
(*Plots of the total selection gradients*)
Totalwtotalz[Na_,τ_, p_, q_] :=
ArrayFlatten[{{Totalwtotalx[Na, τ, p, q]},{Totalwtotaly[Na, τ, p, q]}}]
dwdyxPlot[Na_, T_, p_, q_] := {Export[StringJoin[ToString[NotebookDirectory[]],
StringJoin["Fig.example.social.dwdy.p=", ToString[p]], ".pdf"],
Show[Table[ListLinePlot[Flatten[Totalwtotaly[Na, τ, p, q]],
PlotStyle →PlotStyleFunction[τ, T], AxesStyle →Large,
PlotRange →{-0.4, 0.1}, PlotStyle →{Black, Thickness[0.01]},
PlotMarkers →{"●", Large}],{τ,{1, 2, 5, 10, 20, Finalτ}}]]],
Export[StringJoin[ToString[NotebookDirectory[]],
StringJoin["Fig.example.social.dwdx.p=", ToString[p]], ".pdf"],
Show[Table[ListLinePlot[Flatten[Totalwtotalx[Na, τ, p, q]],
PlotStyle →PlotStyleFunction[τ, T], AxesStyle →Large,
PlotRange →{0, 0.5}, PlotStyle →{Black, Thickness[0.01]},
PlotMarkers →{"●", Large}],{τ,{1, 2, 5, 10, 20, Finalτ}}]]]}
dwdyxPlot[NA, Finalτ, Pp, Qp];
(*Plots of the partial selection gradients*)
Partialwpartialx[Na_,τ_, p_, q_] :=
partialwpartialx[Na, p, q] /. Table[y[a]→ysol[τ][[a, 1]],{a, 1, Na}] /.
Table[x[a]→xsol[τ, a, 1],{a, 1, Na}]
Partialwpartialy[Na_,τ_, p_, q_] :=
partialwpartialy[Na, p, q] /. Table[y[a]→ysol[τ][[a, 1]],{a, 1, Na}] /.
Table[x[a]→xsol[τ, a, 1],{a, 1, Na}]
8
���
Example.nb
PartialwpartialyxPlot[Na_, T_, p_, q_] :=
{Export[StringJoin[ToString[NotebookDirectory[]],
StringJoin["Fig.example.social.partialwpartialy.p=", ToString[p]], ".pdf"],
Show[Table[ListLinePlot[Flatten[Partialwpartialy[Na, τ, p, q]],
PlotStyle →PlotStyleFunction[τ, T], AxesStyle →Large,
PlotRange →{-0.4, 0.1}, PlotStyle →{Black, Thickness[0.01]},
PlotMarkers →{"●", Large}],{τ,{1, 2, 5, 10, 20, Finalτ}}]]],
Export[StringJoin[ToString[NotebookDirectory[]],
StringJoin["Fig.example.social.partialwpartialx.p=", ToString[p]], ".pdf"],
Show[Table[ListLinePlot[Flatten[Partialwpartialx[Na, τ, p, q]],
PlotStyle →PlotStyleFunction[τ, T], AxesStyle →Large,
PlotRange →{0, 0.5}, PlotStyle →{Black, Thickness[0.01]},
PlotMarkers →{"●", Large}],{τ,{1, 2, 5, 10, 20, Finalτ}}]]]}
PartialwpartialyxPlot[NA, Finalτ, Pp, Qp];
Example.nb
���
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A mathematical framework for evo-devo dynamics
Article
December 2023
Mauricio González-Forero