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Clear[Am, As, Aa, \[Alpha], \[Rho], \[Theta]m, \[Theta]s, \
w( P7 `/ [2 V\[CurlyPhi]m, \[CurlyPhi]s, \[Epsilon]]/ t, Z) c) y- F: M& {# K1 i- F
\[Gamma]a = 0.1; \[Gamma]m = 0.15; \[Gamma]s = ' e" v/ [* X' Z7 S5 S
1 - \[Gamma]a - \[Gamma]m;1 V& ]- t# v0 s* _( r7 L* q
\[Epsilon] = 0.04; \[Alpha] = 0.3; \[Rho] = 0.04;
) V$ d% f K/ N7 X0 { l3 a$ s6 B\[Theta]m = 0.75; \[Theta]s = 0.9;% h& I5 x, h9 z) _5 `
gRate = 0.02;/ _* t9 i6 E. u9 ^5 _5 F) M
Am = (gRate + \[Rho])/\[Alpha]; Ba = 4; Bm = 1; Bs = 2.5;
2 }/ D& u. _0 A/ r- f: ~' Jps = Bm/Bs; pa = Bm/Ba;
( C- v9 M1 _/ m1 h0 `! a7 ]; m\[Delta] = 0.03;
! t/ W- L! C: b' ]B = \!\(TraditionalForm\`\*
# |( p' K, { H8 h% U/ Y- fFractionBox[# y- m" R* w( ~+ C6 ~' D, C
RowBox[{. K# C5 R" t; n
RowBox[{9 {1 T/ N+ D7 ]7 s5 y; U& S* R
RowBox[{
; z9 P; y2 C& a5 N8 RStyleBox["(",
7 x& p4 w4 g* z- rSpanMinSize->1.,
: J+ q j9 ~3 y: w2 m- h) k7 [$ D' ^SpanMaxSize->1.], 6 ~1 @9 y8 V+ g
RowBox[{"1", "\[Minus]", "\[Alpha]"}], ; M' [$ w% j! h
StyleBox[")",
/ B, {. K; d9 N0 [5 o TSpanMinSize->1.,% x' ]$ ^1 ?2 {6 D
SpanMaxSize->1.]}], "gRate"}], "+", "\[Rho]"}], - i0 m1 B9 d3 W/ Y
"\[Alpha]"] \[Minus] \[Delta]\);
7 x7 B& n9 B* P2 l6 O8 q5 Ocap = 10;" s. p6 m- [$ V/ ?. [
csp = (pa*cap)/ps;7 [+ h& k- v! C. P; e5 k5 P8 x0 R
D = ((1 \[Minus] \[Alpha])*
0 I6 ]0 w& F" P5 B" X gRate + \[Rho] - \[Alpha]*\[Delta])/(\[Rho] + gRate);
# y* m, T0 }! V; u# A\[CurlyPhi]m = 0.1; \[CurlyPhi]s = 0.1;
& \$ V$ U4 E* O+ ^Print["*** Initial Values ***"]
. o/ t( j0 q0 Q* G- UE0 = 1.5;
* A E+ @* d) Z; f1 D3 oK0 = E0/B;+ s& p$ u4 F% [' [" A3 s
hm0 = 0.25; hs0 = 0.25;(* initial values *)
* h. N: k+ A% F5 w1 C\[Eta]m0 = hm0/K0; \[Eta]s0 = hs0/K0;
) d( k! u# {0 m$ h- ^6 [xm0 = (B*\[Gamma]m^\[Epsilon]*
: o6 A, [ ~2 M- ] hm0^(\[Theta]m*(1 - \[Epsilon])))/(\[Gamma]a^\[Epsilon]*pa^(
0 p" { P B- ?8 i 1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*% {5 }7 [7 j3 y8 l6 C ], O/ n1 u" i$ z& z
hm0^(\[Theta]m*(1 - \[Epsilon])) + \[Gamma]s^\[Epsilon]*(ps*3 e7 B" H5 t% X0 S$ {* d
hs0^\[Theta]s)^(1 - \[Epsilon]));/ b8 P, e7 z# v8 U+ A% a- x; R/ [0 L9 Q
xs0 = (B*\[Gamma]s^\[Epsilon]*(ps*% A9 P" C- a! ]6 O/ J
hs0^\[Theta]s)^(1 - \[Epsilon]))/(\[Gamma]a^\[Epsilon]*pa^(; M) j' C6 H- t2 a% _
