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Clear[Am, As, Aa, \[Alpha], \[Rho], \[Theta]m, \[Theta]s, \
7 Y2 a$ |% J0 q h! ~: W' r! Z \[CurlyPhi]m, \[CurlyPhi]s, \[Epsilon]]. F) P" K9 d9 s8 C4 [
\[Gamma]a = 0.1; \[Gamma]m = 0.15; \[Gamma]s =
& R5 V" {( }" S1 x1 Z 1 - \[Gamma]a - \[Gamma]m; m1 J6 `, i$ D; j4 w
\[Epsilon] = 0.04; \[Alpha] = 0.3; \[Rho] = 0.04;8 F$ t5 l; c; h, k3 s% t
\[Theta]m = 0.75; \[Theta]s = 0.9;( A! X+ Z5 G5 r7 ]. d, A
gRate = 0.02;
: z% a+ l( Z1 A, Y/ _ Am = (gRate + \[Rho])/\[Alpha]; Ba = 4; Bm = 1; Bs = 2.5;; @& Z! q7 u' |3 J: @
ps = Bm/Bs; pa = Bm/Ba;
4 P: q: S0 `3 e \[Delta] = 0.03;
. [; g& T2 C( l1 H6 K3 Q B = \!\(TraditionalForm\`\*
* r j$ ^* R1 X/ Q3 I FractionBox[( D* } ^' Y2 W) n2 C& @
RowBox[{' a1 f' ~( X9 }
RowBox[{
6 f4 O3 V f8 `* G RowBox[{
2 C- @: q1 |" h6 s1 {# S StyleBox["(",
t, ^) _" `8 ?; S" A1 ~9 ` SpanMinSize->1.,& O2 {) B% A n
SpanMaxSize->1.],
" N* H! }! w/ j% }6 ^$ z1 J RowBox[{"1", "\[Minus]", "\[Alpha]"}],
4 D5 N7 l0 R9 j( o StyleBox[")",# m. c& i. b) S( n7 h- `
SpanMinSize->1.,* B! O1 `! o3 @- E; f! v7 ~' V7 y
SpanMaxSize->1.]}], "gRate"}], "+", "\[Rho]"}], 0 k& x9 k% C$ l q5 c$ t
"\[Alpha]"] \[Minus] \[Delta]\);
& ]. r( U z+ P) N3 J' ]2 ^7 p5 K4 ^ cap = 10;
9 [' T2 l: V8 `9 @ csp = (pa*cap)/ps;
- d Y. T( r+ \$ h1 a8 @7 _- f D = ((1 \[Minus] \[Alpha])*
( @+ b& x& [5 t$ z4 f gRate + \[Rho] - \[Alpha]*\[Delta])/(\[Rho] + gRate);
* R. r+ S7 V2 Z+ j/ x \[CurlyPhi]m = 0.1; \[CurlyPhi]s = 0.1;
* b0 ^$ `- X2 C2 I/ A+ t Print["*** Initial Values ***"]/ [" X6 C8 f9 P
E0 = 1.5;* {2 _. v% _1 x; {
K0 = E0/B;
! p. I+ L8 j5 m0 E) u hm0 = 0.25; hs0 = 0.25;(* initial values *)
0 ^- l. P6 b7 F2 y \[Eta]m0 = hm0/K0; \[Eta]s0 = hs0/K0;
7 ~ b6 @% z6 G8 S xm0 = (B*\[Gamma]m^\[Epsilon]*
# d3 M; h9 `" v hm0^(\[Theta]m*(1 - \[Epsilon])))/(\[Gamma]a^\[Epsilon]*pa^(
1 }. u% t) O R+ g/ M r 1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*
0 Y# E: E4 y8 Y: ^3 f hm0^(\[Theta]m*(1 - \[Epsilon])) + \[Gamma]s^\[Epsilon]*(ps*' P6 j/ I* Z I! S) Z' E- N
hs0^\[Theta]s)^(1 - \[Epsilon]));
- ]. E4 M8 g/ r8 m& ? xs0 = (B*\[Gamma]s^\[Epsilon]*(ps*
1 y [' Y, G# Z" d6 ^# A e/ a8 n hs0^\[Theta]s)^(1 - \[Epsilon]))/(\[Gamma]a^\[Epsilon]*pa^(: }* {$ Z1 }- e
1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*
