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Clear[Am, As, Aa, \[Alpha], \[Rho], \[Theta]m, \[Theta]s, \
& ?+ P5 u. {7 p1 y5 k: C* u\[CurlyPhi]m, \[CurlyPhi]s, \[Epsilon]]/ P' Q0 O, o( J5 L
\[Gamma]a = 0.1; \[Gamma]m = 0.15; \[Gamma]s =
3 u5 Z6 m0 e- ~ 1 - \[Gamma]a - \[Gamma]m;; E9 o6 ]4 Z' v& q% i
\[Epsilon] = 0.04; \[Alpha] = 0.3; \[Rho] = 0.04; {8 ~ h( d8 f% v! L$ Q
\[Theta]m = 0.75; \[Theta]s = 0.9;9 {( T( `1 a9 h4 H) v/ w5 A% b
gRate = 0.02;
' i% _* o% q5 G4 xAm = (gRate + \[Rho])/\[Alpha]; Ba = 4; Bm = 1; Bs = 2.5;
" o% w. z' T( k6 Sps = Bm/Bs; pa = Bm/Ba;7 l) e5 @( ], |+ I4 J1 j% I
\[Delta] = 0.03;
" U( F$ @4 h, m) h ~8 JB = \!\(TraditionalForm\`\*
' ?+ e: D' {$ k( T5 H/ B# CFractionBox[
9 Q3 t/ v- R1 H3 \7 W! g2 R2 |6 zRowBox[{
8 Z N, l) p7 ^) C% m, KRowBox[{
7 e$ v! u- X# I. m6 {) S9 tRowBox[{
. F4 w3 `; l+ R- pStyleBox["(",
o: ]8 q* U2 QSpanMinSize->1.,
1 S* O% T; V% b5 C {/ MSpanMaxSize->1.], $ K" Y% R- A' i" b5 W
RowBox[{"1", "\[Minus]", "\[Alpha]"}], ' t' r0 G& t, K: ~
StyleBox[")",
/ I' {8 R8 e: z$ W& \& a/ C& aSpanMinSize->1.,( V, C1 W9 o$ B3 H( {
SpanMaxSize->1.]}], "gRate"}], "+", "\[Rho]"}],
w/ G5 s E- Y- d- n |$ `, u7 j% w "\[Alpha]"] \[Minus] \[Delta]\);% m- G7 P1 _( D
cap = 10;2 I* z) K) C& F5 h/ d" o, ]
csp = (pa*cap)/ps;8 }1 f3 \# L5 q' k; {* Q9 X
D = ((1 \[Minus] \[Alpha])*" w3 c* j" C! A0 O9 E* V; C
gRate + \[Rho] - \[Alpha]*\[Delta])/(\[Rho] + gRate);
$ w! h4 U1 x% L/ W; f\[CurlyPhi]m = 0.1; \[CurlyPhi]s = 0.1;% G. C# ?1 G# x4 ~
Print["*** Initial Values ***"]; f- ~) k2 W c
E0 = 1.5;& g! A1 y% l# }3 Y. A5 Q
K0 = E0/B;
7 e$ E" ?2 f3 D2 _' A g4 ?hm0 = 0.25; hs0 = 0.25;(* initial values *)! Y5 M3 @! V% P, Z! \+ h6 O v
\[Eta]m0 = hm0/K0; \[Eta]s0 = hs0/K0;0 u' B! o' u* |2 p- I2 J5 |
xm0 = (B*\[Gamma]m^\[Epsilon]*" X2 |1 j- s% ~) p. g
hm0^(\[Theta]m*(1 - \[Epsilon])))/(\[Gamma]a^\[Epsilon]*pa^(
# X( r- H" K. `+ I! U' _5 G 1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*; n( e- }: I" i/ [8 Q9 I
hm0^(\[Theta]m*(1 - \[Epsilon])) + \[Gamma]s^\[Epsilon]*(ps*, Z* w: `+ _% s# f' c- c
hs0^\[Theta]s)^(1 - \[Epsilon]));5 j$ f% W. n3 C4 ]! B6 H8 p
xs0 = (B*\[Gamma]s^\[Epsilon]*(ps*
0 F* i$ k4 L" Z; A hs0^\[Theta]s)^(1 - \[Epsilon]))/(\[Gamma]a^\[Epsilon]*pa^(6 K4 k6 \- n6 i
