Clear[Am, As, Aa, \[Alpha], \[Rho], \[Theta]m, \[Theta]s, \ ; g8 E* z, U0 G+ D$ P\[CurlyPhi]m, \[CurlyPhi]s, \[Epsilon]]% M+ P& O! T& z2 p- j' n9 K
\[Gamma]a = 0.1; \[Gamma]m = 0.15; \[Gamma]s = 1 T4 Z j9 T8 T' P8 w+ i" u
1 - \[Gamma]a - \[Gamma]m; O- G! a0 D% r6 m
\[Epsilon] = 0.04; \[Alpha] = 0.3; \[Rho] = 0.04; 2 h- H8 a' {+ n% }\[Theta]m = 0.75; \[Theta]s = 0.9; 6 e; m- D6 c$ c0 X9 M# W0 @7 HgRate = 0.02;* R/ L+ C% ?" p! m4 V- S/ V0 Q
Am = (gRate + \[Rho])/\[Alpha]; Ba = 4; Bm = 1; Bs = 2.5; % K3 |( `* P" ~4 xps = Bm/Bs; pa = Bm/Ba;4 d5 [0 m* _7 c( |5 U( ~
\[Delta] = 0.03; 9 Z7 f* t7 c( o0 KB = \!\(TraditionalForm\`\*! ]: z3 e `% E
FractionBox[ H7 K7 J0 [4 {' C/ r9 R9 Q
RowBox[{: U- f' v( h& `" }' B* ^) P! V
RowBox[{ : {8 x; A9 i5 x5 ` [RowBox[{0 l+ s4 o: x# c% S2 L8 k3 v
StyleBox["(", ) C4 V6 @8 g& ]# Z; v& V4 A$ XSpanMinSize->1.,; j n- C s2 ?" Q
SpanMaxSize->1.], 6 Q; x, Y/ |- |, N2 }: P* t2 [
RowBox[{"1", "\[Minus]", "\[Alpha]"}], 2 A. T* N. A% U( K0 YStyleBox[")", 9 z4 _ ]# x- f- R# V% f* H, c% pSpanMinSize->1.,0 |. o# K4 `$ w3 Z, p
SpanMaxSize->1.]}], "gRate"}], "+", "\[Rho]"}], + v% e3 @ r8 t) J/ {1 V "\[Alpha]"] \[Minus] \[Delta]\);9 n' @* V- [ }0 v. N
cap = 10;0 Z( q: E5 v9 g, r2 \
csp = (pa*cap)/ps;( {# P: s# Q- `1 [
D = ((1 \[Minus] \[Alpha])*& R6 j6 ^% U0 s# j, Z, P3 G) x
gRate + \[Rho] - \[Alpha]*\[Delta])/(\[Rho] + gRate); 7 C _& ?+ n7 ^1 x\[CurlyPhi]m = 0.1; \[CurlyPhi]s = 0.1; U& q: e: \, d
Print["*** Initial Values ***"] 0 F. p8 j& a& p* J2 p- n! W, \E0 = 1.5;! G4 H! a0 I. k0 U5 S& |8 J" G
K0 = E0/B;1 L- H+ j4 ]- u7 _! G) |% S1 k
hm0 = 0.25; hs0 = 0.25;(* initial values *)1 @2 O: k8 y# a: f5 p9 X( _' Y
\[Eta]m0 = hm0/K0; \[Eta]s0 = hs0/K0; : r5 B X5 @' K3 x: d7 i8 R& Kxm0 = (B*\[Gamma]m^\[Epsilon]*/ v$ ?9 |% p U# c$ x
hm0^(\[Theta]m*(1 - \[Epsilon])))/(\[Gamma]a^\[Epsilon]*pa^(5 H5 |! {6 D. F& Z' X5 @
1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*% R2 X9 N: b) e( p7 `
hm0^(\[Theta]m*(1 - \[Epsilon])) + \[Gamma]s^\[Epsilon]*(ps*1 m5 }& H" Z. q& t/ D% v; d
hs0^\[Theta]s)^(1 - \[Epsilon]));3 y } e; w g+ {: d
xs0 = (B*\[Gamma]s^\[Epsilon]*(ps* 3 q% f; E% [7 \) v# v/ g hs0^\[Theta]s)^(1 - \[Epsilon]))/(\[Gamma]a^\[Epsilon]*pa^(3 i: F3 u+ V2 V# @. Y: o
1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]* , Q4 S' w1 {* N+ ] u* j* C hm0^(\[Theta]m*(1 - \[Epsilon])) + \[Gamma]s^\[Epsilon]*(ps*! b8 }- ?6 t6 V
hs0^\[Theta]s)^(1 - \[Epsilon]));9 w; }& }. k, s H+ x( E
Print["\[Eta]_{m,0}=" <> ToString[\[Eta]m0], . J$ |3 J) S3 w3 @+ ? ", \[Eta]_{s,0}=" <> ToString[\[Eta]s0], 2 V0 M {- J2 V) X' Y ", x_{m,0}=" <> ToString[xm0], ", x_{s,0}=" <> ToString[xs0]] 6 j5 X) H$ h& ?7 R" \TT = 100;(* end time *)8 |! P r- F6 n7 p1 M
(* Solve differential equations *)- x% e( H6 A7 N
Sol = NDSolve[{xs'[t] = (1 - \[Epsilon])* $ K; |* a; \6 a, d! ]& k xs[t]*( (1 - xs[t]/4 S) y2 C9 M4 V
B)*(\[Theta]s*\[CurlyPhi]s*(xs[t]/(ps*\[Eta]s[t]) - 1) - ! [8 W5 {2 d6 Q% B, t
xm[t]/B \[Theta]m*\[CurlyPhi]m*(xm[t]/\[Eta]m[t] - 1))), 3 K# Z% ]! ^$ l# E f+ ?8 p& r
xm'[t] == (1 - \[Epsilon])* % E. c/ f. ~/ b( |$ u [: x xm[t]*( (1 - xm[t]/ 1 O; D6 Q0 ?- |, f7 @$ {' v B)*\[Theta]m*\[CurlyPhi]m*(xm[t]/\[Eta]m[t] - 1) - 0 ^$ D! ], I2 X- D6 K. R. Q4 X/ U xs[t]/B*\[Theta]s*\[CurlyPhi]s*(xs[t]/(ps*\[Eta]s[t]) - 6 g4 g# b9 D9 i u. e8 [# H" ?
