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Clear[Am, As, Aa, \[Alpha], \[Rho], \[Theta]m, \[Theta]s, \
$ v- }+ r* u- s$ [( x' d7 W9 N \[CurlyPhi]m, \[CurlyPhi]s, \[Epsilon]]0 y2 U, M1 L3 k5 Y% @" [
\[Gamma]a = 0.1; \[Gamma]m = 0.15; \[Gamma]s =
{$ u* N9 `* _) }% Q6 e 1 - \[Gamma]a - \[Gamma]m;
! T' n5 E! G) b a6 U \[Epsilon] = 0.04; \[Alpha] = 0.3; \[Rho] = 0.04;* L1 d) Y1 l+ ~4 W
\[Theta]m = 0.75; \[Theta]s = 0.9;
9 u# k4 O% `4 k2 W3 Y5 F- `7 K gRate = 0.02;
0 r, `+ [. f9 g0 \' j( B* W# ^2 D Am = (gRate + \[Rho])/\[Alpha]; Ba = 4; Bm = 1; Bs = 2.5;' c: A7 c% r6 |9 t
ps = Bm/Bs; pa = Bm/Ba;
1 c, M, x$ J% ^' v# x5 s4 X8 ~, k \[Delta] = 0.03;
5 j) f; ?4 a+ G- T B = \!\(TraditionalForm\`\*) I" Y3 j' e5 {
FractionBox[
8 M* y7 Z5 Q3 G6 j: g1 v RowBox[{) A$ z' p6 O, R9 D' F
RowBox[{
8 y5 r: `/ T2 A- D( j9 P+ u6 v& @, s RowBox[{
) s h* M$ p! G# a% L) J" V StyleBox["(",
5 \. {7 f: x; v* [4 V; K SpanMinSize->1.,
7 h8 u1 r1 P k. P/ Q! y x SpanMaxSize->1.],
" F) N: j& d6 b$ Y RowBox[{"1", "\[Minus]", "\[Alpha]"}],
) u; ~2 |( Z. d5 M: y StyleBox[")",
* d5 U. e7 c$ R( m2 M6 e7 K SpanMinSize->1.,
- {$ v% W& ?4 c+ U% G" p. g9 i- v SpanMaxSize->1.]}], "gRate"}], "+", "\[Rho]"}], / q, F# G: T* H2 n
"\[Alpha]"] \[Minus] \[Delta]\);
' z' B( j9 g5 G, z1 u& N+ B" N" G cap = 10;
, e, m; L# \8 h! p( y& ~ csp = (pa*cap)/ps;/ e2 M6 Z" r% ~4 {" ~# L& @
D = ((1 \[Minus] \[Alpha])*- }% m" }" {/ g
gRate + \[Rho] - \[Alpha]*\[Delta])/(\[Rho] + gRate);8 M2 U0 P% ?3 v6 W2 W% ^
\[CurlyPhi]m = 0.1; \[CurlyPhi]s = 0.1;* u7 w0 u, |9 r0 `
Print["*** Initial Values ***"]; [6 L+ _3 ]. C2 R
E0 = 1.5;" J# o0 r) q& y. a" v% c
K0 = E0/B;/ t. X% u! Y& t& G- A
hm0 = 0.25; hs0 = 0.25;(* initial values *)
) ~3 t4 x4 S% S. o" E6 q. i) j! o \[Eta]m0 = hm0/K0; \[Eta]s0 = hs0/K0;
6 N$ k! i2 z5 V9 s xm0 = (B*\[Gamma]m^\[Epsilon]*% V# V. f+ f& G6 b" h- o) [! z$ C J
hm0^(\[Theta]m*(1 - \[Epsilon])))/(\[Gamma]a^\[Epsilon]*pa^(
/ i, p& A* I2 F* y% a( _3 M 1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*4 v7 G! E8 V, s/ @2 U+ A# R t
hm0^(\[Theta]m*(1 - \[Epsilon])) + \[Gamma]s^\[Epsilon]*(ps*( O7 y( k/ d! a
hs0^\[Theta]s)^(1 - \[Epsilon]));. h/ _+ Y/ z# P
xs0 = (B*\[Gamma]s^\[Epsilon]*(ps*
6 Q/ |: G) [3 \ hs0^\[Theta]s)^(1 - \[Epsilon]))/(\[Gamma]a^\[Epsilon]*pa^(
- M- w# N% C* ]( t 1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*
