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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 88 "                          \+
                 MATHEMATICAL EVOLUTIONARY THEORY  -  MATH 583" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "plotsetup(default):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "plotsetup(window):" }}}
{SECT 1 {PARA 3 "" 0 "" {TEXT -1 19 "Haploid populations" }}{SECT 1 
{PARA 4 "" 0 "" {TEXT -1 97 "Dynamics of allele frequencies in a one-l
ocus k-allele haploid population with constant fitnesses" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "k:=5:   t_max:=100:                \+
        # number of alleles, time_max" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 81 "f[1]:=.2: f[2]:=.2: f[3]:=.2: f[4]:=.2: f[5]:=.2:    \+
# initial allele frequencies" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 64 "w[1]:=.92: w[2]:=.94: w[3]:=.96: w[4]:=.98: w[5]:=1: # fitnesses
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "wmean:=sum( 'f[i]*w[i]^
t', 'i'=1..k): Wmean:=sum( 'f[i]*w[i]^(t+1)', 'i'=1..k):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "wbar:= Wmean/wmean:         # mean \+
fitness at generation t" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "
for i from 1 to k do p[i]:=f[i]*w[i]^t/wmean od:            # allele f
requencies at generation t" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
180 "plot( \{p[1],p[2],p[3],p[4],p[5],wbar\}, t=0..t_max, title=`p_i a
nd w_mean 1-locus k-allele haploid population with constant fitnesses`
,labels=[generation_number,p_i]);        # plot" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 29 "Frequency-depend
ent selection" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "solving a cubic O
DE" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 208 "### WARNING: `dsolve` has been ext
ensively rewritten, many new result forms can occur and options are sl
ightly different, see help page for details\ndsolve( diff(x(t),t) = -s
*x(t)*(1-x(t))+mu*(1-2*x(t)),x(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#/-%\"xG6#%\"tG*&,(*&-%'RootOfG6#,(*&F'\"\"\"-%%sqrtG6#,&*$)%\"sG\"\"#
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ng solutions" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "eq:=2*s*p(t
)*(1-p(t))*(P-p(t)):    # dynamic equation" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 47 "s:=1:    P:=.5:    t_max:=10:      # parameters" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "init_cond:=\{seq([0,i/10],i
=1..9)\}; # initial conditions" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*i
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> " 0 "" {MPLTEXT 1 0 51 "with(DEtools):                     # load li
braries" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 159 "DEplot(diff(p(t
),t)=eq,p(t),0..t_max, init_cond, p=0..1,title=`Linear frequency-depen
dent selection in 1-locus 2-allele haploid population`);            # \+
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!\"\"F^a[l$\"\"#F`a[l-%%VIEWG6$;$!+++++]F*$\"++++]5F[u;$Fha[lFeu$Fja[l
FF-%+AXESLABELSG6$Q\"t6\"Q%p(t)Fbb[l-%&TITLEG6#%foLinear~frequency-dep
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320 2 0 1 0 2 6 0 4 2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 
0 0 0 0 0 0 0 0 1 1 0 0 0 26 0 0 0 0 0 0 0 }}}}{SECT 1 {PARA 4 "" 0 "
" {TEXT -1 58 "Coevolution of two species: no within-species interacti
ons" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "With no between-species int
eractions, the dynamics are described by" }}{PARA 0 "" 0 "" {TEXT -1 
52 "                           dp1/dt = p1 q1 b (p2-q2)," }}{PARA 0 "
" 0 "" {TEXT -1 52 "                           dp2/dt = p2 q2 c (p1-q1
