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Clear[Am, As, Aa, \[Alpha], \[Rho], \[Theta]m, \[Theta]s, \) a1 i. e- c+ O7 E
\[CurlyPhi]m, \[CurlyPhi]s, \[Epsilon]]6 `/ b7 N- X) J
\[Gamma]a = 0.1; \[Gamma]m = 0.15; \[Gamma]s = 7 b, ?: E. |$ C. E8 D" r$ [
1 - \[Gamma]a - \[Gamma]m;
, u" }: ~' J/ q1 C3 H( U\[Epsilon] = 0.04; \[Alpha] = 0.3; \[Rho] = 0.04;9 u3 Y! p" q3 }
\[Theta]m = 0.75; \[Theta]s = 0.9;! z2 N" D3 e" z8 Y
gRate = 0.02;
2 t. ]2 _( B% A. Z7 R" U {0 ]Am = (gRate + \[Rho])/\[Alpha]; Ba = 4; Bm = 1; Bs = 2.5;* b' I1 V9 \8 z8 X
ps = Bm/Bs; pa = Bm/Ba;
, D; b1 o" S- T- d; b\[Delta] = 0.03;
7 j7 u% R. V8 [B = \!\(TraditionalForm\`\*
" L9 x2 G+ t, ~& B0 ZFractionBox[
/ \& T* c A x- CRowBox[{
0 \3 y$ }! k9 t. J) pRowBox[{
3 w. ? x$ J3 ]. {" E- D, gRowBox[{6 _" L5 E' q0 [1 u6 T
StyleBox["(",' S- N( ~$ O7 H& `' `8 {+ N
SpanMinSize->1.,
$ Z# O9 z# r' L% eSpanMaxSize->1.], ; e2 q5 K- `/ Q) E
RowBox[{"1", "\[Minus]", "\[Alpha]"}], . O! u- c5 V/ J
StyleBox[")",
! j, I( t! t& O- R6 e0 sSpanMinSize->1.,
/ T* T- s1 c+ W# pSpanMaxSize->1.]}], "gRate"}], "+", "\[Rho]"}], ) M* ?. n9 @& E5 F0 g' k
"\[Alpha]"] \[Minus] \[Delta]\);: J2 O: s i5 B$ G$ S
cap = 10;1 \/ K7 x: _8 K% _, c' P i5 g
csp = (pa*cap)/ps;
0 u6 v; F) T h8 D+ dD = ((1 \[Minus] \[Alpha])*
4 m9 h* w7 t5 D! w gRate + \[Rho] - \[Alpha]*\[Delta])/(\[Rho] + gRate);9 c, w2 w" {- e4 m+ U& U) B
\[CurlyPhi]m = 0.1; \[CurlyPhi]s = 0.1;
4 a/ M& \7 P$ {. iPrint["*** Initial Values ***"]
4 q4 t# `/ L5 j2 L9 GE0 = 1.5;
% N: h9 ^, V. |& x# Y: m$ \K0 = E0/B;8 L2 B6 V% U9 Z5 K% C; @; n
hm0 = 0.25; hs0 = 0.25;(* initial values *)4 g: v0 u5 m B T1 ?+ X0 y. j
\[Eta]m0 = hm0/K0; \[Eta]s0 = hs0/K0;) g. f9 z! j! e! D5 ?3 o+ X
xm0 = (B*\[Gamma]m^\[Epsilon]*2 {5 S6 c5 j$ |) \# o+ t8 [# l) R4 i
hm0^(\[Theta]m*(1 - \[Epsilon])))/(\[Gamma]a^\[Epsilon]*pa^(
3 h! F" Y; B1 z* ^' a 1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*
k. Z- p' p# y$ u' o; ^ hm0^(\[Theta]m*(1 - \[Epsilon])) + \[Gamma]s^\[Epsilon]*(ps*
% G/ x% D# u3 k [2 d8 ] hs0^\[Theta]s)^(1 - \[Epsilon]));
! ~, I8 P' r& ?, u% _xs0 = (B*\[Gamma]s^\[Epsilon]*(ps*
9 A- S% s0 @ N7 Z5 r) V1 `, E. G hs0^\[Theta]s)^(1 - \[Epsilon]))/(\[Gamma]a^\[Epsilon]*pa^(
: R4 M" q4 J6 ] 1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*# D+ }- B% F% B$ s% n3 v1 K
