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Clear[Am, As, Aa, \[Alpha], \[Rho], \[Theta]m, \[Theta]s, \
9 ^" b* A9 k/ k3 @\[CurlyPhi]m, \[CurlyPhi]s, \[Epsilon]]2 S; e9 p! e$ y" a- Z( a9 L
\[Gamma]a = 0.1; \[Gamma]m = 0.15; \[Gamma]s = ( O5 m+ b% P1 g& F Q1 _
1 - \[Gamma]a - \[Gamma]m;
4 k3 m c: x- S3 Y; _4 l\[Epsilon] = 0.04; \[Alpha] = 0.3; \[Rho] = 0.04;1 i; D- j0 L7 j' B: N9 P* R+ _$ {
\[Theta]m = 0.75; \[Theta]s = 0.9;
1 @+ V& ?. Z8 D1 z; MgRate = 0.02;
2 |4 Z7 ]* s: }, VAm = (gRate + \[Rho])/\[Alpha]; Ba = 4; Bm = 1; Bs = 2.5;6 S5 k/ Y% b3 D' {
ps = Bm/Bs; pa = Bm/Ba;" F! P! g3 ^) M3 f' U
\[Delta] = 0.03;3 ~0 g2 l. p- W- Z3 H% N9 S
B = \!\(TraditionalForm\`\*2 ~3 i% u2 t, ]( A. H$ u
FractionBox[5 [ D% a9 ]. u# u& Q/ {
RowBox[{
5 H+ O% n! x* X0 zRowBox[{4 C4 }1 d D0 h$ R" O0 O( L& ^- B
RowBox[{3 [( v8 ^, A4 d9 Q9 M/ f
StyleBox["(",+ Z. \$ t. k* h6 B. P/ v' e
SpanMinSize->1.,
& e- U5 b# ?( J1 f# T4 I/ cSpanMaxSize->1.], + Z9 m( b8 G. N! w6 s2 r1 |
RowBox[{"1", "\[Minus]", "\[Alpha]"}], # H2 B( K, `& Y, Y l U
StyleBox[")",% V- N3 I8 s7 I9 @2 B
SpanMinSize->1.,/ n6 K1 G/ q. o5 O! ~/ z! k
SpanMaxSize->1.]}], "gRate"}], "+", "\[Rho]"}],
2 |$ q2 Z7 M4 x! e, ]9 Q "\[Alpha]"] \[Minus] \[Delta]\);* H4 u" K" p e1 Q0 U% P# W; m
cap = 10;
* P$ f) ^9 f) h4 j5 a7 s; Q/ ycsp = (pa*cap)/ps;4 k# c" V% Y9 [3 `# o6 j& ]
D = ((1 \[Minus] \[Alpha])*
+ e" |6 A, j: Z! c& S- o* N gRate + \[Rho] - \[Alpha]*\[Delta])/(\[Rho] + gRate);" J3 V2 j, t$ t9 z% ~: X g) |
\[CurlyPhi]m = 0.1; \[CurlyPhi]s = 0.1;& l" d% ]0 R( g3 x
Print["*** Initial Values ***"]$ _. y3 V8 Q% Z1 e" V
E0 = 1.5;
; C4 \- ?$ E- nK0 = E0/B;( ~# T0 o7 M- @5 y2 K) ~+ O* n
hm0 = 0.25; hs0 = 0.25;(* initial values *)
: t! l2 ^ o5 c) |. ]0 ~\[Eta]m0 = hm0/K0; \[Eta]s0 = hs0/K0;2 y9 h& J. J, X, W7 n* Q
xm0 = (B*\[Gamma]m^\[Epsilon]*
# K2 S. Y0 j# e3 O. i hm0^(\[Theta]m*(1 - \[Epsilon])))/(\[Gamma]a^\[Epsilon]*pa^(, |8 f% L) C. L) G
1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*
1 U+ y! Q! o* o- v hm0^(\[Theta]m*(1 - \[Epsilon])) + \[Gamma]s^\[Epsilon]*(ps*+ p+ O* q0 s; l5 e% j3 t9 j! n
hs0^\[Theta]s)^(1 - \[Epsilon]));& z, G7 X& e4 n9 {6 @& P; m
xs0 = (B*\[Gamma]s^\[Epsilon]*(ps*
3 S: [" P: S6 ]1 f- \+ ^ hs0^\[Theta]s)^(1 - \[Epsilon]))/(\[Gamma]a^\[Epsilon]*pa^(6 u- V# N0 h$ S1 m3 I' q* O
