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Clear[Am, As, Aa, \[Alpha], \[Rho], \[Theta]m, \[Theta]s, \
) t$ j: b1 ?% [( ^3 v\[CurlyPhi]m, \[CurlyPhi]s, \[Epsilon]]2 d6 H( y. u! Z) _; F" ~ v# t! V7 u
\[Gamma]a = 0.1; \[Gamma]m = 0.15; \[Gamma]s =
# w2 b$ o/ R) n) H u( ?2 ] 1 - \[Gamma]a - \[Gamma]m;/ N" s! \' I* _% C7 L F5 u
\[Epsilon] = 0.04; \[Alpha] = 0.3; \[Rho] = 0.04;' I6 e5 M2 d+ P p. |) U) U
\[Theta]m = 0.75; \[Theta]s = 0.9;
0 n! F) y: _7 ^; L E5 b, d3 G5 igRate = 0.02;
5 h6 N4 v( c( r7 ?6 Y4 ~Am = (gRate + \[Rho])/\[Alpha]; Ba = 4; Bm = 1; Bs = 2.5;+ B5 y0 M4 g- [6 F
ps = Bm/Bs; pa = Bm/Ba;
/ F( x, o6 O# m" O8 i* k\[Delta] = 0.03;' T) r2 X, U( q. y- A0 t( M7 \
B = \!\(TraditionalForm\`\*
4 z2 W1 ^2 ^+ f6 e7 F/ rFractionBox[
) l& ]9 A4 m5 s$ I) d1 v9 Y: T( tRowBox[{2 w( y9 O! z. D! q& b4 z0 W: n7 ^/ t( M
RowBox[{
- L* |6 s8 Z* u. ]% h/ n& L- N+ B5 RRowBox[{2 f7 Q ?4 R- H: _7 d8 N
StyleBox["(",
( V& U U: u& s4 M# q, RSpanMinSize->1.,
" }- A( ]& r1 H# P* @SpanMaxSize->1.],
# }* Z4 d6 M9 R/ d% ?: ?RowBox[{"1", "\[Minus]", "\[Alpha]"}], 7 J0 Y+ O0 a+ W ~% Z: l
StyleBox[")"," s( j+ p# `8 |/ q
SpanMinSize->1.," _8 ` }( E& y. X% z1 ~: x
SpanMaxSize->1.]}], "gRate"}], "+", "\[Rho]"}], $ ] }9 e; ~0 C: o, a- e4 g) J! l% i ]$ B% g
"\[Alpha]"] \[Minus] \[Delta]\);
. _5 E" M4 ~% T3 w9 H& z0 L2 N3 icap = 10; W4 C7 X" s5 D
csp = (pa*cap)/ps;
( p1 K) N- j% xD = ((1 \[Minus] \[Alpha])*! e6 l1 Q* V" ~, q9 @: p
gRate + \[Rho] - \[Alpha]*\[Delta])/(\[Rho] + gRate);
0 h+ @, l- ~6 [) O\[CurlyPhi]m = 0.1; \[CurlyPhi]s = 0.1;; B/ g) {* ~/ g. M( P5 {
Print["*** Initial Values ***"]6 U1 X5 H7 @. x, O9 K+ H& R
E0 = 1.5;
' u5 `% L% f5 h8 O, rK0 = E0/B;( \9 C7 }; {( w; A. r0 r, q
hm0 = 0.25; hs0 = 0.25;(* initial values *)
n4 E; a+ ]% \. j; @- _" B$ i\[Eta]m0 = hm0/K0; \[Eta]s0 = hs0/K0;9 E4 J9 q5 I- \3 F8 L, |0 O/ P
xm0 = (B*\[Gamma]m^\[Epsilon]*
+ {. A$ \8 c7 m' _0 x hm0^(\[Theta]m*(1 - \[Epsilon])))/(\[Gamma]a^\[Epsilon]*pa^(
) e1 E0 X! |2 [* p( U7 o 1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*
1 D' n' q2 l# l) J. q- S: ` hm0^(\[Theta]m*(1 - \[Epsilon])) + \[Gamma]s^\[Epsilon]*(ps*
9 L+ ~- n: J" E4 ?) g# ]6 k2 } hs0^\[Theta]s)^(1 - \[Epsilon]));5 c: V3 |- {1 G N; p- d
xs0 = (B*\[Gamma]s^\[Epsilon]*(ps*
9 S/ v$ r! `5 V hs0^\[Theta]s)^(1 - \[Epsilon]))/(\[Gamma]a^\[Epsilon]*pa^(* G% G5 |8 I/ l Y+ c: ]