1 p, E2 n5 q$ ?. D5 U/ [ hm0^(\[Theta]m*(1 - \[Epsilon])) + \[Gamma]s^\[Epsilon]*(ps*, t) Q1 x$ k+ F
hs0^\[Theta]s)^(1 - \[Epsilon]));
- H9 z6 I: b: W$ b1 z. n Print["\[Eta]_{m,0}=" <> ToString[\[Eta]m0], 4 w9 w: A: y, \+ e. H1 ]; T& J
", \[Eta]_{s,0}=" <> ToString[\[Eta]s0],
) u5 f6 t1 p; q/ } N5 R) y: F ", x_{m,0}=" <> ToString[xm0], ", x_{s,0}=" <> ToString[xs0]]/ s& k y3 { M2 ?; I) p
TT = 100;(* end time *)% A- _1 y6 C+ R$ s+ M& Z
(* Solve differential equations *)
: q9 {: [3 T0 c+ p8 O" z3 I' a Sol = NDSolve[{xs'[t] = (1 - \[Epsilon])*' D# D/ |! K' }( h1 X5 Z N$ `
xs[t]*( (1 - xs[t]/3 p7 u% Y2 p5 ]
B)*(\[Theta]s*\[CurlyPhi]s*(xs[t]/(ps*\[Eta]s[t]) - 1) -
% U$ c5 d8 ]/ f5 v# t/ Q) E N xm[t]/B \[Theta]m*\[CurlyPhi]m*(xm[t]/\[Eta]m[t] - 1))), - C4 ]" E- N0 G- V( x
xm'[t] == (1 - \[Epsilon])*: W- |4 V1 U3 q' p4 ?7 b( l9 ]
xm[t]*( (1 - xm[t]/
- g) t6 H9 ?8 [ B)*\[Theta]m*\[CurlyPhi]m*(xm[t]/\[Eta]m[t] - 1) - " r0 A* {3 f$ n/ G' @! n
xs[t]/B*\[Theta]s*\[CurlyPhi]s*(xs[t]/(ps*\[Eta]s[t]) - ; P9 x8 z1 N$ q# n2 V0 r
1) ), \[Eta]m'[) p- c% {7 _' B7 o+ @) Y( l8 Z3 k
t] == \[CurlyPhi]m*" b" P/ F% z$ z6 C r! X
xm[t] - (\[CurlyPhi]m + gRate)*\[Eta]m[t], \[Eta]s'[7 A9 E% e4 y7 }* \
t] == \[CurlyPhi]s*xs[t]/ps - (\[CurlyPhi]s + gRate)*\[Eta]s[t], # l6 B3 j" f3 M2 s
K'[t] == gRate*K[t], hm[t] == \[Eta]m[t]*K[t],
3 W/ Z$ n' N/ B* Q5 I* o, m Z hs[t] == \[Eta]s[t]*K[t],
+ u5 z! j. B3 r0 U Sa[t] == (\[Gamma]a^\[Epsilon]*(pa)^(1 - \[Epsilon]))/(\[Gamma]a^\
7 j3 D) W( w( Q" t% i* N7 P; u \[Epsilon]*pa^(1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*
8 B9 J6 s/ R7 h: s hm[t]^(\[Theta]m*(1 - \[Epsilon])) + \[Gamma]s^\[Epsilon]*(ps*: Q$ E+ N1 e2 q4 ~5 i3 w
hs[t]^\[Theta]s)^(1 - \[Epsilon])) + (\[Gamma]m^\[Epsilon]*5 w" j: i( Y4 b# I
hm[t]^(\[Theta]m*(1 - \[Epsilon]))*pa*
. v T# {) b) w% M; L( u# g7 O, m cap)/((\[Gamma]a^\[Epsilon]*pa^(
9 M8 J( S0 n& W& I: I 1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*
6 x8 f* w8 P0 L: A- B/ l4 ? h hm[t]^(\[Theta]m*(1 - \[Epsilon])) + \0 T6 u/ X+ i- @! w
\[Gamma]s^\[Epsilon]*(ps*hs[t]^\[Theta]s)^(1 - \[Epsilon]))*k (t)*5 P7 j/ @. Q9 w0 m6 }/ e/ h
xm (t)),
# O& C" }' ?( ]# j7 o$ d Sm[t] == (\[Gamma]m^\[Epsilon]*5 J! ?( R8 ?3 F
hm[t]^(\[Theta]m*(1 - \[Epsilon])))/(\[Gamma]a^\[Epsilon]*pa^(
1 ]" w: U8 w- R$ Y" q1 T6 ]+ E 1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*8 C) C: S) I: M- B0 K9 R, G: G