)." }}{PARA 0 "" 0 "" {TEXT -1 104 "The solutions of the former satisf
y to  (p1 q1)^c / (p2 q2)^b = const. The following plots these curves.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 
"with(plots): z:= (x*(1-x))^c/(y*(1-y))^b;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"zG*&)*&%\"xG\"\"\",&F)F)F(!\"\"F)%\"cG\"\"\")*&%\"y
GF),&F)F)F0F+F)%\"bG!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "Let \+
b and c have the same sign (competitive or cooperative interactions)" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "b:=1: c:=1: contourplot(z
, x=0.1..0.9,y=0.1..0.9);" }}{PARA 13 "" 1 "" {INLPLOT "6*-%'CURVESG6M
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7$Fiel7$$\"1qi0ula&[&F*$\"1LP%fU`WJ\"F*7$7$FiuFf`lFfgl7$Fdgl7$$\"1])Q
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18!QW@r<H\"F*7$7$$\"1/+++++?hF*Fb_lFfhl7$Fdhl7$$\"11lGX#R,2'F*$\"1-Nra
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0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 27 0 0 0 0 0 0 0 }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 
-1 34 "Solving a system of two linear ODE" }}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 27 "A system of two linear ODE:" }}{PARA 0 "" 0 "" {TEXT -1 
59 "                                              x' = ax + by," }}
{PARA 0 "" 0 "" {TEXT -1 59 "                                         \+
     y' = cx + dy." }}{PARA 0 "" 0 "" {TEXT -1 63 "Its solutions depen
d on the eigenvalues of the stability matrix" }}{PARA 0 "" 0 "" {TEXT 
-1 59 "                                                    [ a  b]" }}
{PARA 0 "" 0 "" {TEXT -1 58 "                                         \+
 S  =    [ c  d]." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "S := l
inalg[matrix](2,2,[a,b,c,d]);            # define a 2x2 matrix S" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "eq := [ diff(x(t),t)=a*x(t)+
b*y(t), diff(y(t),t) = c*x(t)+d*y(t)]; with(DEtools):" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 26 "Several cases to consider." }}{PARA 0 "" 
0 "" {TEXT -1 35 "Both eigenvalues real and positive." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "a := 1: b := 0: c := 1: d := 1.5: e
valf(Eigenvals(S)); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 222 "ph
aseportrait([ diff(x(t),t)=a*x(t)+b*y(t), diff(y(t),t) = c*x(t)+d*y(t)
],[x(t),y(t)],-2..1,\{[0,.5,.5],[0,.5,-.5],[0,-.5,-.5],[0,-.5,.5]\},st
epsize=.2,x=-1..1,y=-1..1,title=`unstable node (real and positive)`,ar
rows=THICK);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Both eigenvalues \+
real and negative" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "a := -
.9: b := 0: c := 1: d := -1.5: evalf(Eigenvals(S));" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 220 "phaseportrait([ diff(x(t),t)=a*x(t)+b*y(
t), diff(y(t),t) = c*x(t)+d*y(t)],[x(t),y(t)],-2..1,\{[0,.5,.5],[0,.5,
-.5],[0,-.5,-.5],[0,-.5,.5]\},stepsize=.2,x=-1..1,y=-1..1,title=`stabl
e node (real and negative)`,arrows=THICK);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 51 "One positive eigenvalue and one negative eigenvalue" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "a := 1: b := 0: c := 1: d :=
 -1.5: evalf(Eigenvals(S));   " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 224 "phaseportrait([ diff(x(t),t)=a*x(t)+b*y(t), diff(y(t),t) = c*
x(t)+d*y(t)],[x(t),y(t)],-2..1,\{[0,.5,.5],[0,.5,-.5],[0,-.5,-.5],[0,-
.5,.5]\},stepsize=.2,x=-1..1,y=-1..1,title=`saddle (one negative, one \+
positive)`,arrows=THICK);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "Comp
lex eigenvalues, Re(l)>0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 
"a := .1: b := 1: c := -1: d := .5: evalf(Eigenvals(S));" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 222 "phaseportrait([ diff(x(t),t)=a*x(t
