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Clear[Am, As, Aa, \[Alpha], \[Rho], \[Theta]m, \[Theta]s, \
8 g/ j& M, F( ] j- i\[CurlyPhi]m, \[CurlyPhi]s, \[Epsilon]]: F- Y: Z2 R# x' j& [2 s
\[Gamma]a = 0.1; \[Gamma]m = 0.15; \[Gamma]s = # d9 v0 L1 @; m5 a$ E; b) J# f
1 - \[Gamma]a - \[Gamma]m;
o& K# v7 H0 A/ x; {$ `& Q( r2 @\[Epsilon] = 0.04; \[Alpha] = 0.3; \[Rho] = 0.04;
0 Q& B* z: {' g8 i& Q m8 ?\[Theta]m = 0.75; \[Theta]s = 0.9;9 |; f* t5 G) I/ ^
gRate = 0.02;
7 \7 {7 z) t4 d$ g8 Q" P% uAm = (gRate + \[Rho])/\[Alpha]; Ba = 4; Bm = 1; Bs = 2.5;
( G+ H4 q* f; I8 \1 J$ F4 F/ cps = Bm/Bs; pa = Bm/Ba;/ L0 Q9 O$ N& K- ~ W9 v5 B
\[Delta] = 0.03;
" g w9 J" x: k/ F. u/ {4 J) U4 ?B = \!\(TraditionalForm\`\*
2 S, S* y. ?1 b6 z) @& FFractionBox[% Y, W1 R# }$ K0 {( W( o
RowBox[{# s" g: G/ t' Y: q% e) Z% x
RowBox[{
) u' C" f- ?; }% ?7 I: h, aRowBox[{, j/ e9 [+ `$ r& @9 C" o) y
StyleBox["(",. N7 W& Q- Z& h8 j. b( D, R. |0 u$ j, u
SpanMinSize->1.,
- O" d, B/ R7 }SpanMaxSize->1.], # x, f0 D4 E( I/ i P
RowBox[{"1", "\[Minus]", "\[Alpha]"}],
* `1 f9 E/ E" [7 m3 p$ Y( QStyleBox[")",
& r. q' C. j: ~: k# k! j: HSpanMinSize->1.,% D- [0 S" _; \* O/ _4 h
SpanMaxSize->1.]}], "gRate"}], "+", "\[Rho]"}],
2 o- {# N/ |+ Z C% ~: K5 ` "\[Alpha]"] \[Minus] \[Delta]\);
& H, O R4 @) X4 F1 B* Zcap = 10;/ q* l: J3 b" u+ j% t2 J
csp = (pa*cap)/ps;
+ I" o- t; l* @6 L& T3 [$ F& y5 UD = ((1 \[Minus] \[Alpha])*
$ }& |7 N- j6 n8 @ gRate + \[Rho] - \[Alpha]*\[Delta])/(\[Rho] + gRate);; p4 \( f5 a% u& B: a! q* F
\[CurlyPhi]m = 0.1; \[CurlyPhi]s = 0.1;
8 g# c/ J, Y) x( e! E) vPrint["*** Initial Values ***"]
" M4 c# L* j( U+ l. ]8 n u4 FE0 = 1.5;
- z- a% B9 e$ wK0 = E0/B;
; m# u0 h7 u& M- T* Qhm0 = 0.25; hs0 = 0.25;(* initial values *)' W. ~% b# K/ Q& d) A$ a
\[Eta]m0 = hm0/K0; \[Eta]s0 = hs0/K0;
! J' H. `$ y. ]$ `2 Rxm0 = (B*\[Gamma]m^\[Epsilon]*
, y6 T6 ]# N9 W# s1 m hm0^(\[Theta]m*(1 - \[Epsilon])))/(\[Gamma]a^\[Epsilon]*pa^(
; O E' ]* B5 |/ o 1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*
) f/ S! g" A9 d; a! C3 F. P hm0^(\[Theta]m*(1 - \[Epsilon])) + \[Gamma]s^\[Epsilon]*(ps*