1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*' ^1 ^1 ?3 \( n. _! W4 J
hm0^(\[Theta]m*(1 - \[Epsilon])) + \[Gamma]s^\[Epsilon]*(ps** x: n9 R7 R' M1 T( Z
hs0^\[Theta]s)^(1 - \[Epsilon]));) ]" E6 i* w( [' W9 f
Print["\[Eta]_{m,0}=" <> ToString[\[Eta]m0],
! X; V* P, u9 } ", \[Eta]_{s,0}=" <> ToString[\[Eta]s0],
, d" f1 i, E+ x% o- n0 i ", x_{m,0}=" <> ToString[xm0], ", x_{s,0}=" <> ToString[xs0]]& h$ R# [, s; E; H" I- K: Y
TT = 100;(* end time *)
2 X' Q0 Q* s7 m# b, E" i- D(* Solve differential equations *)
" H* o7 y$ `. X6 q! FSol = NDSolve[{xs'[t] = (1 - \[Epsilon])*( i9 B1 f6 t$ f
xs[t]*( (1 - xs[t]/
8 G, c; O: H6 H: m B)*(\[Theta]s*\[CurlyPhi]s*(xs[t]/(ps*\[Eta]s[t]) - 1) - # S* A& S# A5 f3 O
xm[t]/B \[Theta]m*\[CurlyPhi]m*(xm[t]/\[Eta]m[t] - 1))), ' I# {. r t( y- @4 s3 V, ~
xm'[t] == (1 - \[Epsilon])*8 h" z* K( K/ Y5 q# F
xm[t]*( (1 - xm[t]/
3 W2 m% {1 H' X4 ~ B)*\[Theta]m*\[CurlyPhi]m*(xm[t]/\[Eta]m[t] - 1) -
2 B& h( Q, e8 {2 @! g$ C! y/ W xs[t]/B*\[Theta]s*\[CurlyPhi]s*(xs[t]/(ps*\[Eta]s[t]) -
) z S+ o$ T9 y" u% i8 W 1) ), \[Eta]m'[7 Z+ T$ A% i, d# U' P& Q
t] == \[CurlyPhi]m*0 U8 V& Y+ e* W+ r4 E
xm[t] - (\[CurlyPhi]m + gRate)*\[Eta]m[t], \[Eta]s'[
9 P8 ?6 j0 J1 o3 Z2 s- i& B% r' x t] == \[CurlyPhi]s*xs[t]/ps - (\[CurlyPhi]s + gRate)*\[Eta]s[t], 5 r+ M3 `, z" Y
K'[t] == gRate*K[t], hm[t] == \[Eta]m[t]*K[t], 1 c9 I* I$ J* R+ V# v$ A; w4 W
hs[t] == \[Eta]s[t]*K[t],
6 q. @8 U+ U2 @! n3 g0 i' k* D Sa[t] == (\[Gamma]a^\[Epsilon]*(pa)^(1 - \[Epsilon]))/(\[Gamma]a^\
4 r- s5 u ?+ h$ ?\[Epsilon]*pa^(1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*
L6 q* c0 a/ {* ^3 G: M hm[t]^(\[Theta]m*(1 - \[Epsilon])) + \[Gamma]s^\[Epsilon]*(ps*
0 z. U- Q% n* x9 h$ h4 t& E A9 Y hs[t]^\[Theta]s)^(1 - \[Epsilon])) + (\[Gamma]m^\[Epsilon]*( U( J, C+ p% B# F
hm[t]^(\[Theta]m*(1 - \[Epsilon]))*pa*
9 c7 j. [/ s V8 R1 R. j cap)/((\[Gamma]a^\[Epsilon]*pa^(0 H. \- M- a' s$ f
1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*. N' o7 F! a5 }2 e
hm[t]^(\[Theta]m*(1 - \[Epsilon])) + \% U! C6 S* c. E4 n8 L
\[Gamma]s^\[Epsilon]*(ps*hs[t]^\[Theta]s)^(1 - \[Epsilon]))*k (t)*
2 R6 O* n8 l j' H xm (t)), : {+ [4 V" J; u5 C8 a9 L0 d
Sm[t] == (\[Gamma]m^\[Epsilon]*
) L* M" s7 a. a9 C hm[t]^(\[Theta]m*(1 - \[Epsilon])))/(\[Gamma]a^\[Epsilon]*pa^(