)+b*y(t), diff(y(t),t) = c*x(t)+d*y(t)],[x(t),y(t)],-2..1,\{[0,.5,.5],
[0,.5,-.5],[0,-.5,-.5],[0,-.5,.5]\},stepsize=.2,x=-1..1,y=-1..1,title=
`unstable focus (complex, Re(l)>0)`,arrows=THICK);" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 28 "Complex eigenvalues, Re(l)<0" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 57 "a :=- .5: b := 1: c := -1: d := -.5: evalf(E
igenvals(S));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 220 "phaseport
rait([ diff(x(t),t)=a*x(t)+b*y(t), diff(y(t),t) = c*x(t)+d*y(t)],[x(t)
,y(t)],-2..1,\{[0,.5,.5],[0,.5,-.5],[0,-.5,-.5],[0,-.5,.5]\},stepsize=
.2,x=-1..1,y=-1..1,title=`stable focus (complex, Re(l)<0)`,arrows=THIC
K);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "Complex eigenvalues, Re(l)
=0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "a :=0: b := 1: c := -
1: d := .0: evalf(Eigenvals(S)); " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 214 "phaseportrait([ diff(x(t),t)=a*x(t)+b*y(t), diff(y(t
),t) = c*x(t)+d*y(t)],[x(t),y(t)],-2..1,\{[0,.5,.5],[0,.5,-.5],[0,-.5,
-.5],[0,-.5,.5]\},stepsize=.2,x=-1..1,y=-1..1,title=`center (complex, \+
Re(l)=0)`,arrows=THICK);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 52 "Stability analysis for syst
ems of two non-linear ODE" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "The f
ollowing system will be used as an example" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
67 " f1:=x*(a-b*x-c*y);                             # specify   f1(x,y
)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f1G*&%\"xG\"\"\",(%\"aG\"\"\"*
&%\"bGF*F&F*!\"\"*&%\"cGF*%\"yGF*F-F*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 67 " f2:=y*(d*x-e-f*y);                            # spec
ify   f2(x,y) " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f2G*&%\"yG\"\"\",
(*&%\"dG\"\"\"%\"xGF+F+%\"eG!\"\"*&%\"fGF+F&F+F.F+" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 64 "with(linalg):                              \+
     # load libraries" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning, new def
inition for norm" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new definiti
on for trace" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "S:=array([[
diff(f1,x),diff(f1,y)],[diff(f2,x),diff(f2,y)]]);     # define stabili
ty matrix  S" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SG-%'matrixG6#7$7$
,(%\"aG\"\"\"*&%\"bGF,%\"xGF,!\"#*&%\"cGF,%\"yGF,!\"\",$*&F/\"\"\"F2F7
F47$*&F3F7%\"dGF,,(*&F:F7F/F7F,%\"eGF4*&%\"fGF,F3F7F0" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "S0:=subs( [x=0,y=0],evalm(S));   # \+
substitute equilibrium values in S" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%#S0G-%'matrixG6#7$7$%\"aG\"\"!7$F+,$%\"eG!\"\"" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 51 "lambda0:=eigenvals(evalm(S0));   # find eigen
values" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(lambda0G6$%\"aG,$%\"eG!\"
\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "S1:=subs( [x=a/b,y=0]
, evalm(S));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#S1G-%'matrixG6#7$7$
,$%\"aG!\"\",$*&*&F+\"\"\"%\"cGF0\"\"\"%\"bG!\"\"F,7$\"\"!,&*&*&%\"dGF
0F+F2F2F3F4F0%\"eGF," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "lam
bda1:=eigenvals( evalm(S1));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(lam
bda1G6$,$%\"aG!\"\"*&,&*&%\"dG\"\"\"F'F-F-*&%\"eGF-%\"bGF-F(\"\"\"F0!
\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "nontriv:=solve( \{a
-b*x-c*y=0,d*x-e-f*y=0\},\{x,y\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%(nontrivG<$/%\"xG*&,&*&%\"eG\"\"\"%\"cGF,F,*&%\"fGF,%\"aGF,F,\"\"\",
&*&%\"bGF,F/F1F,*&F-F1%\"dGF,F,!\"\"/%\"yG*&,&*&F6F1F0F1F,*&F+F1F4F1!