: U c# k; Y. G hm0^(\[Theta]m*(1 - \[Epsilon])) + \[Gamma]s^\[Epsilon]*(ps*
+ M9 X) o9 N3 z; ?2 ~ hs0^\[Theta]s)^(1 - \[Epsilon]));
. m" z! j4 u; j& f6 R, T Print["\[Eta]_{m,0}=" <> ToString[\[Eta]m0],
. |+ [; B- @5 v! Z e) Y+ h ", \[Eta]_{s,0}=" <> ToString[\[Eta]s0], 0 M0 u% P/ y6 Q
", x_{m,0}=" <> ToString[xm0], ", x_{s,0}=" <> ToString[xs0]]
5 `# o, A z6 Y; {2 D9 { TT = 100;(* end time *)$ G8 v* |8 z: F* a, h) E: N$ I
(* Solve differential equations *)
3 C6 ~# e( Z: Z3 h6 `0 U Sol = NDSolve[{xs'[t] = (1 - \[Epsilon])*+ W6 ]# m' P8 M1 ^- g
xs[t]*( (1 - xs[t]/
" K0 T2 X# j/ F7 M5 z+ U- S B)*(\[Theta]s*\[CurlyPhi]s*(xs[t]/(ps*\[Eta]s[t]) - 1) - & x' M) U8 E; d, g' [2 R+ [
xm[t]/B \[Theta]m*\[CurlyPhi]m*(xm[t]/\[Eta]m[t] - 1))), 8 Y% T; J h; W) c
xm'[t] == (1 - \[Epsilon])*4 D* u7 ]2 A E; c0 w
xm[t]*( (1 - xm[t]/4 B# x! r# q1 x, L, k% E# D
B)*\[Theta]m*\[CurlyPhi]m*(xm[t]/\[Eta]m[t] - 1) -
& \& K ?* W% @. ~3 p) s. L; { xs[t]/B*\[Theta]s*\[CurlyPhi]s*(xs[t]/(ps*\[Eta]s[t]) -
w) j; I1 t! U) J* H+ U 1) ), \[Eta]m'[! ?, P. k0 d) P5 H. j
t] == \[CurlyPhi]m*
, I5 y* `# d+ b5 `+ F" V xm[t] - (\[CurlyPhi]m + gRate)*\[Eta]m[t], \[Eta]s'[" Z0 I9 n8 h# v" A. M
t] == \[CurlyPhi]s*xs[t]/ps - (\[CurlyPhi]s + gRate)*\[Eta]s[t], ) t7 I/ ~# C! k+ ?7 |1 |
K'[t] == gRate*K[t], hm[t] == \[Eta]m[t]*K[t],
( u; J/ Z% h4 a hs[t] == \[Eta]s[t]*K[t],
0 c) o- ]/ D) s2 `' z* D7 j" P# \ Sa[t] == (\[Gamma]a^\[Epsilon]*(pa)^(1 - \[Epsilon]))/(\[Gamma]a^\3 k& V0 h4 F; C& I3 U- \, ]# C
\[Epsilon]*pa^(1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*: i' _1 R% V5 D- R7 |
hm[t]^(\[Theta]m*(1 - \[Epsilon])) + \[Gamma]s^\[Epsilon]*(ps*
: e( s, u+ I$ t& U5 j6 Y hs[t]^\[Theta]s)^(1 - \[Epsilon])) + (\[Gamma]m^\[Epsilon]*3 ?0 U- ?5 J% w5 q! }5 y( x
hm[t]^(\[Theta]m*(1 - \[Epsilon]))*pa*# [- F* j( u% ?: D' O
cap)/((\[Gamma]a^\[Epsilon]*pa^(. o/ _5 G7 N+ H* P9 K% m
1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*+ m5 j: \! T, f( D/ t& _
hm[t]^(\[Theta]m*(1 - \[Epsilon])) + \
, N+ P S! n* B% i9 m \[Gamma]s^\[Epsilon]*(ps*hs[t]^\[Theta]s)^(1 - \[Epsilon]))*k (t)*
$ m, A6 j2 n, }" I% u xm (t)),
, S7 @& g. N7 y+ ?# r' c8 a8 h+ K Sm[t] == (\[Gamma]m^\[Epsilon]*6 W7 w7 E; Z3 J( N* w5 `7 }2 I
hm[t]^(\[Theta]m*(1 - \[Epsilon])))/(\[Gamma]a^\[Epsilon]*pa^(9 w. V4 Y) o! t" x3 p2 Z0 {! Z9 b