1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*4 L/ M, k L" [# B
hm0^(\[Theta]m*(1 - \[Epsilon])) + \[Gamma]s^\[Epsilon]*(ps*, y7 ]' E( |6 O) `
hs0^\[Theta]s)^(1 - \[Epsilon]));; s/ e; ]$ _" N& D8 D4 q
Print["\[Eta]_{m,0}=" <> ToString[\[Eta]m0], 0 E& A7 o" ]# K% M# j' D E
", \[Eta]_{s,0}=" <> ToString[\[Eta]s0], & D/ `# w+ U1 {0 g, l8 g Q# E" F
", x_{m,0}=" <> ToString[xm0], ", x_{s,0}=" <> ToString[xs0]]0 s5 S& |& D5 U: l
TT = 100;(* end time *)
T- p" g# ]* @9 Z: e+ U(* Solve differential equations *)" d$ Q. X1 b7 l' n% G6 x+ ^ H
Sol = NDSolve[{xs'[t] = (1 - \[Epsilon])*
! u2 L8 A) t. @2 E9 H4 {+ l xs[t]*( (1 - xs[t]/* q% b8 w$ w/ L
B)*(\[Theta]s*\[CurlyPhi]s*(xs[t]/(ps*\[Eta]s[t]) - 1) - 4 ?; O! i5 C6 u, e. i
xm[t]/B \[Theta]m*\[CurlyPhi]m*(xm[t]/\[Eta]m[t] - 1))), : G3 s0 e p2 K- Z) t: q
xm'[t] == (1 - \[Epsilon])*
5 P" A# }$ m/ Q xm[t]*( (1 - xm[t]/
: l# x2 s& F- d4 ]" U1 b( y B)*\[Theta]m*\[CurlyPhi]m*(xm[t]/\[Eta]m[t] - 1) - & H+ I6 t: V& I+ D) n( {
xs[t]/B*\[Theta]s*\[CurlyPhi]s*(xs[t]/(ps*\[Eta]s[t]) - . N: H- E4 V/ x
1) ), \[Eta]m'[9 j) b" e; N. p- `' b6 u8 E
t] == \[CurlyPhi]m*
) ~. N" u! U+ n& P6 x3 i' b: O xm[t] - (\[CurlyPhi]m + gRate)*\[Eta]m[t], \[Eta]s'[4 i) B7 }( [" Q9 f5 ~' K6 n) M% }
t] == \[CurlyPhi]s*xs[t]/ps - (\[CurlyPhi]s + gRate)*\[Eta]s[t],
) w) S4 p6 t( K" A K'[t] == gRate*K[t], hm[t] == \[Eta]m[t]*K[t], 8 q. {7 O$ s: q; p
hs[t] == \[Eta]s[t]*K[t],
) h( z4 o2 p: y. U- o- O& u. S Sa[t] == (\[Gamma]a^\[Epsilon]*(pa)^(1 - \[Epsilon]))/(\[Gamma]a^\ H/ @. W0 z( ^4 Y5 ^ w
\[Epsilon]*pa^(1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*
4 K; n5 d3 z2 t3 K3 X! T" U( N hm[t]^(\[Theta]m*(1 - \[Epsilon])) + \[Gamma]s^\[Epsilon]*(ps*
; R: ?6 q7 D1 f0 K& q hs[t]^\[Theta]s)^(1 - \[Epsilon])) + (\[Gamma]m^\[Epsilon]*
/ ]7 n1 \/ @9 Z) S7 [ hm[t]^(\[Theta]m*(1 - \[Epsilon]))*pa* A2 |! I1 f$ s: P9 n
cap)/((\[Gamma]a^\[Epsilon]*pa^(' a* { F `2 C( g. y# L
1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*
! ^3 P" E) i; V: V' r- B hm[t]^(\[Theta]m*(1 - \[Epsilon])) + \* }1 V2 z1 _, K: r, T9 U* @3 u q8 n
\[Gamma]s^\[Epsilon]*(ps*hs[t]^\[Theta]s)^(1 - \[Epsilon]))*k (t)*- |2 X( k4 A; |* |6 F! h- ^
xm (t)), , K4 Q0 @: O: L* d! R1 y: Z) w. m- ]
Sm[t] == (\[Gamma]m^\[Epsilon]*; F, r* [8 a7 A$ V
hm[t]^(\[Theta]m*(1 - \[Epsilon])))/(\[Gamma]a^\[Epsilon]*pa^(( X' D+ D' X3 n