5 H7 B1 `! l) M. K% v hs0^\[Theta]s)^(1 - \[Epsilon]));; s, X3 j7 ^5 [: q$ x* Z9 ?
xs0 = (B*\[Gamma]s^\[Epsilon]*(ps*
3 d! M8 O- B8 t: T: B hs0^\[Theta]s)^(1 - \[Epsilon]))/(\[Gamma]a^\[Epsilon]*pa^(
3 b( g$ g/ n- H9 O/ {$ I3 S# \ 1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*
, G# U& W' x& P3 Z hm0^(\[Theta]m*(1 - \[Epsilon])) + \[Gamma]s^\[Epsilon]*(ps*, U. K, g6 y6 e
hs0^\[Theta]s)^(1 - \[Epsilon]));7 S; ?. n- J# O0 `4 Q( r; k# Q
Print["\[Eta]_{m,0}=" <> ToString[\[Eta]m0], ! e) ^: B. Z1 B8 @
", \[Eta]_{s,0}=" <> ToString[\[Eta]s0],
' D! R8 [/ O- y8 S$ f0 L0 s ", x_{m,0}=" <> ToString[xm0], ", x_{s,0}=" <> ToString[xs0]]' c/ W+ O; U1 b' R- L) j
TT = 100;(* end time *)
+ ~ I/ P s. R' ]4 @(* Solve differential equations *)
4 M7 D9 M2 |% L. k( v+ v ]Sol = NDSolve[{xs'[t] = (1 - \[Epsilon])*; [7 }2 G3 Y' K# N0 l5 j
xs[t]*( (1 - xs[t]/
" Q2 I( O1 E. @# a/ Y( D8 q B)*(\[Theta]s*\[CurlyPhi]s*(xs[t]/(ps*\[Eta]s[t]) - 1) - 9 j- _$ u& h3 R/ v% g0 a3 T
xm[t]/B \[Theta]m*\[CurlyPhi]m*(xm[t]/\[Eta]m[t] - 1))), ; }0 t. e4 r/ B- k& {, e& C A
xm'[t] == (1 - \[Epsilon])*- w- i' M0 \: c$ |9 S _
xm[t]*( (1 - xm[t]/0 \" q/ m' ^+ n4 A( H3 d% }
B)*\[Theta]m*\[CurlyPhi]m*(xm[t]/\[Eta]m[t] - 1) -
+ W9 k7 f; H% R$ j xs[t]/B*\[Theta]s*\[CurlyPhi]s*(xs[t]/(ps*\[Eta]s[t]) -
2 W( J! O3 e: n0 } 1) ), \[Eta]m'[
2 O2 K' O( x& v! f t] == \[CurlyPhi]m*+ e; Q) n3 ~ g6 [' ~' i, t' k
xm[t] - (\[CurlyPhi]m + gRate)*\[Eta]m[t], \[Eta]s'[
7 ?2 P, c/ k+ y, V* `; Z4 L t] == \[CurlyPhi]s*xs[t]/ps - (\[CurlyPhi]s + gRate)*\[Eta]s[t],
: e/ v W: ~+ b2 a5 [5 \ K'[t] == gRate*K[t], hm[t] == \[Eta]m[t]*K[t], / }3 ?" y6 _1 O% \/ F$ @
hs[t] == \[Eta]s[t]*K[t], $ s5 N0 b- ]% k" H" H" G4 ^, ^8 y) |
Sa[t] == (\[Gamma]a^\[Epsilon]*(pa)^(1 - \[Epsilon]))/(\[Gamma]a^\9 G* V. `* t$ U. F' G8 D: f3 h
\[Epsilon]*pa^(1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]* d l6 L; N/ n, p ^
hm[t]^(\[Theta]m*(1 - \[Epsilon])) + \[Gamma]s^\[Epsilon]*(ps*
B% z- x' r7 J, X$ b2 S$ b hs[t]^\[Theta]s)^(1 - \[Epsilon])) + (\[Gamma]m^\[Epsilon]*% i5 L' G$ v b, e% S
hm[t]^(\[Theta]m*(1 - \[Epsilon]))*pa* z) J6 N B7 B/ m# W3 K
cap)/((\[Gamma]a^\[Epsilon]*pa^(1 R4 p3 {7 T/ v' y6 x9 k) @ I9 W+ @
1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*7 d' Q, o7 n+ k( A( n% [
hm[t]^(\[Theta]m*(1 - \[Epsilon])) + \6 ]: n; E" l/ l: q& u: X: [/ g8 g
\[Gamma]s^\[Epsilon]*(ps*hs[t]^\[Theta]s)^(1 - \[Epsilon]))*k (t)*
2 J( P1 k) n; u6 M. G& Q6 b; z0 r8 z( R5 F xm (t)),
