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Clear[Am, As, Aa, \[Alpha], \[Rho], \[Theta]m, \[Theta]s, \3 Y: b1 i Z& Q0 ~! o# ]
\[CurlyPhi]m, \[CurlyPhi]s, \[Epsilon]]
& l6 b; y! ?1 d. [/ V\[Gamma]a = 0.1; \[Gamma]m = 0.15; \[Gamma]s =
* |/ |' B( W6 n% ]4 b4 U 1 - \[Gamma]a - \[Gamma]m;
) C% g/ D4 ^' E' `3 ]\[Epsilon] = 0.04; \[Alpha] = 0.3; \[Rho] = 0.04;
0 G! t' ^$ q* _- s# \0 B) q\[Theta]m = 0.75; \[Theta]s = 0.9;7 E1 O9 f6 N( M9 }6 a
gRate = 0.02;. ?1 ]6 b" g0 }4 H! I8 O) }
Am = (gRate + \[Rho])/\[Alpha]; Ba = 4; Bm = 1; Bs = 2.5;5 l2 \5 Y" A+ ` r/ M+ j
ps = Bm/Bs; pa = Bm/Ba;, r" I8 y( {$ E1 V3 n3 a- W
\[Delta] = 0.03;$ H1 |( B2 o$ x2 y' D3 ]( w$ V
B = \!\(TraditionalForm\`\*
8 _3 w. k" Y" q) q/ z: k) D, jFractionBox[
: s$ u8 r4 u; pRowBox[{
& E! }0 l; o+ }. eRowBox[{6 w8 d8 x+ V8 y. F) a
RowBox[{: r1 {+ @, B7 K: L/ I
StyleBox["(",
9 H( m9 O1 O1 |! a! B' h, j1 vSpanMinSize->1.,
" S, U0 \4 s4 p9 O1 {8 t0 t! MSpanMaxSize->1.], ( t6 T! b+ }1 a# z( c o
RowBox[{"1", "\[Minus]", "\[Alpha]"}],
# J9 b( Z3 }$ f) O- ^StyleBox[")",
+ p* W+ N2 E M$ Q8 c" Y3 ESpanMinSize->1.,
. z, l6 t% k3 K. ZSpanMaxSize->1.]}], "gRate"}], "+", "\[Rho]"}], 5 y/ E' Y8 g# U4 T0 F3 q) f
"\[Alpha]"] \[Minus] \[Delta]\);
$ N, g) K5 x' w5 w! e# o W v3 qcap = 10;
# e; t. l5 X( ~0 X- a& Q+ rcsp = (pa*cap)/ps;
9 y/ x5 Q1 A' Q/ l$ Y$ ]D = ((1 \[Minus] \[Alpha])*
# H2 T: c2 J/ c6 H gRate + \[Rho] - \[Alpha]*\[Delta])/(\[Rho] + gRate);
& f# j0 D5 h6 D$ K; L. ?5 M5 ^\[CurlyPhi]m = 0.1; \[CurlyPhi]s = 0.1;. G7 E8 M3 m8 {7 ~3 h7 R: w) A
Print["*** Initial Values ***"]
+ q$ H. i" m; E' ^E0 = 1.5;
5 j( r4 }( v; hK0 = E0/B;
/ `: m. x- w& j' ~4 Jhm0 = 0.25; hs0 = 0.25;(* initial values *)) E) t, t. \% w' ]% v% ^% X
\[Eta]m0 = hm0/K0; \[Eta]s0 = hs0/K0;6 S& m+ `. e2 a3 o
xm0 = (B*\[Gamma]m^\[Epsilon]*+ s2 f0 m! s6 o# F& Q {$ W
hm0^(\[Theta]m*(1 - \[Epsilon])))/(\[Gamma]a^\[Epsilon]*pa^(
7 Z; B& w0 J( S 1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*6 f0 C `# s, I
hm0^(\[Theta]m*(1 - \[Epsilon])) + \[Gamma]s^\[Epsilon]*(ps*
5 q" y/ [# f' N7 c8 q8 O6 F, S w hs0^\[Theta]s)^(1 - \[Epsilon]));/ y. S2 E$ U6 h' j2 @
xs0 = (B*\[Gamma]s^\[Epsilon]*(ps*
; A# \3 E, k4 M& ` hs0^\[Theta]s)^(1 - \[Epsilon]))/(\[Gamma]a^\[Epsilon]*pa^(* y" X9 o+ Z _' U' K9 H% z9 K4 i
