Clear[Am, As, Aa, \[Alpha], \[Rho], \[Theta]m, \[Theta]s, \ , I7 [' l% `* m+ y/ b$ `\[CurlyPhi]m, \[CurlyPhi]s, \[Epsilon]], M, ]* l: F8 H x
\[Gamma]a = 0.1; \[Gamma]m = 0.15; \[Gamma]s = . W: x) [) f8 {' W6 E2 }' j1 P% A
1 - \[Gamma]a - \[Gamma]m; / q) Q& v) z. B) W\[Epsilon] = 0.04; \[Alpha] = 0.3; \[Rho] = 0.04;- g- b# ~1 f! J6 ~7 g1 S+ f. v" I. O! T
\[Theta]m = 0.75; \[Theta]s = 0.9; ( R% n. o! b B) o3 j2 K( t3 @! q# u8 LgRate = 0.02;7 h& n& W3 V P
Am = (gRate + \[Rho])/\[Alpha]; Ba = 4; Bm = 1; Bs = 2.5;4 V$ `% x* j9 B& ]5 n2 Q
ps = Bm/Bs; pa = Bm/Ba; 0 S3 z' Y, }5 A, |8 d2 k) f* l\[Delta] = 0.03;: B F: i( J, j) W3 L9 t
B = \!\(TraditionalForm\`\* * V) \. C# V; {% }3 x( DFractionBox[ 3 b* J5 z) ~3 R) n7 p" H/ B, T6 yRowBox[{ & t9 h, w6 d3 T: x# U" t2 D& ARowBox[{6 W# O# i$ l3 p1 f' x& Z# X1 n, P
RowBox[{ ( F9 F" ^) H& v# pStyleBox["(",4 s2 _1 j) s" N: A: u# p! m
SpanMinSize->1.,2 [) U# ^) R) K/ G; }; S4 E! Y
SpanMaxSize->1.], 1 u, M/ q B* jRowBox[{"1", "\[Minus]", "\[Alpha]"}], ; e# K8 n4 \5 bStyleBox[")",0 i* e9 s# S# ~( t/ s
SpanMinSize->1.,* x% w* |9 A) w) i `# @2 |. _
SpanMaxSize->1.]}], "gRate"}], "+", "\[Rho]"}], \* [5 k& v5 w, C4 n
"\[Alpha]"] \[Minus] \[Delta]\);5 \: t$ V6 K6 ]2 Z7 x0 h
cap = 10;5 N$ l1 L0 Z! U4 }2 V8 D |; O
csp = (pa*cap)/ps; 9 N$ s5 s2 D* |; o) V4 {2 ED = ((1 \[Minus] \[Alpha])*# g: z' h. u. v P9 r
gRate + \[Rho] - \[Alpha]*\[Delta])/(\[Rho] + gRate);4 X! O$ t# m: t1 W" [
\[CurlyPhi]m = 0.1; \[CurlyPhi]s = 0.1;+ g" H2 M& Z: N& B% O
Print["*** Initial Values ***"] # H0 q! M! N- ?E0 = 1.5; 5 J# ?: n8 y: d; ]4 M3 Y4 o4 }K0 = E0/B; ; L2 j, c9 A) z3 }9 N: [+ Rhm0 = 0.25; hs0 = 0.25;(* initial values *) * _/ u5 ] L+ v\[Eta]m0 = hm0/K0; \[Eta]s0 = hs0/K0; 5 f2 E) i/ `. S( pxm0 = (B*\[Gamma]m^\[Epsilon]* K! m" N* {! Q: O
hm0^(\[Theta]m*(1 - \[Epsilon])))/(\[Gamma]a^\[Epsilon]*pa^(& Y* B# e9 U2 P ^6 ~/ V$ A
1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*( g6 ?8 t/ P- x E* O3 w
hm0^(\[Theta]m*(1 - \[Epsilon])) + \[Gamma]s^\[Epsilon]*(ps*1 X: U4 A0 T6 x3 I3 P# K
hs0^\[Theta]s)^(1 - \[Epsilon]));1 R9 K+ n V( y
xs0 = (B*\[Gamma]s^\[Epsilon]*(ps*! Y* P& w! w( n- x
hs0^\[Theta]s)^(1 - \[Epsilon]))/(\[Gamma]a^\[Epsilon]*pa^(; ^1 E5 s+ J9 ~ b