1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*6 }+ s1 o7 S5 _/ T+ g: R
hm0^(\[Theta]m*(1 - \[Epsilon])) + \[Gamma]s^\[Epsilon]*(ps*3 v- m' P* U4 D8 C1 ~3 \5 Y& l7 }
hs0^\[Theta]s)^(1 - \[Epsilon]));- Y2 E/ x* ]' Q3 ]; r
Print["\[Eta]_{m,0}=" <> ToString[\[Eta]m0],
8 l9 X* k/ w) M0 j ", \[Eta]_{s,0}=" <> ToString[\[Eta]s0], & H U0 c" g; z! o) E1 A- a
", x_{m,0}=" <> ToString[xm0], ", x_{s,0}=" <> ToString[xs0]]" a3 E9 X4 z8 n8 R" j
TT = 100;(* end time *)
3 J1 z, E% {1 H, S2 T(* Solve differential equations *)
6 \( r% N& Z7 T# ^( V% t1 C0 |) QSol = NDSolve[{xs'[t] = (1 - \[Epsilon])*
0 h7 K& h* S/ u: p! ^$ h$ y/ p% c' ? xs[t]*( (1 - xs[t]/) F6 @3 m3 \0 ?7 H) g, O% ^
B)*(\[Theta]s*\[CurlyPhi]s*(xs[t]/(ps*\[Eta]s[t]) - 1) - ) e( }! B- J; k2 i: Q
xm[t]/B \[Theta]m*\[CurlyPhi]m*(xm[t]/\[Eta]m[t] - 1))), 6 i/ g8 @3 r. b
xm'[t] == (1 - \[Epsilon])*7 |7 J7 R' V6 E$ Z; O; }" U
xm[t]*( (1 - xm[t]/
- U/ B3 p/ F& ^ w. s* i B)*\[Theta]m*\[CurlyPhi]m*(xm[t]/\[Eta]m[t] - 1) - " d) X- n3 r% `2 Z/ @
xs[t]/B*\[Theta]s*\[CurlyPhi]s*(xs[t]/(ps*\[Eta]s[t]) -
( n0 d' b) `8 Z. ] 1) ), \[Eta]m'[
4 Q- n5 H! H$ b: B: r t] == \[CurlyPhi]m*1 C0 v' |5 Z B' B$ A4 j
xm[t] - (\[CurlyPhi]m + gRate)*\[Eta]m[t], \[Eta]s'[/ N8 E! o; W+ j
t] == \[CurlyPhi]s*xs[t]/ps - (\[CurlyPhi]s + gRate)*\[Eta]s[t],
* W0 g: T/ }4 u6 F) t# G/ C2 N K'[t] == gRate*K[t], hm[t] == \[Eta]m[t]*K[t], . _7 U8 k4 n/ I, ^3 m7 `6 W7 P) B
hs[t] == \[Eta]s[t]*K[t],
! v y6 r- Q2 y0 [4 _. ?0 A7 d Sa[t] == (\[Gamma]a^\[Epsilon]*(pa)^(1 - \[Epsilon]))/(\[Gamma]a^\
: ^: Q) ?/ A* g; `\[Epsilon]*pa^(1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*
+ ^# Q1 j+ }! Q) T1 s4 R8 W. m hm[t]^(\[Theta]m*(1 - \[Epsilon])) + \[Gamma]s^\[Epsilon]*(ps*9 `5 j8 P2 B0 i; g
hs[t]^\[Theta]s)^(1 - \[Epsilon])) + (\[Gamma]m^\[Epsilon]*( x: n5 f& U. N" A
hm[t]^(\[Theta]m*(1 - \[Epsilon]))*pa*
. T4 v8 e" ^( u cap)/((\[Gamma]a^\[Epsilon]*pa^(
) i+ J3 ^7 a6 B 1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*$ i! M. s- u+ D1 _: ]
hm[t]^(\[Theta]m*(1 - \[Epsilon])) + \; P0 h K- _* S1 {7 |) B+ Y2 c
\[Gamma]s^\[Epsilon]*(ps*hs[t]^\[Theta]s)^(1 - \[Epsilon]))*k (t)*7 W& S+ a9 r9 [4 R* F+ U4 _! b W5 R
xm (t)), - o% S5 i6 d6 z
Sm[t] == (\[Gamma]m^\[Epsilon]*
4 m% d, u! h& G hm[t]^(\[Theta]m*(1 - \[Epsilon])))/(\[Gamma]a^\[Epsilon]*pa^(