1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*
8 e) W0 h% z- @ hm0^(\[Theta]m*(1 - \[Epsilon])) + \[Gamma]s^\[Epsilon]*(ps*, P9 E+ Z( B& T* `/ P# q$ I
hs0^\[Theta]s)^(1 - \[Epsilon]));
9 K$ R9 M8 ^0 c. \Print["\[Eta]_{m,0}=" <> ToString[\[Eta]m0],
/ ]3 s- d8 M+ W- g& B ", \[Eta]_{s,0}=" <> ToString[\[Eta]s0], j) f2 X! ]# Q- H! O
", x_{m,0}=" <> ToString[xm0], ", x_{s,0}=" <> ToString[xs0]]+ k8 }% C+ A! k; X5 p! G4 ]
TT = 100;(* end time *)
$ m+ F- y* P6 U) z( H) B5 W: F( f(* Solve differential equations *)' h7 b0 e2 z. i3 D
Sol = NDSolve[{xs'[t] = (1 - \[Epsilon])*( V4 X. z$ J+ y0 L Q4 Z* e/ V
xs[t]*( (1 - xs[t]/- c& b0 c5 `( o' x8 B
B)*(\[Theta]s*\[CurlyPhi]s*(xs[t]/(ps*\[Eta]s[t]) - 1) - 5 u; R7 e6 k8 S. v% M
xm[t]/B \[Theta]m*\[CurlyPhi]m*(xm[t]/\[Eta]m[t] - 1))),
+ f) o7 w0 J% b6 E9 ~; j! B xm'[t] == (1 - \[Epsilon])*1 E$ Z$ j6 i/ U9 _2 Y' } y
xm[t]*( (1 - xm[t]/
' n. M0 D1 K( J$ Q. z B)*\[Theta]m*\[CurlyPhi]m*(xm[t]/\[Eta]m[t] - 1) - & j' x' a7 G( M7 f) S3 ^
xs[t]/B*\[Theta]s*\[CurlyPhi]s*(xs[t]/(ps*\[Eta]s[t]) -
; |; r4 S! S# ]9 v 1) ), \[Eta]m'[
! `' k$ v) }; n3 m b; O9 c t] == \[CurlyPhi]m*
" U/ m4 P0 R9 F" J xm[t] - (\[CurlyPhi]m + gRate)*\[Eta]m[t], \[Eta]s'[
8 |$ D0 E: o+ ~3 o* { t] == \[CurlyPhi]s*xs[t]/ps - (\[CurlyPhi]s + gRate)*\[Eta]s[t],
1 z) |0 ~+ b4 B6 I K'[t] == gRate*K[t], hm[t] == \[Eta]m[t]*K[t],
$ ~" }( h$ u4 P& R( J: y3 p6 D hs[t] == \[Eta]s[t]*K[t], $ E# z% r" u5 N! B
Sa[t] == (\[Gamma]a^\[Epsilon]*(pa)^(1 - \[Epsilon]))/(\[Gamma]a^\ [* b$ K* O' P; }# i8 D
\[Epsilon]*pa^(1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*
7 \, v7 ]+ a0 ] hm[t]^(\[Theta]m*(1 - \[Epsilon])) + \[Gamma]s^\[Epsilon]*(ps*1 D1 w4 {: A& u" @; e
hs[t]^\[Theta]s)^(1 - \[Epsilon])) + (\[Gamma]m^\[Epsilon]*' h6 ]8 k' z3 r
hm[t]^(\[Theta]m*(1 - \[Epsilon]))*pa*
, M$ ~: \0 [. `2 J; Y( I( }5 { cap)/((\[Gamma]a^\[Epsilon]*pa^(
9 p0 F! T) U1 a) X 1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*) ?3 q4 l6 j. e& p: V8 O$ j q
hm[t]^(\[Theta]m*(1 - \[Epsilon])) + \
6 k& I! u; O9 S6 ]7 z8 C( q/ E, K\[Gamma]s^\[Epsilon]*(ps*hs[t]^\[Theta]s)^(1 - \[Epsilon]))*k (t)*
: s8 P. q) Y5 k$ `. _" W) k' D xm (t)),
4 x, p! O) t" S D0 y. d" S7 s1 ]! Z. }2 E Sm[t] == (\[Gamma]m^\[Epsilon]*4 ?/ [* _' m* t/ u A( W
hm[t]^(\[Theta]m*(1 - \[Epsilon])))/(\[Gamma]a^\[Epsilon]*pa^(
