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Clear[Am, As, Aa, \[Alpha], \[Rho], \[Theta]m, \[Theta]s, \
3 S; z5 c# r |4 @+ D8 j\[CurlyPhi]m, \[CurlyPhi]s, \[Epsilon]]
; u) B' F8 Z8 f% V0 b9 ?\[Gamma]a = 0.1; \[Gamma]m = 0.15; \[Gamma]s =
- b3 e7 B. N4 h# B, Q C% G 1 - \[Gamma]a - \[Gamma]m;( O/ U" P0 O. R" a% M/ w* h
\[Epsilon] = 0.04; \[Alpha] = 0.3; \[Rho] = 0.04;
g: Z, q0 k$ i' p\[Theta]m = 0.75; \[Theta]s = 0.9;; v" t2 e. N( D( k3 k3 n
gRate = 0.02;2 X& M8 L0 ]' {
Am = (gRate + \[Rho])/\[Alpha]; Ba = 4; Bm = 1; Bs = 2.5;
. v- Z0 |, R M: x7 D7 A& L3 ]ps = Bm/Bs; pa = Bm/Ba;/ p0 c! ?1 s, ?# r
\[Delta] = 0.03;5 r- a2 T& H% K0 u! s9 l' Q, R6 ?
B = \!\(TraditionalForm\`\*
2 Z) y4 W2 @3 k/ P1 _, S) W0 J' AFractionBox[
7 J! c! ~9 g. E! ERowBox[{
6 `0 d, s" z/ K/ u' w* LRowBox[{
& D: x% M9 e( w) ?+ ]RowBox[{: n) r& [5 `# b4 ^- r
StyleBox["(",7 l7 g; b) }6 M9 |/ L5 e
SpanMinSize->1.,2 P7 y- l! C2 b* {# Y3 k# g
SpanMaxSize->1.],
, N2 o& Y! c6 N: k/ uRowBox[{"1", "\[Minus]", "\[Alpha]"}],
- T/ L7 W7 }8 }( j& ^) J3 a, @StyleBox[")",: l2 r: i2 ~; g8 X
SpanMinSize->1.,/ o( {+ X' l6 W! ^5 L4 q, `5 r; p
SpanMaxSize->1.]}], "gRate"}], "+", "\[Rho]"}], 0 R( R6 p8 s5 T2 B5 O
"\[Alpha]"] \[Minus] \[Delta]\);
' e5 @% p. `' X9 wcap = 10;6 p1 u) u2 O. `$ p* }
csp = (pa*cap)/ps;
- C2 r5 F0 g% T0 T7 O! lD = ((1 \[Minus] \[Alpha])*" r, U7 H: {) i1 z: @: o
gRate + \[Rho] - \[Alpha]*\[Delta])/(\[Rho] + gRate);
6 D" S/ o* ~2 f\[CurlyPhi]m = 0.1; \[CurlyPhi]s = 0.1;
4 s- B# z9 [7 o, c( B8 HPrint["*** Initial Values ***"]' s5 S$ e: R4 \
E0 = 1.5;" s0 @2 W$ Y0 ]! X
K0 = E0/B;6 y: `( J# w, @/ G u @- N, C' Z7 M
hm0 = 0.25; hs0 = 0.25;(* initial values *)
5 j G; t6 {; O( @" y4 M) r* O' C\[Eta]m0 = hm0/K0; \[Eta]s0 = hs0/K0;
/ H* |3 C( j0 I3 ]4 V, z) N( Txm0 = (B*\[Gamma]m^\[Epsilon]*
/ ^) ?: a% E u$ }5 Q% W, a hm0^(\[Theta]m*(1 - \[Epsilon])))/(\[Gamma]a^\[Epsilon]*pa^(# {$ M K, F h. v- l6 S+ A
1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*
/ N7 ~/ i2 a3 D hm0^(\[Theta]m*(1 - \[Epsilon])) + \[Gamma]s^\[Epsilon]*(ps*
) r& L- b* P4 d, W1 E; m. l hs0^\[Theta]s)^(1 - \[Epsilon]));( k1 T9 F- h9 t$ ~5 j8 ]! @
xs0 = (B*\[Gamma]s^\[Epsilon]*(ps*$ ^/ S- ~- P& Z7 _4 H
hs0^\[Theta]s)^(1 - \[Epsilon]))/(\[Gamma]a^\[Epsilon]*pa^(
% r$ ]$ X8 I1 l$ J Q 1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*
% ^3 b, s. G- a' B5 T9 M hm0^(\[Theta]m*(1 - \[Epsilon])) + \[Gamma]s^\[Epsilon]*(ps*
* a* F! I9 y% ?. o hs0^\[Theta]s)^(1 - \[Epsilon]));: [, p. G. j3 X! o# j6 z5 |
Print["\[Eta]_{m,0}=" <> ToString[\[Eta]m0],
1 S$ @# }+ t' R- D$ P: S ", \[Eta]_{s,0}=" <> ToString[\[Eta]s0], ; C) @% z$ K+ T% k- N
", x_{m,0}=" <> ToString[xm0], ", x_{s,0}=" <> ToString[xs0]]) b r a$ V: Z% s0 ?