2 i: w5 G& v8 Q5 Z4 |7 v 1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*# }3 L% n( f% C# r$ s& h
hm[t]^(\[Theta]m*(1 - \[Epsilon])) + \[Gamma]s^\[Epsilon]*(ps*
5 Y1 L, Q6 i" }6 b& L hs[t]^\[Theta]s)^(1 - \[Epsilon])),
. A* Q. n2 ]/ y# c& M/ g/ h2 p$ M Ss[t] == (\[Gamma]s^\[Epsilon]*(ps*) g( i3 f5 m* T/ ^5 M
hs[t]^\[Theta]s)^(1 - \[Epsilon]))/(\[Gamma]a^\[Epsilon]*pa^(8 ^8 u% X. D" n7 ?& ^
1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]** x1 W& [3 v. W; ^
hm[t]^(\[Theta]m*(1 - \[Epsilon])) + \[Gamma]s^\[Epsilon]*(ps*0 P2 [0 A! Q; ^9 u( Z9 q) k! H
hs[t]^\[Theta]s)^(1 - \[Epsilon])) - (\[Gamma]m^\[Epsilon]*
6 ]8 D2 {* \* l: Q8 f; O, P' Y5 } hm[t]^(\[Theta]m*(1 - \[Epsilon]))*ps*
! [% K$ j( E8 ` csp)/((\[Gamma]a^\[Epsilon]*pa^(
0 Z; O- w X# `; X' Y7 } 1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*# r/ @5 T1 w1 N i- c e+ D
hm[t]^(\[Theta]m*(1 - \[Epsilon])) + \
, h4 H" X0 p0 g3 X\[Gamma]s^\[Epsilon]*(ps*hs[t]^\[Theta]s)^(1 - \[Epsilon]))*k (t)*
: f( e/ n2 V" K6 B& O xm (t)), xm[0] == xm0, 5 k! f# y, c( j; i
xs[0] == xs0, \[Eta]m[0] == \[Eta]m0, \[Eta]s[0] == \[Eta]s0, , _0 I5 \4 x$ z ?
K[0] == K0}, {xm, xs, \[Eta]m, \[Eta]s, K, hm, hs, Sa, Sm, Ss}, {t,
3 K( v& `0 o, d9 `: `9 l8 h 0, TT}]6 h+ w/ B& y: _
Plot[{Evaluate[Sa[t] /. Sol], Evaluate[Sm[t] /. Sol],
5 o) N# {1 p/ E5 @; Y1 s, n Evaluate[Ss[t] /. Sol]}, {t, 0, TT}, AxesOrigin -> {0, 0},
9 J7 W- A* `) e4 n2 ~ PlotRange -> {0., 0.8}, PlotStyle -> {Blue, Dashed, Dashing[{0.05}]}]
0 W# L, v! U8 g" w% ^5 _% ]Plot[{Evaluate[D*Sa[t] /. Sol], ) u* l V/ }+ L7 i9 D p8 O
Evaluate[(D*Sm[t] + (\[Alpha]*(gRate + \[Delta]))/(\[Rho] + . m8 G) @' S5 x
gRate)) /. Sol], Evaluate[D*Ss[t] /. Sol]}, {t, 0, TT}, 1 l7 K1 T! j0 n; Y: b- }
AxesOrigin -> {0, 0}, PlotRange -> {0., 0.8},
3 O! t. v& ?& r$ ~, { PlotStyle -> {Blue, Dashed, Dashing[{0.05}]}]% |) K$ X% l, |7 b; P' {
6 D) R8 m% p+ |3 n, g( D
2 w6 l) [5 `, M, H7 ?" {1 T/ }
+ V+ y: P8 R4 {% o- s! a! A
2 ^8 o* X/ [. H0 ^9 Y( ~$ B
Set::wrsym: Symbol D is Protected.
! T; f2 J/ d. t5 o0 E8 m8 p% O( i* {* R5 m. ]
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" p1 w) v/ \
/ ]+ |! X; W7 ]$ c2 s) C* {4 U
" N! W6 V& ]# g; L" M w. ^9 Q: ]- j0 Z& V
7 C3 _9 u! X: r2 C0 V |
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