\"\"F1F2F7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "S2:=subs( non
triv, evalm(S));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#S2G-%'matrixG6#
7$7$,(%\"aG\"\"\"*&*&%\"bG\"\"\",&*&%\"eGF,%\"cGF,F,*&%\"fGF,F+F,F,F,F
0,&*&F/F,F6F0F,*&F4F0%\"dGF,F,!\"\"!\"#*&*&F4F0,&*&F:F0F+F0F,*&F3F0F/F
0!\"\"F,F0F7F;FB,$*&*&F1F0F4F0F0F7F;FB7$*&*&F?F0F:F0F0F7F;,(*&*&F:F0F1
F0F0F7F;F,F3FB*&*&F6F0F?F0F0F7F;F<" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 31 "lambda2:=eigenvals( evalm(S2));" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%(lambda2G6$,$*&,,*(%\"bG\"\"\"%\"eG\"\"\"%\"cGF+!\"\"
*(%\"aGF-F*F+%\"fGF+F/*(F,F+F*F+F2F+F-*(F2F+%\"dGF+F1F+F/*$-%%sqrtG6#,
<*()F*\"\"#F+)F,F=F+)F.F=F+F-*,F<F+F,F+F.F+F1F+F2F+F=**F<F+F>F+F.F+F2F
+F=*.F*F+F,F+F.F+F2F+F5F+F1F+F=*()F1F=F+F<F+)F2F=F+F-**F1F+F<F+FEF+F,F
+F=**FDF+F*F+FEF+F5F+!\"#*(F>F+F<F+FEF+F-*,F,F+F*F+FEF+F5F+F1F+FH*(FEF
+)F5F=F+FDF+F-**F?F+FLF+F,F+F1F+!\"%**F?F+F5F+F>F+F*F+\"\"%**F.F+FLF+F
2F+FDF+FNF+F-F+,&*&F*F-F2F-F-*&F.F-F5F-F-!\"\"#F-F=,$*&,,F)F/F0F/F3F-F
4F/F6F/F+FRFUFV" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "det(S2);
                   # find determinant" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#*&*&,&*&%\"eG\"\"\"%\"cGF(F(*&%\"fGF(%\"aGF(F(F(,&*&%\"dGF(F,\"\"\"
F(*&F'F0%\"bGF(!\"\"F(F0,&*&F2F0F+F0F(*&F)F0F/F0F(!\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "factor( trace(S2));        # find t
race" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,**(%\"bG\"\"\"%\"eGF(%\"c
GF(F(*(%\"aGF(F'\"\"\"%\"fGF(F(*(F)F-F'F-F.F-!\"\"*(F.F-%\"dGF(F,F-F(F
-,&*&F'F-F.F-F(*&F*F-F2F-F(!\"\"F0" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 40 "Coevolution
 of two species: general case" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 8 "restart:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 98 "Coevolutionary d
ynamics of two haploid species    (linear symmetric frequency-dependen
t selection)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "eq:=[diff(x(t),t)=
x(t)*(1-x(t))*(a*(2*x(t)-1)+b*(2*y(t)-1)),       diff(y(t),t)=y(t)*(1-
y(t))*(c*(2*x(t)-1)+d*(2*y(t)-1))]:     # dynamic equations" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 79 "a:=-1.0 : b:=1.2: c:=-1.3: d:=1.01: t_max
:=500:                    # parameters" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 114 "init_cond:=\{seq([0,1/4,j/4],j=1..3),seq([0,3/4,j/4]
,j=1..3)\};                                 # initial conditions" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*init_condG<(7%\"\"!#\"\"\"\"\"%F(7%
F'F(#F)\"\"#7%F'F(#\"\"$F*7%F'F/F(7%F'F/F,7%F'F/F/" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 35 "init_cond:=\{[0,1/4,3/4],[0,.5,.6]\}:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "with(DEtools):             \+
                                                                      \+
                   # load libraries" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 204 "phaseportrait( eq, [x(t),y(t)], 0..t_max, init_cond,
stepsize=.1,x=0..1, y=0..1, arrows=THIN, title=`Coevolutionary dynamic
s of two haploid species`);                                           \+
      # plot" }}{PARA 13 "" 1 "" {INLPLOT "6(-%'CURVESG6]dl7%7$\"\"!F(
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 binomial sampling" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "N:=50: p:=.4: T:=100: X:=arr