1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*$ s. j5 c- d1 L1 v* B- o- n
hm0^(\[Theta]m*(1 - \[Epsilon])) + \[Gamma]s^\[Epsilon]*(ps*& r2 q; Y: j" z$ I' T$ }0 p+ v
hs0^\[Theta]s)^(1 - \[Epsilon]));
' `' _7 u( K7 r2 EPrint["\[Eta]_{m,0}=" <> ToString[\[Eta]m0], 2 m' n* f: |* }9 ~5 M; ]
", \[Eta]_{s,0}=" <> ToString[\[Eta]s0],
# d8 x( {( R& ^: B1 r' D ", x_{m,0}=" <> ToString[xm0], ", x_{s,0}=" <> ToString[xs0]]* W+ P$ j3 y: k3 ]
TT = 100;(* end time *) U* F0 h: {% d% x1 C
(* Solve differential equations *)8 ]5 r2 S4 ]# Q# M; I
Sol = NDSolve[{xs'[t] = (1 - \[Epsilon])*
. n/ Q- W8 y$ V$ }( | xs[t]*( (1 - xs[t]/
U' t' t; _! a7 I0 C; B' S B)*(\[Theta]s*\[CurlyPhi]s*(xs[t]/(ps*\[Eta]s[t]) - 1) - , H% w9 z1 R1 `, y' J
xm[t]/B \[Theta]m*\[CurlyPhi]m*(xm[t]/\[Eta]m[t] - 1))),
6 P' S3 a' S' z0 t/ ` xm'[t] == (1 - \[Epsilon])*0 f/ y) U' B/ q3 Q. T, A( ^ A; F
xm[t]*( (1 - xm[t]/. Q( _$ a, R6 l& Z! W$ m
B)*\[Theta]m*\[CurlyPhi]m*(xm[t]/\[Eta]m[t] - 1) - 6 m! L) V2 |5 _' O9 I7 a- [
xs[t]/B*\[Theta]s*\[CurlyPhi]s*(xs[t]/(ps*\[Eta]s[t]) -
& I, m3 w0 N0 r+ H1 s 1) ), \[Eta]m'[5 v9 M7 j/ D+ {
t] == \[CurlyPhi]m*8 @2 g' d; R0 G4 B7 j6 Q
xm[t] - (\[CurlyPhi]m + gRate)*\[Eta]m[t], \[Eta]s'[0 ]" M% @0 x. g* u) l
t] == \[CurlyPhi]s*xs[t]/ps - (\[CurlyPhi]s + gRate)*\[Eta]s[t],
2 G: j3 }2 z+ U* U K'[t] == gRate*K[t], hm[t] == \[Eta]m[t]*K[t],
, W5 }7 G3 g. V ]8 b hs[t] == \[Eta]s[t]*K[t],
& @; y9 c( Q5 @3 \( k: G Sa[t] == (\[Gamma]a^\[Epsilon]*(pa)^(1 - \[Epsilon]))/(\[Gamma]a^\
; H5 I4 ]4 Z& M6 }) h6 x+ v\[Epsilon]*pa^(1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*9 M/ V$ ^, F0 `" m/ u
hm[t]^(\[Theta]m*(1 - \[Epsilon])) + \[Gamma]s^\[Epsilon]*(ps*) }8 o, K) x* D; K* X% V% H
hs[t]^\[Theta]s)^(1 - \[Epsilon])) + (\[Gamma]m^\[Epsilon]*
! V6 D a8 S( N3 A: @& a9 a9 } hm[t]^(\[Theta]m*(1 - \[Epsilon]))*pa*0 j9 |( j, u+ @ Q9 }- \
cap)/((\[Gamma]a^\[Epsilon]*pa^(
3 e9 Z5 v3 |7 e8 K$ T. W 1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*7 \* X. T7 F4 D" G- D5 E
hm[t]^(\[Theta]m*(1 - \[Epsilon])) + \) C0 ~ W( U( ~/ P, b
\[Gamma]s^\[Epsilon]*(ps*hs[t]^\[Theta]s)^(1 - \[Epsilon]))*k (t)*7 j$ P; E ?0 e h9 U* [
xm (t)), ) Y& R. ~) @ e2 W4 T: _3 Z3 r
Sm[t] == (\[Gamma]m^\[Epsilon]*
. S/ C; h: b" d7 X( K& ~9 D hm[t]^(\[Theta]m*(1 - \[Epsilon])))/(\[Gamma]a^\[Epsilon]*pa^(