1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]* % P5 ~0 L: E% H* H7 f3 @% E ^ hm0^(\[Theta]m*(1 - \[Epsilon])) + \[Gamma]s^\[Epsilon]*(ps* 5 J* ?) j+ j" t q8 U* \5 G hs0^\[Theta]s)^(1 - \[Epsilon]));3 z" t3 A3 G- B) B0 H
Print["\[Eta]_{m,0}=" <> ToString[\[Eta]m0], + E1 c7 S2 o4 C8 P/ B
", \[Eta]_{s,0}=" <> ToString[\[Eta]s0], + a2 z; c$ T' ~. {
", x_{m,0}=" <> ToString[xm0], ", x_{s,0}=" <> ToString[xs0]] - G( g# c- ]7 A8 q! pTT = 100;(* end time *) # S/ j4 E) v* }(* Solve differential equations *) 8 V9 ?$ w. \# L M! ]* ZSol = NDSolve[{xs'[t] = (1 - \[Epsilon])*6 N [; e/ x7 \ n+ J S
xs[t]*( (1 - xs[t]/) K4 _9 x4 t' l I% z2 L
B)*(\[Theta]s*\[CurlyPhi]s*(xs[t]/(ps*\[Eta]s[t]) - 1) - 7 t6 ?4 V% m0 P9 m+ \9 g
xm[t]/B \[Theta]m*\[CurlyPhi]m*(xm[t]/\[Eta]m[t] - 1))), 1 ]" G7 _2 S/ }) g: s$ r, K3 G
xm'[t] == (1 - \[Epsilon])*' m0 c5 m7 [0 o3 J7 h. L" U
xm[t]*( (1 - xm[t]/ ' @# i: l& o, I: u% v. ?4 L# ] B)*\[Theta]m*\[CurlyPhi]m*(xm[t]/\[Eta]m[t] - 1) - 4 E, |* F2 F# D! F. c% D
xs[t]/B*\[Theta]s*\[CurlyPhi]s*(xs[t]/(ps*\[Eta]s[t]) - : c, j5 u a* v$ m2 I/ x
1) ), \[Eta]m'[# P3 K9 K* M8 R: [/ k
t] == \[CurlyPhi]m*; |9 \/ h5 b% N% I3 v" Z: o$ f
xm[t] - (\[CurlyPhi]m + gRate)*\[Eta]m[t], \[Eta]s'[7 h: B6 Q+ R1 r" k7 h
t] == \[CurlyPhi]s*xs[t]/ps - (\[CurlyPhi]s + gRate)*\[Eta]s[t], 2 V+ O, u( c, a8 \0 }$ s K'[t] == gRate*K[t], hm[t] == \[Eta]m[t]*K[t], 1 M4 R/ J( S6 W0 j" m2 o
hs[t] == \[Eta]s[t]*K[t], ) q8 q L6 i8 s0 d5 k$ z$ c) c! n
Sa[t] == (\[Gamma]a^\[Epsilon]*(pa)^(1 - \[Epsilon]))/(\[Gamma]a^\ 4 ?) A& G2 p: v2 Z8 n\[Epsilon]*pa^(1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]* 3 ?- ]- O6 ^4 l9 V) ^8 R0 } hm[t]^(\[Theta]m*(1 - \[Epsilon])) + \[Gamma]s^\[Epsilon]*(ps* 3 o! Q, X) z* s. {. G hs[t]^\[Theta]s)^(1 - \[Epsilon])) + (\[Gamma]m^\[Epsilon]* & S1 p1 F9 K$ E) w7 [ hm[t]^(\[Theta]m*(1 - \[Epsilon]))*pa* ! Y* l, H; R7 _ cap)/((\[Gamma]a^\[Epsilon]*pa^(; s3 f, p$ W7 O( |5 w9 j3 J+ n
1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]* , W, e3 P" t% n; k+ p% Q hm[t]^(\[Theta]m*(1 - \[Epsilon])) + \ , y3 G5 A7 _7 u1 D9 X6 X" Z. H\[Gamma]s^\[Epsilon]*(ps*hs[t]^\[Theta]s)^(1 - \[Epsilon]))*k (t)* 7 e; Z& P8 [9 @ xm (t)), . C2 t0 B3 U9 j; X. Z