hm[t]^(\[Theta]m*(1 - \[Epsilon])) + \[Gamma]s^\[Epsilon]*(ps*
( B. O3 d2 g; k% J hs[t]^\[Theta]s)^(1 - \[Epsilon])),
3 j2 ?7 @9 J% I1 p0 \* O! { Ss[t] == (\[Gamma]s^\[Epsilon]*(ps*% q/ C& z7 i8 ~# [
hs[t]^\[Theta]s)^(1 - \[Epsilon]))/(\[Gamma]a^\[Epsilon]*pa^(
3 V( t3 M0 L3 u 1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*; e7 ]+ u6 Q. y3 g" \- H6 X
hm[t]^(\[Theta]m*(1 - \[Epsilon])) + \[Gamma]s^\[Epsilon]*(ps*
$ m* [# F' i) b! s1 N8 O hs[t]^\[Theta]s)^(1 - \[Epsilon])) - (\[Gamma]m^\[Epsilon]*
: N5 i9 y9 ]# S4 _6 V% u5 e hm[t]^(\[Theta]m*(1 - \[Epsilon]))*ps*8 C- Z5 Z- Z& d1 L7 g
csp)/((\[Gamma]a^\[Epsilon]*pa^(" S* F) u0 N) W
1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*; W( P: \6 P) M) p; o0 T
hm[t]^(\[Theta]m*(1 - \[Epsilon])) + \
1 T j9 [% ^( P, p: E3 p( I \[Gamma]s^\[Epsilon]*(ps*hs[t]^\[Theta]s)^(1 - \[Epsilon]))*k (t)*
& G/ G, E5 k4 x( F# ?. m, v xm (t)), xm[0] == xm0, # J# ~7 H$ {8 s- K( k: Y
xs[0] == xs0, \[Eta]m[0] == \[Eta]m0, \[Eta]s[0] == \[Eta]s0,
2 A) P! \. ]/ q L& f, u K[0] == K0}, {xm, xs, \[Eta]m, \[Eta]s, K, hm, hs, Sa, Sm, Ss}, {t,
' }; e. H6 H! L& V& | 0, TT}]6 n" X+ j5 d9 [( C: p
Plot[{Evaluate[Sa[t] /. Sol], Evaluate[Sm[t] /. Sol], % C2 _8 A6 P& _0 f* v/ x5 ?+ I
Evaluate[Ss[t] /. Sol]}, {t, 0, TT}, AxesOrigin -> {0, 0}, ( X* W/ V, j. ]& |4 h$ O2 f
PlotRange -> {0., 0.8}, PlotStyle -> {Blue, Dashed, Dashing[{0.05}]}]
( n6 j7 l* q# s- i- }9 L, \9 E ~6 L0 w3 y Plot[{Evaluate[D*Sa[t] /. Sol], " j; G% i9 q3 R( {
Evaluate[(D*Sm[t] + (\[Alpha]*(gRate + \[Delta]))/(\[Rho] +
7 P8 l8 p" N0 c9 V6 z# X3 Z0 l- ~% Q gRate)) /. Sol], Evaluate[D*Ss[t] /. Sol]}, {t, 0, TT},
/ z8 U+ c; g; _' i" e, @ AxesOrigin -> {0, 0}, PlotRange -> {0., 0.8},
. d0 W x; }* V( d* o. c6 I PlotStyle -> {Blue, Dashed, Dashing[{0.05}]}]8 v1 s4 n! ~4 T% a" B7 _
/ H* Y$ n( Q- j
' Y9 V8 U, z2 E+ w3 ^2 r
5 K W5 _" l9 m3 p' b8 K
6 Y: z( R8 K! F2 v* Y Set::wrsym: Symbol D is Protected.
5 Q1 c! x: n" _' o & t7 E( R- K* K) K' Q l
NDSolve::deqn: Equation or list of equations expected instead of 0.96 (1-6.66667 xs[t]) xs[t] (-0.5 xm[t] (-1+xm[t]/\[Eta]m[t])+0.09 (-1+(2.5 xs[t])/\[Eta]s[t])) in the first argument {0.96 (1-6.66667 xs[t]) xs[t] (-0.5 xm[t] (-1+xm[t]/\[Eta]m[<<1>>])+0.09 (-1+(2.5 xs[t])/\[Eta]s[<<1>>])),<<13>>,K[0]==10.}.
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5 u5 D& c: i) l# a6 _- t
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