( u, P* Y$ k" g K0 g- R 1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*% ], [; o7 e% e+ C4 G4 k* B
hm[t]^(\[Theta]m*(1 - \[Epsilon])) + \[Gamma]s^\[Epsilon]*(ps*6 C$ w1 i5 i% h3 u+ W' K3 T- P
hs[t]^\[Theta]s)^(1 - \[Epsilon])),
) Q+ A5 y* X) j Ss[t] == (\[Gamma]s^\[Epsilon]*(ps*% T* E V- ?, ?6 K! e$ \
hs[t]^\[Theta]s)^(1 - \[Epsilon]))/(\[Gamma]a^\[Epsilon]*pa^(8 z) T( o. e* L+ v
1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*
1 ~2 p. B* y. W hm[t]^(\[Theta]m*(1 - \[Epsilon])) + \[Gamma]s^\[Epsilon]*(ps*
, d7 c/ H+ f u, I, l hs[t]^\[Theta]s)^(1 - \[Epsilon])) - (\[Gamma]m^\[Epsilon]*
% G [! h# f) P A5 @ hm[t]^(\[Theta]m*(1 - \[Epsilon]))*ps*6 l) t7 a( |) h* T9 B1 X3 R" L
csp)/((\[Gamma]a^\[Epsilon]*pa^(' d- W# W5 I1 T1 _4 y3 A
1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*
4 A5 y/ O5 q! K& s; B hm[t]^(\[Theta]m*(1 - \[Epsilon])) + \
8 I- r7 E' `+ L( y! R9 }\[Gamma]s^\[Epsilon]*(ps*hs[t]^\[Theta]s)^(1 - \[Epsilon]))*k (t)*4 v& | J, @. x6 F4 @4 z
xm (t)), xm[0] == xm0, 0 \- c0 w: O6 C3 M9 |# Y% {; r
xs[0] == xs0, \[Eta]m[0] == \[Eta]m0, \[Eta]s[0] == \[Eta]s0, ( C7 e, w$ q" x8 U
K[0] == K0}, {xm, xs, \[Eta]m, \[Eta]s, K, hm, hs, Sa, Sm, Ss}, {t,
1 V6 E" T7 C( f& x) Y 0, TT}]
0 `8 K* o' g- w ]5 v% \Plot[{Evaluate[Sa[t] /. Sol], Evaluate[Sm[t] /. Sol], 9 m+ @! ?7 O& S* Z; L# x
Evaluate[Ss[t] /. Sol]}, {t, 0, TT}, AxesOrigin -> {0, 0}, $ E& J# _( \- k7 X$ H3 |7 W% t- T2 p
PlotRange -> {0., 0.8}, PlotStyle -> {Blue, Dashed, Dashing[{0.05}]}]
! Q$ b, r) }. g! APlot[{Evaluate[D*Sa[t] /. Sol], - l' _) R7 w; O# v: I' L- Q- k9 s" j
Evaluate[(D*Sm[t] + (\[Alpha]*(gRate + \[Delta]))/(\[Rho] + 3 b4 @5 c7 A/ E7 ?" w* S
gRate)) /. Sol], Evaluate[D*Ss[t] /. Sol]}, {t, 0, TT},
9 S _1 q2 [! } b AxesOrigin -> {0, 0}, PlotRange -> {0., 0.8},
! ^; q" b1 d6 y* b! }) l! S PlotStyle -> {Blue, Dashed, Dashing[{0.05}]}]
( K& ^( T: r! B* G/ x w+ }) F1 ?# t0 k& b# }7 @
3 q# V) b3 q$ v2 H" C; i; U
- ~$ _6 M/ P1 _7 ]# |
$ s# a) r- H: b" `6 c: SSet::wrsym: Symbol D is Protected.
/ ?, X3 [8 w, g* \8 {5 u1 E) S5 F. K5 p' M* D
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.}.( c2 N1 W4 O! E7 x! G) d! j- G0 |
' E, C% B- K8 Q/ }
7 }- n* J+ n$ j# T: p% k0 |; [' T6 ^# d
8 z1 Z. P; Z$ e. V. v8 O |
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