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o T do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "n:=0: for j from 1 to N d
o" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "                   x:=random[u
niform](1):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "                   i
f x<p then n:=n+1: fi od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "      \+
p:=n/N: X[i]:=p; od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "F:=
seq([i,X[i]],i=0..T): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "
plot(  \{F\}, style=point, labels=[ `generation number`,`frequency` ],
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\"#fF*F+7$$\"#gF*F+7$$\"#hF*F+7$$\"#iF*F+7$$\"#jF*F+7$$\"#kF*F+7$$\"#l
F*F+7$$\"#ZF*F+7$$\"#[F*F+7$$\"#\\F*F+7$$\"#]F*F+7$$\"#^F*F+7$$\"#_F*F
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\"#?F*$\"+++++qF[v7$$\"#@F*$\"+++++!)F[v7$$\"#AF*$\"+++++%)F[v7$$\"#BF
*$\"+++++#)F[v7$$\"#CF*$\"+++++yF[v7$$\"#DF*F^w7$$\"#EF*Fcw7$$\"#FF*Fd
v7$$\"#GF*F^w7$$\"#HF*Fiv7$$\"#IF*$\"+++++!*F[v7$$\"#JF*Fgx7$$\"#KF*Fg
x7$$\"#LF*$\"+++++')F[v7$$\"#MF*Fby7$$\"#NF*Fby7$$\"#OF*$\"+++++))F[v7
$$\"#PF*Fgx7$$\"#QF*Fby7$$\"#RF*Fiv7$$\"#SF*$\"+++++#*F[v7$$\"#TF*F[[l
7$$\"#UF*$\"+++++)*F[v7$$\"#VF*F+7$$\"#WF*F+7$$\"#XF*F+7$$\"#YF*F+7$$
\"\"%F*$\"+++++kF[v7$$\"\"&F*$\"+++++gF[v7$$\"\"'F*Fiu7$$\"\"(F*$\"+++
++YF[v7$$\"\")F*Fa]l7$$\"\"*F*Fa]l7$$\"#5F*$\"+++++aF[v7$$\"#6F*$\"+++
++QF[v7$$\"#7F*$\"+++++]F[v7$$\"#8F*Ff^l7$$\"#9F*Fi\\l7$$\"#:F*Fiu7$$
\"#;F*$\"+++++iF[v7$$\"#<F*$\"+++++_F[v7$$\"#=F*F\\^l7$F+$\"+++++WF[v7
$$\"\"#F*$\"+++++UF[v7$$\"\"$F*F\\^l7$%%FAILGFj`l-%'COLOURG6&%$RGBG$\"
*++++\"!\")F*F*-%*LINESTYLEG6#F,-%&STYLEG6#%&POINTG-%+AXESLABELSG6$%2g
eneration~numberG%*frequencyG-%&TITLEG6#%.Genetic~DriftG-%%VIEWG6$%(DE
FAULTGFebl" 2 320 320 320 5 0 1 0 2 9 0 4 2 1.000000 45.000000 
45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 22260 95 0 0 0 
0 0 0 }}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "
" {TEXT -1 20 "Fixation probability" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 
"with(DEtools):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "  General solu
tion for the probability of fixation" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 44 "eq:=m(p)*diff(u(p),p)+v(p)*diff(u(p),p$2)/2;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#eqG,&*&-%\"mG6#%\"pG\"\"\"-%%diffG6
$-%\"uGF)F*F+F+*&-%\"vGF)F+-F-6$F/-%\"$G6$F*\"\"#F+#F+F9" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "dsolve(eq=0,u(p));" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/-%\"uG6#%\"pG,&%$_C1G\"\"\"*&%$_C2GF*-%$IntG6$-%$
expG6#,$-F.6$*&-%\"mGF&\"\"\"-%\"vGF&!\"\"F'!\"#F'F*F*" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "F:=exp(-2*int(m(p)/v(p),p));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG-%$expG6#,$-%$intG6$*&-%\"mG6#%
\"pG\"\"\"-%\"vGF/!\"\"F0!\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 31 "U:=int(F,p=0..x)/int(F,p=0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%\"UG*&-%$intG6$-%$expG6#,$-F'6$*&-%\"mG6#%\"pG\"\"\"-%\"vGF2!\"\"
F3!\"#/F3;\"\"!%\"xGF4-F'6$F)/F3;F;\"\"\"F7" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 19 " Additive fitnesses" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 37 "m(p):=s*p*(1-p); v(p):=p*(1-p)/(2*N);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>-%\"mG6#%\"pG*(%\"sG\"\"\"F'F*,&F*F*F'!\"\"F*" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"vG6#%\"pG,$*&*&F'\"\"\",&F+F+F'!