1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*
' G- H$ G9 ?. r, R( J/ q- A( o hm[t]^(\[Theta]m*(1 - \[Epsilon])) + \[Gamma]s^\[Epsilon]*(ps*
1 w9 ^; K- ?1 `! S4 ? hs[t]^\[Theta]s)^(1 - \[Epsilon])), 1 i% E5 h- H. S7 b
Ss[t] == (\[Gamma]s^\[Epsilon]*(ps*7 [* S: \8 a+ @" E5 |
hs[t]^\[Theta]s)^(1 - \[Epsilon]))/(\[Gamma]a^\[Epsilon]*pa^(9 y% f( h& M5 S% L: |! Q
1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*
+ S& L7 e! E9 q9 z9 @2 m hm[t]^(\[Theta]m*(1 - \[Epsilon])) + \[Gamma]s^\[Epsilon]*(ps*
8 x" I( o, w8 W% w/ N hs[t]^\[Theta]s)^(1 - \[Epsilon])) - (\[Gamma]m^\[Epsilon]*, O# y3 U+ T% a# c* n
hm[t]^(\[Theta]m*(1 - \[Epsilon]))*ps*2 P' p% a" S0 n4 H7 o5 c! m
csp)/((\[Gamma]a^\[Epsilon]*pa^($ W, f! ^0 e0 p3 w4 f5 j
1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*. A, Q! Z: M3 {
hm[t]^(\[Theta]m*(1 - \[Epsilon])) + \+ Z% q# A6 ]( S
\[Gamma]s^\[Epsilon]*(ps*hs[t]^\[Theta]s)^(1 - \[Epsilon]))*k (t)*. L/ f7 `1 C0 n7 u4 A
xm (t)), xm[0] == xm0, v+ e; N5 Z9 J# O7 D& i9 L
xs[0] == xs0, \[Eta]m[0] == \[Eta]m0, \[Eta]s[0] == \[Eta]s0,
9 @: I1 D' l- Y K[0] == K0}, {xm, xs, \[Eta]m, \[Eta]s, K, hm, hs, Sa, Sm, Ss}, {t,
! r2 p4 o: D: l' j 0, TT}]' r4 ~( f3 I" `8 k! b+ }5 e2 X
Plot[{Evaluate[Sa[t] /. Sol], Evaluate[Sm[t] /. Sol],
- l" B7 e: c f8 \& z Evaluate[Ss[t] /. Sol]}, {t, 0, TT}, AxesOrigin -> {0, 0},
* v, z0 Y8 p2 E/ i. P2 s7 A. i' R, h PlotRange -> {0., 0.8}, PlotStyle -> {Blue, Dashed, Dashing[{0.05}]}]
' [* `' g, Z/ q8 ^) e2 zPlot[{Evaluate[D*Sa[t] /. Sol],
+ k8 ]$ C5 B- {6 Y% q! z/ D8 ^ Evaluate[(D*Sm[t] + (\[Alpha]*(gRate + \[Delta]))/(\[Rho] + + O& h! \' X) w/ K. a! e; d/ D- C$ F
gRate)) /. Sol], Evaluate[D*Ss[t] /. Sol]}, {t, 0, TT},
3 j$ J3 U% e) j; }: q AxesOrigin -> {0, 0}, PlotRange -> {0., 0.8},
& N# `0 y( Z4 R. y3 y" F PlotStyle -> {Blue, Dashed, Dashing[{0.05}]}]4 E9 F9 f& v) h4 t
% s& z4 m, a$ X
; K: j4 w2 ]3 T4 B! T
2 z& D- f& k6 w5 i. Y1 n# Q& i
: J Z4 ?% J/ z
Set::wrsym: Symbol D is Protected.
9 q6 g3 w1 V+ c- R! y
+ B& L/ p8 [( w3 V0 F) WNDSolve::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.}.
4 w O) s6 A0 b' ?6 q5 N
1 F. O- t9 e ~0 ~3 R2 P! m* @
( f2 H7 L) G1 Z: B/ X" C" d7 o* A
, Y1 w- I c6 z+ Q5 ~/ \- \
" S0 A, Y& _# \/ q1 f3 E! r; U |
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