hm0^(\[Theta]m*(1 - \[Epsilon])) + \[Gamma]s^\[Epsilon]*(ps*
# |6 Z( A* S" y$ y* { hs0^\[Theta]s)^(1 - \[Epsilon]));
8 {9 i2 \ [, YPrint["\[Eta]_{m,0}=" <> ToString[\[Eta]m0],
3 j4 ~ C/ t0 \3 \ ", \[Eta]_{s,0}=" <> ToString[\[Eta]s0],
, g& e0 u' d; v N5 _! a ", x_{m,0}=" <> ToString[xm0], ", x_{s,0}=" <> ToString[xs0]]
( C: t* ]5 @" ?6 b; o+ wTT = 100;(* end time *)% p$ |# ], b- `9 m L
(* Solve differential equations *)
/ B; M! w6 B, I: R5 ~7 v3 |Sol = NDSolve[{xs'[t] = (1 - \[Epsilon])*
3 L$ h9 F+ ^' M( v3 @7 G9 X; J$ W xs[t]*( (1 - xs[t]/
0 `2 _% |& M4 \2 n B)*(\[Theta]s*\[CurlyPhi]s*(xs[t]/(ps*\[Eta]s[t]) - 1) -
/ O( f- q0 c- ]/ u9 L! x' e xm[t]/B \[Theta]m*\[CurlyPhi]m*(xm[t]/\[Eta]m[t] - 1))),
, p; M' I6 b5 M9 N! q% Q xm'[t] == (1 - \[Epsilon])*6 ~% X0 n, l3 N' O& j- K
xm[t]*( (1 - xm[t]/' S, A$ ^3 ?' e% L+ ~/ f) u0 g/ ]
B)*\[Theta]m*\[CurlyPhi]m*(xm[t]/\[Eta]m[t] - 1) -
, v# n8 k! l r' W$ v. V xs[t]/B*\[Theta]s*\[CurlyPhi]s*(xs[t]/(ps*\[Eta]s[t]) -
7 ?- n5 r" c# b" w1 a 1) ), \[Eta]m'[9 ~2 z, k: s6 O+ C7 b( W9 p
t] == \[CurlyPhi]m*+ B% B4 S7 k1 x5 R8 @
xm[t] - (\[CurlyPhi]m + gRate)*\[Eta]m[t], \[Eta]s'[* c; W2 k/ n. f% f+ d" ]
t] == \[CurlyPhi]s*xs[t]/ps - (\[CurlyPhi]s + gRate)*\[Eta]s[t],
2 b) k j# i) n# O K'[t] == gRate*K[t], hm[t] == \[Eta]m[t]*K[t], 6 ?: }. |/ g8 r5 |7 B7 q8 Z
hs[t] == \[Eta]s[t]*K[t], ; u I" W7 w2 w) _& @& |3 v, e
Sa[t] == (\[Gamma]a^\[Epsilon]*(pa)^(1 - \[Epsilon]))/(\[Gamma]a^\
2 P+ A/ [+ P5 I) Y\[Epsilon]*pa^(1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*
4 v, L" l' V. ^ hm[t]^(\[Theta]m*(1 - \[Epsilon])) + \[Gamma]s^\[Epsilon]*(ps*% r3 p Z. |/ D$ m" d$ G
hs[t]^\[Theta]s)^(1 - \[Epsilon])) + (\[Gamma]m^\[Epsilon]*" n5 {; P5 Z) [3 G/ T9 A
hm[t]^(\[Theta]m*(1 - \[Epsilon]))*pa*
' l6 H7 `8 y' X% q cap)/((\[Gamma]a^\[Epsilon]*pa^(& k3 T/ I" ]5 L; M |
1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]* O. E# Q$ C2 u8 E. ]' o$ [
hm[t]^(\[Theta]m*(1 - \[Epsilon])) + \8 D s) Y: e" q8 J s2 a
\[Gamma]s^\[Epsilon]*(ps*hs[t]^\[Theta]s)^(1 - \[Epsilon]))*k (t)** `' [3 D5 t9 j% j" j& e& h
xm (t)),
% m$ f/ ^: z! j/ Z" Y) }2 | Sm[t] == (\[Gamma]m^\[Epsilon]*
+ v8 {0 x8 v* x `' }; v' {9 B hm[t]^(\[Theta]m*(1 - \[Epsilon])))/(\[Gamma]a^\[Epsilon]*pa^(