* ?( V" e( s# P" T: f# q h 1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*8 k9 v8 K7 _" l+ r3 ]3 X
hm[t]^(\[Theta]m*(1 - \[Epsilon])) + \[Gamma]s^\[Epsilon]*(ps*
. H/ t+ @* y9 _# Y7 r7 s' N hs[t]^\[Theta]s)^(1 - \[Epsilon])),
0 _) C7 _: Q# `. F1 q7 s Ss[t] == (\[Gamma]s^\[Epsilon]*(ps*$ t9 ]: ?$ u7 C
hs[t]^\[Theta]s)^(1 - \[Epsilon]))/(\[Gamma]a^\[Epsilon]*pa^(
( m2 {. n$ p7 Y) N6 P( j; C 1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*
' r2 K, v0 ?5 a9 ` hm[t]^(\[Theta]m*(1 - \[Epsilon])) + \[Gamma]s^\[Epsilon]*(ps*# y- x& t' _+ W
hs[t]^\[Theta]s)^(1 - \[Epsilon])) - (\[Gamma]m^\[Epsilon]*
, O( J' W8 H' A6 I hm[t]^(\[Theta]m*(1 - \[Epsilon]))*ps*
1 j; V+ }. {! ^" I9 w6 r csp)/((\[Gamma]a^\[Epsilon]*pa^(: }& |: @; L" D+ O- e
1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*, [! G. W) z# h- b
hm[t]^(\[Theta]m*(1 - \[Epsilon])) + \' T8 z) i" F9 D" O0 Q0 d- n$ l* ~
\[Gamma]s^\[Epsilon]*(ps*hs[t]^\[Theta]s)^(1 - \[Epsilon]))*k (t)*+ z+ H5 E) v1 m5 v) a1 }
xm (t)), xm[0] == xm0, ( w2 `5 e- N; ], t; w# z u) _
xs[0] == xs0, \[Eta]m[0] == \[Eta]m0, \[Eta]s[0] == \[Eta]s0,
% `+ Z: u% r5 q& K K[0] == K0}, {xm, xs, \[Eta]m, \[Eta]s, K, hm, hs, Sa, Sm, Ss}, {t,
! A. b! B, t, k6 ]) N 0, TT}]9 p; x# x1 @2 ?5 j1 `% \
Plot[{Evaluate[Sa[t] /. Sol], Evaluate[Sm[t] /. Sol],
" a- k3 j! H8 O9 w5 b9 {0 D" W Evaluate[Ss[t] /. Sol]}, {t, 0, TT}, AxesOrigin -> {0, 0}, ( A W, c* I, W
PlotRange -> {0., 0.8}, PlotStyle -> {Blue, Dashed, Dashing[{0.05}]}]
* ^( j0 {9 @* g$ ~Plot[{Evaluate[D*Sa[t] /. Sol], 1 _" l% g/ o) q8 p
Evaluate[(D*Sm[t] + (\[Alpha]*(gRate + \[Delta]))/(\[Rho] + : e9 I% v3 d& k) k% R2 `1 H/ F
gRate)) /. Sol], Evaluate[D*Ss[t] /. Sol]}, {t, 0, TT}, " p+ D* j: u3 b5 Z
AxesOrigin -> {0, 0}, PlotRange -> {0., 0.8},
0 O- d7 [7 u0 f. r7 k; q PlotStyle -> {Blue, Dashed, Dashing[{0.05}]}]8 I9 B+ W* b$ I8 w1 Q
) ~5 m1 V. J$ P, B
5 i- O8 |; i8 M4 }# N
2 i m k% t& G
~. @- `) [7 E% i8 ~8 v2 l
Set::wrsym: Symbol D is Protected.0 q; H" b' ?1 y: v+ U/ M# n
\0 t8 F/ l( J7 f0 j& @: }) DNDSolve::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.}.% p# E* ?& Z( |; n2 a
/ E1 H3 w+ {7 L- G( F5 z! ^
: J4 ]8 Q6 L' h* w% U% o3 E- a' ?, H+ n3 G& ]7 P% F9 S9 R+ L
O8 c$ d! `1 Z/ b+ F/ [ |
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