0 `" T2 G! H) D 1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*
7 f, m) [" O* A4 s% H6 J hm[t]^(\[Theta]m*(1 - \[Epsilon])) + \[Gamma]s^\[Epsilon]*(ps*4 Z7 I. k$ x* T, o3 u
hs[t]^\[Theta]s)^(1 - \[Epsilon])), / a' Q! K) p. p) _5 `! f' {
Ss[t] == (\[Gamma]s^\[Epsilon]*(ps*+ Q, S+ G+ n8 [1 B
hs[t]^\[Theta]s)^(1 - \[Epsilon]))/(\[Gamma]a^\[Epsilon]*pa^(( G8 @9 N) r4 }) z8 d- V6 `
1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*
& Y1 W# H2 @ b* K9 G hm[t]^(\[Theta]m*(1 - \[Epsilon])) + \[Gamma]s^\[Epsilon]*(ps*
+ D3 {1 b. {6 `# s; w/ r0 a+ W# }1 @ hs[t]^\[Theta]s)^(1 - \[Epsilon])) - (\[Gamma]m^\[Epsilon]*
, A5 ^: U1 V+ M' x% A% q$ y1 X hm[t]^(\[Theta]m*(1 - \[Epsilon]))*ps*
) B9 f; x$ M3 l, W8 Q" B csp)/((\[Gamma]a^\[Epsilon]*pa^(
' U2 o! B* \8 w. ^7 O6 @ 1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*
0 ]& o% q) W' s+ P" K hm[t]^(\[Theta]m*(1 - \[Epsilon])) + \
[; e1 E4 d% P ]: V0 ^: n( a\[Gamma]s^\[Epsilon]*(ps*hs[t]^\[Theta]s)^(1 - \[Epsilon]))*k (t)*
" i5 g z6 f0 Y* _% I; }2 m4 w xm (t)), xm[0] == xm0,
! M% G5 J; X: |8 ^9 x1 k" Z xs[0] == xs0, \[Eta]m[0] == \[Eta]m0, \[Eta]s[0] == \[Eta]s0,
0 }! @! ~! A: T* x! i3 T K[0] == K0}, {xm, xs, \[Eta]m, \[Eta]s, K, hm, hs, Sa, Sm, Ss}, {t," ]- s* x" r/ Q( o/ D# R8 d2 a7 X
0, TT}]
/ `% |* P' k3 h/ h0 q2 Q, [Plot[{Evaluate[Sa[t] /. Sol], Evaluate[Sm[t] /. Sol],
4 y* [/ ~# y9 m" G Evaluate[Ss[t] /. Sol]}, {t, 0, TT}, AxesOrigin -> {0, 0}, O! M4 O5 `( r3 r2 t6 z0 }
PlotRange -> {0., 0.8}, PlotStyle -> {Blue, Dashed, Dashing[{0.05}]}]/ H# q. h/ ]7 b% l! j% J* }0 \' L& N* @
Plot[{Evaluate[D*Sa[t] /. Sol], , r- W2 @+ u, b( {, _" c d% q
Evaluate[(D*Sm[t] + (\[Alpha]*(gRate + \[Delta]))/(\[Rho] + $ o& X2 |/ p$ a* o" J2 k; e
gRate)) /. Sol], Evaluate[D*Ss[t] /. Sol]}, {t, 0, TT}, - ~: w* T3 i8 j/ @1 v; s+ p
AxesOrigin -> {0, 0}, PlotRange -> {0., 0.8},
" W3 g7 |/ G4 z: z PlotStyle -> {Blue, Dashed, Dashing[{0.05}]}]6 v7 ^/ t) `* F6 V, N
6 p7 L& e& V; Z' n
, q k5 Y) \3 _- _9 r# H* D
8 ^. t$ G( p" s( Y9 ~
' {- [* C; `7 _Set::wrsym: Symbol D is Protected.1 G/ i9 Q% | T p0 Z
1 I* x0 @9 s1 hNDSolve::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.}.2 N$ o; \5 t% g8 @( O
) P5 v7 B" `( v3 P3 C. r/ i
0 ~' k6 u2 _9 Z6 V& U5 _: n4 q2 C% @
3 _+ {/ Z) n6 U/ s2 g' s
; ^) e6 J, F3 `: _ |
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