TT = 100;(* end time *)
+ V1 g$ l6 E, O' F, D o9 F1 t(* Solve differential equations *)( s0 ]' a P4 H( \5 c
Sol = NDSolve[{xs'[t] = (1 - \[Epsilon])*
- v( e' }7 G# F, _. p xs[t]*( (1 - xs[t]/% a5 V' x& }0 W- s5 n2 l
B)*(\[Theta]s*\[CurlyPhi]s*(xs[t]/(ps*\[Eta]s[t]) - 1) -
; Z# a! C* g9 @& b4 k* { a xm[t]/B \[Theta]m*\[CurlyPhi]m*(xm[t]/\[Eta]m[t] - 1))), ) U0 @7 b7 Y8 {+ z; D/ v2 R. V
xm'[t] == (1 - \[Epsilon])*
" I4 D( j3 F6 n( h xm[t]*( (1 - xm[t]/
, T% \8 A# B$ R1 j2 O( T B)*\[Theta]m*\[CurlyPhi]m*(xm[t]/\[Eta]m[t] - 1) -
# K( {( R; C% @ xs[t]/B*\[Theta]s*\[CurlyPhi]s*(xs[t]/(ps*\[Eta]s[t]) -
E' }9 {6 q9 h, O* Z 1) ), \[Eta]m'[5 \4 H# m w) d- N8 A6 I- s
t] == \[CurlyPhi]m*7 L, |, j5 F8 y) y6 u: {
xm[t] - (\[CurlyPhi]m + gRate)*\[Eta]m[t], \[Eta]s'[0 M# U; f! T1 P9 m( ?
t] == \[CurlyPhi]s*xs[t]/ps - (\[CurlyPhi]s + gRate)*\[Eta]s[t], 7 [% i7 l. ]8 `7 p6 A B
K'[t] == gRate*K[t], hm[t] == \[Eta]m[t]*K[t],
& a" x' y4 S( Q4 q. Y8 i6 Z9 K hs[t] == \[Eta]s[t]*K[t], 7 Z+ C' f# a3 v" m
Sa[t] == (\[Gamma]a^\[Epsilon]*(pa)^(1 - \[Epsilon]))/(\[Gamma]a^\3 Z. E/ X. Y9 s( R
\[Epsilon]*pa^(1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*
e5 _* z4 O; s/ l( D8 N6 b3 P q hm[t]^(\[Theta]m*(1 - \[Epsilon])) + \[Gamma]s^\[Epsilon]*(ps*
) W, Y* X9 t; P# W- _' K hs[t]^\[Theta]s)^(1 - \[Epsilon])) + (\[Gamma]m^\[Epsilon]*3 f. n5 f ?% v6 v2 M: ~
hm[t]^(\[Theta]m*(1 - \[Epsilon]))*pa*
4 c$ E1 f( w) n5 y8 z' [) ? cap)/((\[Gamma]a^\[Epsilon]*pa^(. g e. @! r- l8 G
1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*
8 J2 T* _. X; K3 N" z) D, L# b hm[t]^(\[Theta]m*(1 - \[Epsilon])) + \
/ t( _% |+ m, p# L\[Gamma]s^\[Epsilon]*(ps*hs[t]^\[Theta]s)^(1 - \[Epsilon]))*k (t)*
+ e3 G) P) C0 L% P. a) X xm (t)),
, q: p/ X3 T0 T. J Q! C Sm[t] == (\[Gamma]m^\[Epsilon]*
7 v4 P" N& U1 n0 f+ X hm[t]^(\[Theta]m*(1 - \[Epsilon])))/(\[Gamma]a^\[Epsilon]*pa^(( K# H3 }, y4 f, I
1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*
. v& l; r8 z9 u( P0 M6 J' H hm[t]^(\[Theta]m*(1 - \[Epsilon])) + \[Gamma]s^\[Epsilon]*(ps* s" o' G3 v- ]0 ]. B5 h
hs[t]^\[Theta]s)^(1 - \[Epsilon])),