Sm[t] == (\[Gamma]m^\[Epsilon]* & X2 W9 t* W: w: ]& V' O& C5 H hm[t]^(\[Theta]m*(1 - \[Epsilon])))/(\[Gamma]a^\[Epsilon]*pa^( 9 J9 ^4 g0 j( |& ^6 G2 U, H$ r 1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]* % y* h/ {1 W7 M hm[t]^(\[Theta]m*(1 - \[Epsilon])) + \[Gamma]s^\[Epsilon]*(ps*9 {% i' ]! |8 M; r$ i# G7 I
hs[t]^\[Theta]s)^(1 - \[Epsilon])), ; Q# Z7 C. a+ u/ O# c2 o4 S$ E Ss[t] == (\[Gamma]s^\[Epsilon]*(ps*$ `8 ]5 r8 _1 {9 ?2 W! Q
hs[t]^\[Theta]s)^(1 - \[Epsilon]))/(\[Gamma]a^\[Epsilon]*pa^(! [0 N- k0 Q9 F! [
1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]* , } A4 r; O" ?" g$ b/ Y hm[t]^(\[Theta]m*(1 - \[Epsilon])) + \[Gamma]s^\[Epsilon]*(ps*' o4 F3 B7 q2 j0 h, Q
hs[t]^\[Theta]s)^(1 - \[Epsilon])) - (\[Gamma]m^\[Epsilon]* ! F. Q" t9 d- l D. H hm[t]^(\[Theta]m*(1 - \[Epsilon]))*ps*3 b) h0 W+ R5 i* G- p
csp)/((\[Gamma]a^\[Epsilon]*pa^( $ F; H6 h$ W' G2 \ 1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*3 B# B* a- K: M0 }. H6 m7 G, @
hm[t]^(\[Theta]m*(1 - \[Epsilon])) + \ ; x, c) y! Z, v8 x& z( P- @1 w\[Gamma]s^\[Epsilon]*(ps*hs[t]^\[Theta]s)^(1 - \[Epsilon]))*k (t)* S" P- Q2 ]: w+ }: r
xm (t)), xm[0] == xm0, 8 W/ M6 x0 W. _" A$ g xs[0] == xs0, \[Eta]m[0] == \[Eta]m0, \[Eta]s[0] == \[Eta]s0, / f8 q% @ x0 z1 i K[0] == K0}, {xm, xs, \[Eta]m, \[Eta]s, K, hm, hs, Sa, Sm, Ss}, {t,9 U2 G3 d5 B7 Y( B* s5 q5 Y& O
0, TT}]1 Q1 @: ^. r* C1 `# A5 k, K! [8 x
Plot[{Evaluate[Sa[t] /. Sol], Evaluate[Sm[t] /. Sol], : d t \+ f. Y0 l, C Evaluate[Ss[t] /. Sol]}, {t, 0, TT}, AxesOrigin -> {0, 0}, * w2 W1 [5 U, E PlotRange -> {0., 0.8}, PlotStyle -> {Blue, Dashed, Dashing[{0.05}]}]" B2 A- Y5 l9 m( L0 H
Plot[{Evaluate[D*Sa[t] /. Sol], 8 M" {! F+ h5 D! E! i* M
Evaluate[(D*Sm[t] + (\[Alpha]*(gRate + \[Delta]))/(\[Rho] + + r" Q% x+ \. e1 f! u
gRate)) /. Sol], Evaluate[D*Ss[t] /. Sol]}, {t, 0, TT}, " N$ E( R: k. M2 A; g AxesOrigin -> {0, 0}, PlotRange -> {0., 0.8}, ! y8 }+ B7 T/ U3 J
PlotStyle -> {Blue, Dashed, Dashing[{0.05}]}]6 ^& \; H6 q& m) O9 h) M( B
8 d/ n+ c1 L% w 1 M- b- d$ s6 L/ F$ Q5 k9 P1 m- {7 h# f1 p
' D2 g% \( G. {8 s$ m
Set::wrsym: Symbol D is Protected. 2 I2 X: i% R' n/ S2 p 6 @" P: Y. X/ f6 }' p# n" FNDSolve::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.}. + s' f/ i$ s$ T7 M1 S & a, E3 k! G: z* j9 I* d$ h ]) T* t: d
. J: ~' J1 h/ G5 l& C: ^/ u. ]( h; L1 Y9 x( m. ~7 i2 T