\"\"F+\"\"\"%\"NG!\"\"#F+\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 2 "U;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&-%$expG6#,$*(%\"sG\"
\"\"%\"NGF+%\"xGF+!\"%F+!\"\"F+\"\"\",&-F&6#,$*&F*F0F,F0F.F+F/F+!\"\"
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "subs(x=1/(2*N),%);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&-%$expG6#,$%\"sG!\"#\"\"\"!\"\"F+
\"\"\",&-F&6#,$*&F)F+%\"NGF+!\"%F+F,F+!\"\"" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 25 "  Underdominant selection" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 24 "m(p):=s*p*(1-p)*(2*p-1);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>-%\"mG6#%\"pG**%\"sG\"\"\"F'F*,&F*F*F'!\"\"F*,&F'\"\"#
F,F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "U;" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#,$*&,&-%$erfG6#*&*(%\"sG\"\"\"%\"NGF,,&%\"xG\"\"#!
\"\"\"\"\"F2F,*$-%%sqrtG6#*&F+F2F-F2F,!\"\"F2-F'6#*&*&F+F,F-F,F,*$-F56
#F7F,F8F2F,F9F8#F2F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "sub
s(x=1/(2*N),%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&-%$erfG6#*&*(
%\"sG\"\"\"%\"NGF,,&*&F,F,F-!\"\"\"\"\"!\"\"F1F1F,*$-%%sqrtG6#*&F+F1F-
F1F,F0F1-F'6#*&*&F+F,F-F,F,*$-F56#F7F,F0F1F,F8F0#F1\"\"#" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "X:=(erf(Y*(e-1))+erf(Y))/(2*erf(Y))
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"XG,$*&,&-%$erfG6#*&%\"YG\"\"
\",&%\"eGF-!\"\"F-F-F--F)6#F,F-\"\"\"F1!\"\"#F-\"\"#" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 16 "series(X,e=0,2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#+'%\"eG*&*&-%$expG6#,$*$)%\"YG\"\"#\"\"\"!\"\"\"\"\"F-F
1F/*&-%%sqrtG6#%#PiGF/-%$erfG6#F-\"\"\"!\"\"\"\"\"-%\"OG6#F1\"\"#" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "subs(\{Y=sqrt(S),e=1/N\},exp
(-Y^2)*Y/(sqrt(Pi)*erf(Y))*e);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&*&
-%$expG6#,$%\"SG!\"\"\"\"\"-%%sqrtG6#F)\"\"\"F/*(-F-6#%#PiGF/-%$erfG6#
*$F,F/\"\"\"%\"NG\"\"\"!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 49 "F:=subs(x=1/N,(exp(-4*s*N*x)-1)/(exp(-4*s*N)-1));" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%\"FG*&,&-%$expG6#,$%\"sG!\"%\"\"\"!\"\"F-\"\"
\",&-F(6#,$*&F+F-%\"NGF-F,F-F.F-!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 91 "plot( \{subs(N=25,log10(F)),subs(N=50,log10(F)),subs(
N=100,log10(F))\},s=-0.1..0.1,y=-17..0);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "
First Lyapunov value" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 15 "General f
ormula" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "restart: readlib(m
taylor):   readlib(coeftayl):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 41 "A:=(i,j)->coeftayl(f1,[x,y]=[0,0],[i,j]);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 41 "B:=(i,j)->coeftayl(f2,[x,y]=[0,0],[i,j]);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 94 "     The first Lyapunov value (as \+
in Bautin 1938; see also p. 207 in Bautin & Leontovich 1976)" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "L:= -1/(A(0,1))*" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 56 " (A(1,0)*B(1,0) * (A(1,1)^2+A(1,1)*B(0,2)+A
(0,2)*B(1,1))" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "+A(1,0)*A(0,1) * (
B(1,1)^2+A(2,0)*B(1,1)+A(1,1)*B(2,0))" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 48 "+B(1,0)^2      * (A(1,1)*A(0,2)+2*A(0,2)*B(0,2))" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 82 "-2*A(1,0)*B(1,0)*(B(0,2)^2-A(2,0)*A(0,2))-2*A(
1,0)*A(0,1)*(A(2,0)^2-B(2,0)*B(0,2))" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 43 "-A(0,1)^2 * (2*A(2,0)*B(2,0)+B(1,1)*B(2,0))" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 57 "+(A(0,1)*B(1,0)-2*A(1,0)^2)*(B(1,1)*B(0,2)-A(1,1)*A
(2,0))" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "-(A(1,0)^2+A(0,1)*B(1,0))