/ `+ s/ W4 G, m9 b. `7 N Ss[t] == (\[Gamma]s^\[Epsilon]*(ps*
6 }( G3 d; [8 |- e& @! C, t! ~" { hs[t]^\[Theta]s)^(1 - \[Epsilon]))/(\[Gamma]a^\[Epsilon]*pa^(
& u5 ]( W4 l' n& c, u0 [. W I* x 1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*8 z" ~6 m1 Q& \& ]8 a
hm[t]^(\[Theta]m*(1 - \[Epsilon])) + \[Gamma]s^\[Epsilon]*(ps*) G2 q+ U& `1 D1 ~
hs[t]^\[Theta]s)^(1 - \[Epsilon])) - (\[Gamma]m^\[Epsilon]*/ i- j% d1 Q+ V" V! ?) h. z
hm[t]^(\[Theta]m*(1 - \[Epsilon]))*ps*$ z$ b+ q, Y. S0 F2 P4 N3 J
csp)/((\[Gamma]a^\[Epsilon]*pa^(- t* e! l" h* B- s/ @6 o
1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*
6 S0 Z. Q0 x9 N6 N2 e* c hm[t]^(\[Theta]m*(1 - \[Epsilon])) + \/ l2 T5 h0 G8 A ]0 \# T7 s, C
\[Gamma]s^\[Epsilon]*(ps*hs[t]^\[Theta]s)^(1 - \[Epsilon]))*k (t)*
+ u( I) h& h' q1 T: f xm (t)), xm[0] == xm0, * }$ Y- [7 ~* n# j; E
xs[0] == xs0, \[Eta]m[0] == \[Eta]m0, \[Eta]s[0] == \[Eta]s0,
2 R- F7 [ W2 w6 D/ ]8 n/ o K[0] == K0}, {xm, xs, \[Eta]m, \[Eta]s, K, hm, hs, Sa, Sm, Ss}, {t, S: S; @; V2 g4 ]8 W
0, TT}]
3 R* O& e/ \: {Plot[{Evaluate[Sa[t] /. Sol], Evaluate[Sm[t] /. Sol], % @- {" m+ |- X B% Y/ ^
Evaluate[Ss[t] /. Sol]}, {t, 0, TT}, AxesOrigin -> {0, 0},
5 i7 m1 _& A$ Z' d. Y9 b" O3 D PlotRange -> {0., 0.8}, PlotStyle -> {Blue, Dashed, Dashing[{0.05}]}]3 @$ b9 a' D; e" N7 N7 a/ h
Plot[{Evaluate[D*Sa[t] /. Sol],
# J. |0 V @4 Y6 a Evaluate[(D*Sm[t] + (\[Alpha]*(gRate + \[Delta]))/(\[Rho] +
$ Z7 f+ F4 ]2 W7 S6 x# |1 J gRate)) /. Sol], Evaluate[D*Ss[t] /. Sol]}, {t, 0, TT}, 3 p+ |0 l+ A, p
AxesOrigin -> {0, 0}, PlotRange -> {0., 0.8}, ( N; j+ \9 t' \. k- p7 A9 a/ i
PlotStyle -> {Blue, Dashed, Dashing[{0.05}]}]4 c$ _9 a; g' g6 o" b7 M1 [/ D1 }
, O$ m% T2 {6 e5 ]: ~5 |, e) n/ v8 t$ b& s9 \( U2 Y- r
' M7 C! P# m, R/ O0 _8 X8 ]7 M3 K6 d2 H4 s3 C
Set::wrsym: Symbol D is Protected.
6 i* ?. [' |* q5 ~# b! P
8 { X. O+ @: d6 n, p$ oNDSolve::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.}.. i" C$ u7 e# f! l6 p+ l
8 {0 Z% W! V/ B$ l
; X. [9 U$ E' [9 `% q+ Q S/ b9 b- R4 S# ~- i* d1 q3 z; K
1 y h! y6 a. y: x# k6 i
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