% ~/ t" S2 d7 V& C 1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*
" y# S( Q: I$ x) f- u! Z hm[t]^(\[Theta]m*(1 - \[Epsilon])) + \[Gamma]s^\[Epsilon]*(ps*
- `5 U" y/ I) g hs[t]^\[Theta]s)^(1 - \[Epsilon])),
, L! z8 ~& b1 {: x: L, @ Ss[t] == (\[Gamma]s^\[Epsilon]*(ps*
$ P9 N; u: g' y* n0 Z hs[t]^\[Theta]s)^(1 - \[Epsilon]))/(\[Gamma]a^\[Epsilon]*pa^(; I; D( n# S3 B5 Q% O( J, n! d! W; J
1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*! A$ q! {/ Y& L* V* G
hm[t]^(\[Theta]m*(1 - \[Epsilon])) + \[Gamma]s^\[Epsilon]*(ps*3 X* K) K' D9 g; u# [
hs[t]^\[Theta]s)^(1 - \[Epsilon])) - (\[Gamma]m^\[Epsilon]*. n/ w% e( m# u- V5 z3 C
hm[t]^(\[Theta]m*(1 - \[Epsilon]))*ps*; K: y- g0 q" w/ k$ p3 O; X% {
csp)/((\[Gamma]a^\[Epsilon]*pa^(- N$ D& O8 q; ?3 z+ H' R1 {% X' Q3 O
1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*
3 ?9 ^5 V/ h. x2 U hm[t]^(\[Theta]m*(1 - \[Epsilon])) + \0 ~! q& P2 K1 r7 A& n* ^
\[Gamma]s^\[Epsilon]*(ps*hs[t]^\[Theta]s)^(1 - \[Epsilon]))*k (t)*" Y) \$ ]# @, g/ l$ g1 x' j
xm (t)), xm[0] == xm0, ( j* n( o& i, r
xs[0] == xs0, \[Eta]m[0] == \[Eta]m0, \[Eta]s[0] == \[Eta]s0,
8 R! }8 N* I A% o% s, N K[0] == K0}, {xm, xs, \[Eta]m, \[Eta]s, K, hm, hs, Sa, Sm, Ss}, {t,
5 j$ \+ R0 M; f8 k! X' Z& N3 V 0, TT}]( I* u" q' o- U8 i- V7 t
Plot[{Evaluate[Sa[t] /. Sol], Evaluate[Sm[t] /. Sol], E) w1 Z1 [! S- t- x
Evaluate[Ss[t] /. Sol]}, {t, 0, TT}, AxesOrigin -> {0, 0},
7 L2 j5 W5 S1 U1 ?- j( D PlotRange -> {0., 0.8}, PlotStyle -> {Blue, Dashed, Dashing[{0.05}]}]6 L5 T7 {. S% w2 B
Plot[{Evaluate[D*Sa[t] /. Sol], e, L+ F7 X. |, I
Evaluate[(D*Sm[t] + (\[Alpha]*(gRate + \[Delta]))/(\[Rho] +
% O2 }+ Y! k7 c4 W6 V. J gRate)) /. Sol], Evaluate[D*Ss[t] /. Sol]}, {t, 0, TT}, - o2 n& D! s6 x% C. B( [
AxesOrigin -> {0, 0}, PlotRange -> {0., 0.8},
7 W! M) f( M- a PlotStyle -> {Blue, Dashed, Dashing[{0.05}]}]/ f5 z2 Y" W1 G; m! g7 V2 U
' Y& i2 I) J& y1 [ B# ^4 m- q
* Q7 o x8 i' j) U9 U/ x5 k1 L1 d2 w5 ^% r( h* G& O- @; T v
3 w6 z7 n. S# L- m8 t: b7 }: ISet::wrsym: Symbol D is Protected.
& |4 J: R9 `$ H8 F
: Y0 p# I8 `/ `# C" |7 r* PNDSolve::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.}.
& ?6 C$ E$ H) Z- }1 W) x, v4 N; d" w, x( y8 U8 ~
9 n- N9 |! Y9 s
; k# R& D3 O% d6 J f; I- l8 P
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