) V" v- q) c0 }: l! E( w9 z/ D; M Sm[t] == (\[Gamma]m^\[Epsilon]*
+ a6 L7 Q9 K% D: `2 r5 F hm[t]^(\[Theta]m*(1 - \[Epsilon])))/(\[Gamma]a^\[Epsilon]*pa^(
1 `' d2 I \2 b5 j4 W+ n9 s 1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*
# Q3 Z. d( @$ q% @ hm[t]^(\[Theta]m*(1 - \[Epsilon])) + \[Gamma]s^\[Epsilon]*(ps*, o( r. }' p$ o- M; a+ }% s
hs[t]^\[Theta]s)^(1 - \[Epsilon])), & H8 o7 a: _; o% H4 X+ p8 X: v
Ss[t] == (\[Gamma]s^\[Epsilon]*(ps*1 f- z2 h) H) B) q% z1 r2 a
hs[t]^\[Theta]s)^(1 - \[Epsilon]))/(\[Gamma]a^\[Epsilon]*pa^(' ]9 D/ [, Z* w& `/ H
1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*/ ]0 z4 U y! ~6 r
hm[t]^(\[Theta]m*(1 - \[Epsilon])) + \[Gamma]s^\[Epsilon]*(ps*1 m2 k7 }: ~* N. b" z
hs[t]^\[Theta]s)^(1 - \[Epsilon])) - (\[Gamma]m^\[Epsilon]*( e* I" L$ _5 ~/ ]# U
hm[t]^(\[Theta]m*(1 - \[Epsilon]))*ps*
7 [: @4 t0 Q! s9 Z8 @3 I* O' b2 v csp)/((\[Gamma]a^\[Epsilon]*pa^(
* j+ c0 B+ z# y2 j 1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*
! J( n/ F8 y1 {! A7 V, o hm[t]^(\[Theta]m*(1 - \[Epsilon])) + \3 w$ t+ D& r, W
\[Gamma]s^\[Epsilon]*(ps*hs[t]^\[Theta]s)^(1 - \[Epsilon]))*k (t)*7 [, K5 L9 C; S
xm (t)), xm[0] == xm0, 4 j* b& t% z% e7 L6 g2 i) Y
xs[0] == xs0, \[Eta]m[0] == \[Eta]m0, \[Eta]s[0] == \[Eta]s0,
/ G' p; L4 ]3 }1 @' `) n/ V3 A1 y K[0] == K0}, {xm, xs, \[Eta]m, \[Eta]s, K, hm, hs, Sa, Sm, Ss}, {t,
2 a" v$ ?6 y+ F( Z& g 0, TT}]
# ^% f; u; A8 V+ a8 U0 dPlot[{Evaluate[Sa[t] /. Sol], Evaluate[Sm[t] /. Sol],
; k& \ J- E; W# R+ _. \5 G% S" E Evaluate[Ss[t] /. Sol]}, {t, 0, TT}, AxesOrigin -> {0, 0}, $ X& g8 I' V. u1 j7 c% a% u
PlotRange -> {0., 0.8}, PlotStyle -> {Blue, Dashed, Dashing[{0.05}]}]
2 z( t, m& Z9 e: ~Plot[{Evaluate[D*Sa[t] /. Sol],
( A# Y( H% H8 v9 z2 k6 X Evaluate[(D*Sm[t] + (\[Alpha]*(gRate + \[Delta]))/(\[Rho] + 2 c* E& o1 Q' l4 C
gRate)) /. Sol], Evaluate[D*Ss[t] /. Sol]}, {t, 0, TT},
3 }8 @( a+ W' T, _& ?$ g7 t0 ] AxesOrigin -> {0, 0}, PlotRange -> {0., 0.8},
; Y" J6 `3 C# l& U) a3 h: Z- n PlotStyle -> {Blue, Dashed, Dashing[{0.05}]}]
) {3 r8 V- m3 U5 i8 o6 Z5 l$ w; N" j: M( ]
7 h0 \8 _5 Z. t' \1 h
5 K9 T- A& p0 _$ d/ d9 N) C) x- P. z) l' C7 b$ I; _5 P$ K+ n' G
Set::wrsym: Symbol D is Protected., e0 W2 \1 Q! {$ N5 t/ l* }
' ?! O+ j- q4 K% `& L5 y
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.}.
/ N" c5 d8 N5 S( A
P+ H- x8 h. n, d" l; ]% ?" {
: S% w7 {, c- z( @$ [
) h; x9 z+ g- Q( J3 T- ~+ T9 \6 Q# r7 F' r* n3 O$ H: T! g1 H
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