1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*( H6 i# H# [# L- h/ T
hm[t]^(\[Theta]m*(1 - \[Epsilon])) + \[Gamma]s^\[Epsilon]*(ps*
: i1 m; [* h' J( z" ?( r hs[t]^\[Theta]s)^(1 - \[Epsilon])),
3 g- [$ e7 ?4 F& n X# W7 B1 P Ss[t] == (\[Gamma]s^\[Epsilon]*(ps*
; I: M* C1 p7 q) U3 D hs[t]^\[Theta]s)^(1 - \[Epsilon]))/(\[Gamma]a^\[Epsilon]*pa^(
9 K& v- E6 |, y; R/ G/ Q* d; a 1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*- W- Z' a4 Q! e( `) R
hm[t]^(\[Theta]m*(1 - \[Epsilon])) + \[Gamma]s^\[Epsilon]*(ps*: Z2 t/ J# j5 v% m
hs[t]^\[Theta]s)^(1 - \[Epsilon])) - (\[Gamma]m^\[Epsilon]*
. S4 N+ h# Y- _9 T3 d hm[t]^(\[Theta]m*(1 - \[Epsilon]))*ps*
( ~6 Y( m! h/ f$ C* M! Z j csp)/((\[Gamma]a^\[Epsilon]*pa^(, l- q/ s7 k" r" J: L! N, u
1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*( a x. K, O7 x0 N7 v$ T R
hm[t]^(\[Theta]m*(1 - \[Epsilon])) + \/ O. A: R3 I' [3 i
\[Gamma]s^\[Epsilon]*(ps*hs[t]^\[Theta]s)^(1 - \[Epsilon]))*k (t)*. O8 v* T) {0 O
xm (t)), xm[0] == xm0,
; K4 q+ e+ A/ I+ d, X5 _' l* N xs[0] == xs0, \[Eta]m[0] == \[Eta]m0, \[Eta]s[0] == \[Eta]s0,
% P, L, W5 ^6 M+ ^; f4 _9 t( t- h! [ K[0] == K0}, {xm, xs, \[Eta]m, \[Eta]s, K, hm, hs, Sa, Sm, Ss}, {t,
1 T+ B7 ~6 X3 \9 a' f/ R) C 0, TT}]' `, ?" Z3 U5 B4 r7 {% w+ }1 x
Plot[{Evaluate[Sa[t] /. Sol], Evaluate[Sm[t] /. Sol],
5 R9 V3 b8 C$ x4 U Evaluate[Ss[t] /. Sol]}, {t, 0, TT}, AxesOrigin -> {0, 0},
2 T0 ?9 E" o3 q* A; e# x8 r PlotRange -> {0., 0.8}, PlotStyle -> {Blue, Dashed, Dashing[{0.05}]}]
3 ~" g' z) F3 e. |' n* E: Y0 W8 j Plot[{Evaluate[D*Sa[t] /. Sol],
+ I- q- s& |, s, H; g Evaluate[(D*Sm[t] + (\[Alpha]*(gRate + \[Delta]))/(\[Rho] + # q" Q2 E+ I4 d; v# B* X
gRate)) /. Sol], Evaluate[D*Ss[t] /. Sol]}, {t, 0, TT},
9 w" ?; ~) k+ d8 a AxesOrigin -> {0, 0}, PlotRange -> {0., 0.8},
3 J2 }0 t8 U& ~1 l: v PlotStyle -> {Blue, Dashed, Dashing[{0.05}]}]
- O7 }. G) Y) A* n . Y: g% K' X( H( Y
6 x, E% s" i* S# `, E3 F
& B/ M. r3 G0 w. }, E+ d' _7 {+ E' N
* {# M! ]- P8 L8 s Set::wrsym: Symbol D is Protected.
5 y; i m+ z- I+ ^
/ }4 Q# ^- V" H5 }$ T, P, _ k8 C 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.}.
& H. i J+ A1 d9 N9 \3 p& o( r
" @% k3 S# F1 @5 c9 W ' Y/ X1 U0 i( f5 T1 h, [/ o
* v D6 U: q% l" b2 L5 h" M9 T: S ' @; ? C. E2 ~" Y& {1 m1 C, a
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