* ( 3*(B(1,0)*B(0,3)-A(0,1)*A(3,0))+2*A(1,0)*(A(2,1)+B(1,2))" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 32 "+ B(1,0)*A(1,2)-A(0,1)*B(2,1))):" }}}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 11 "Example 1: " }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 28 "P:=mu*x-omega*y-(x^2+y^2)*x;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 28 "Q:=omega*x+mu*y-(x^2+y^2)*y;" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 38 "readlib(mtaylor):   readlib(coeftayl):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "A:=(i,j)->coeftayl(P,[x,y]=[
0,0],[i,j]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "B:=(i,j)->c
oeftayl(Q,[x,y]=[0,0],[i,j]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 46 "L:= -1/(A(0,1)*(A(1,0)*B(0,1)-A(0,1)*B(1,0)))*" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 56 " (A(1,0)*B(1,0) * (A(1,1)^2+A(1,1)*B(0,2)+A(0,2)
*B(1,1))" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "+A(1,0)*A(0,1) * (B(1,1
)^2+A(2,0)*B(1,1)+A(1,1)*B(2,0))" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 
"+B(1,0)^2      * (A(1,1)*A(0,2)+2*A(0,2)*B(0,2))" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 82 "-2*A(1,0)*B(1,0)*(B(0,2)^2-A(2,0)*A(0,2))-2*A(1,0)*
A(0,1)*(A(2,0)^2-B(2,0)*B(0,2))" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "
-A(0,1)^2 * (2*A(2,0)*B(2,0)+B(1,1)*B(2,0))" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 57 "+(A(0,1)*B(1,0)-2*A(1,0)^2)*(B(1,1)*B(0,2)-A(1,1)*A(2
,0))" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "-(A(1,0)^2+A(0,1)*B(1,0))* \+
( 3*(B(1,0)*B(0,3)-A(0,1)*A(3,0))+2*A(1,0)*(A(2,1)+B(1,2))" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 32 "+ B(1,0)*A(1,2)-A(0,1)*B(2,1)));" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "factor(subs(mu=0,L));" }}}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 30 "Example 2: predator-prey model" }
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 31 "f1:=r*x*(1-x/K)-beta*x*y/(x+C);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "f2:=(k*beta*x/(x+C)-m)*y;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "solve(\{f1=0,f2=0\},\{x,y\})
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "with(linalg):         \+
                # load libraries" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 93 "S:=array([[diff(f1,x),diff(f1,y)],[diff(f2,x),diff(f2
,y)]]);     # define stability matrix  S" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 69 "S0:=subs( [x=0,y=0],evalm(S));   # substitute equilib
rium values in S" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "lambda0
:=eigenvals(evalm(S0));   # find eigenvalues" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 31 "S1:=subs( [x=K,y=0], evalm(S));" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "lambda1:=eigenvals( evalm(S1));" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "nontriv:=solve( \{(k*beta*x
/(x+C)-m)=0,-f1=0\},\{x,y\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 80 "X:=m*C/(k*beta-m); Y:=r*C*k*(-K*m+K*k*beta-m*C)/(K*(k^2*beta^2-2
*k*beta*m+m^2));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "S2:=sub
s( nontriv, evalm(S));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "l
ambda2:=eigenvals( evalm(S2));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 45 "det(S2);                   # find determinant" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "(-K*m+K*k*beta-m*C)*m*r/(K*k*beta);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "tr:=factor( trace(S2));
        # find trace" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "rea
dlib(mtaylor):   readlib(coeftayl):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 41 "A:=(i,j)->coeftayl(f1,[x,y]=[X,Y],[i,j]);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "B:=(i,j)->coeftayl(f2,[x,y]=[X,Y],[
i,j]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 94 "     The first Lyapunov
 value (as in Bautin 1938; see also p. 207 in Bautin & Leontovich 1976
)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "L:= -1/(A(0,1))*" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 " (A(1,0)*B(1,0) * (A(1,1)^2+A(1,1)*
B(0,2)+A(0,2)*B(1,1))" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "+A(1,0)*A(
0,1) * (B(1,1)^2+A(2,0)*B(1,1)+A(1,1)*B(2,0))" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 48 "+B(1,0)^2      * (A(1,1)*A(0,2)+2*A(0,2)*B(0,2))" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "-2*A(1,0)*B(1,0)*(B(0,2)^2-A(2,0)*A
(0,2))-2*A(1,0)*A(0,1)*(A(2,0)^2-B(2,0)*B(0,2))" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 43 "-A(0,1)^2 * (2*A(2,0)*B(2,0)+B(1,1)*B(2,0))" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 57 "+(A(0,1)*B(1,0)-2*A(1,0)^2)*(B(1,1)*B(0,2
)-A(1,1)*A(2,0))" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "-(A(1,0)^2+A(0,
1)*B(1,0))* ( 3*(B(1,0)*B(0,3)-A(0,1)*A(3,0))+2*A(1,0)*(A(2,1)+B(1,2))
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "+ B(1,0)*A(1,2)-A(0,1)*B(2,1)))
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "factor(L);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "subs(m=k*beta*(K-C)/(K+C),%);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "factor(%);" }}}}{SECT 1 
{PARA 4 "" 0 "" {TEXT -1 31 "Example 3: coevolutionary model" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 33 "P:=x*(1-x)*(a*(2*x-1)+b*(2*y-1));" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Q:=y*(1-y)*(c*(2*x-1)+d*(2*y
-1));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "readlib(mtaylor): \+
  readlib(coeftayl):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "A:=
(i,j)->coeftayl(P,[x,y]=[1/2,1/2],[i,j]);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 44 "B:=(i,j)->coeftayl(Q,[x,y]=[1/2,1/2],[i,j]);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "L:= -1/(A(0,1)*(A(1,0)*B(0,1
)-A(0,1)*B(1,0)))*" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 " (A(1,0)*B(1,
0) * (A(1,1)^2+A(1,1)*B(0,2)+A(0,2)*B(1,1))" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 55 "+A(1,0)*A(0,1) * (B(1,1)^2+A(2,0)*B(1,1)+A(1,1)*B(2,0
))" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "+B(1,0)^2      * (A(1,1)*A(0,
2)+2*A(0,2)*B(0,2))" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "-2*A(1,0)*B(
1,0)*(B(0,2)^2-A(2,0)*A(0,2))-2*A(1,0)*A(0,1)*(A(2,0)^2-B(2,0)*B(0,2))
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "-A(0,1)^2 * (2*A(2,0)*B(2,0)+B(
1,1)*B(2,0))" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "+(A(0,1)*B(1,0)-2*A
(1,0)^2)*(B(1,1)*B(0,2)-A(1,1)*A(2,0))" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 85 "-(A(1,0)^2+A(0,1)*B(1,0))* ( 3*(B(1,0)*B(0,3)-A(0,1)*A(3,0))+2
*A(1,0)*(A(2,1)+B(1,2))" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "+ B(1,0)
*A(1,2)-A(0,1)*B(2,1)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 
"factor(L);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "subs(d=-a,%)
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "factor(%);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 4 "" 0 "" 
{TEXT -1 62 "Complete stability analysis and dynamics for the mimicry \+
model" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 17 "Dynamic equations" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 
" f1:=x*(1-x)*(a*(2*x-1)+b*(2*y-1));                             " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 " f2:=y*(1-y)*(c*(2*x-1)+d*(2
*y-1));                            " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 10 "Equilibria" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "solve(
\{f1=0,f2=0\},\{x,y\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "
factor(resultant(f1,f2,x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
42 "with(linalg):                             " }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 23 "Define stability matrix" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 65 "S:=array([[diff(f1,x),diff(f1,y)],[diff(f2,x),diff(
f2,y)]]);     " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "S1:=subs(
[x=0,y=0],evalm(S)):  lambda1:=eigenvals(evalm(S1));" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 61 "S2:=subs([x=1,y=0],evalm(S)):  lambda2:=e
igenvals(evalm(S2));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "S3:
=subs([x=0,y=1],evalm(S)):             lambda3:=eigenvals( evalm(S3));
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "S4:=subs([x=1,y=1],eval
m(S)):             lambda4:=