1) ), \[Eta]m'[ 8 _& h* V, v, a9 G t] == \[CurlyPhi]m* , }" Q" R% a0 }8 X0 @: a xm[t] - (\[CurlyPhi]m + gRate)*\[Eta]m[t], \[Eta]s'[ / y8 Q. q G9 g0 M3 f" \! Q! w. Y t] == \[CurlyPhi]s*xs[t]/ps - (\[CurlyPhi]s + gRate)*\[Eta]s[t], X6 U& u& A. W& i K'[t] == gRate*K[t], hm[t] == \[Eta]m[t]*K[t], 8 v, Y* |& U1 D* q* t: N hs[t] == \[Eta]s[t]*K[t], # W# H+ g: [/ D. }5 W9 J Sa[t] == (\[Gamma]a^\[Epsilon]*(pa)^(1 - \[Epsilon]))/(\[Gamma]a^\6 d+ j5 S( Y2 D/ F' D
\[Epsilon]*pa^(1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]* : m2 F5 x' ~: B* @/ R: x hm[t]^(\[Theta]m*(1 - \[Epsilon])) + \[Gamma]s^\[Epsilon]*(ps* 1 n8 J l+ g& M hs[t]^\[Theta]s)^(1 - \[Epsilon])) + (\[Gamma]m^\[Epsilon]*2 g) f7 L% z, i7 s+ {
hm[t]^(\[Theta]m*(1 - \[Epsilon]))*pa* \1 d" x4 Q" l cap)/((\[Gamma]a^\[Epsilon]*pa^(8 k4 w1 F @) l# y+ o% P% }/ O
1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*7 t8 O# m! H/ |1 [
hm[t]^(\[Theta]m*(1 - \[Epsilon])) + \ " K# t: A; k5 W- [4 K\[Gamma]s^\[Epsilon]*(ps*hs[t]^\[Theta]s)^(1 - \[Epsilon]))*k (t)*+ u: n4 E* u9 [6 t. I
xm (t)), 9 P' b1 c1 O" A$ ? Sm[t] == (\[Gamma]m^\[Epsilon]*+ n- i: w9 }" ~# X4 j: L
hm[t]^(\[Theta]m*(1 - \[Epsilon])))/(\[Gamma]a^\[Epsilon]*pa^( 6 Z' F* |7 `5 F+ i3 M% r 1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]* : T) H# g! D" ~% E: S$ O hm[t]^(\[Theta]m*(1 - \[Epsilon])) + \[Gamma]s^\[Epsilon]*(ps* ! G' H- Q# Y" F' l; e hs[t]^\[Theta]s)^(1 - \[Epsilon])), ) L( T& g8 r3 ]) q
Ss[t] == (\[Gamma]s^\[Epsilon]*(ps* - C# s+ U0 p7 h/ h: Q% F, k" {0 { hs[t]^\[Theta]s)^(1 - \[Epsilon]))/(\[Gamma]a^\[Epsilon]*pa^( ' Y# @ k% G1 R' ^ 1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]*# \* U8 ]- ~: }! J, M* C6 v
hm[t]^(\[Theta]m*(1 - \[Epsilon])) + \[Gamma]s^\[Epsilon]*(ps*7 l! D8 H; r8 r' t& ~) K; D. p8 O
hs[t]^\[Theta]s)^(1 - \[Epsilon])) - (\[Gamma]m^\[Epsilon]*1 m9 a! U1 ?1 o! M
hm[t]^(\[Theta]m*(1 - \[Epsilon]))*ps*' k) D: p+ ]2 S5 r& R& M, ~9 R1 x& E
csp)/((\[Gamma]a^\[Epsilon]*pa^( 0 x. @: \. s- a3 V& B: y 1 - \[Epsilon]) + \[Gamma]m^\[Epsilon]* / l w! u! h G' h: C hm[t]^(\[Theta]m*(1 - \[Epsilon])) + \ * D, e2 q( a. J% c2 s/ y7 Q\[Gamma]s^\[Epsilon]*(ps*hs[t]^\[Theta]s)^(1 - \[Epsilon]))*k (t)*9 S9 Q4 @! J+ y1 T0 ^
xm (t)), xm[0] == xm0, $ w* _7 X1 k8 U
xs[0] == xs0, \[Eta]m[0] == \[Eta]m0, \[Eta]s[0] == \[Eta]s0, 5 c5 F' q* Z9 [" M# g9 H( O
K[0] == K0}, {xm, xs, \[Eta]m, \[Eta]s, K, hm, hs, Sa, Sm, Ss}, {t,4 o( X$ y7 [- ~% {( [
0, TT}]8 q( e8 v) D( o T! u. T7 N' [
Plot[{Evaluate[Sa[t] /. Sol], Evaluate[Sm[t] /. Sol], $ t0 w: _/ T. H) W Evaluate[Ss[t] /. Sol]}, {t, 0, TT}, AxesOrigin -> {0, 0}, , X# S: i5 y3 \, U
PlotRange -> {0., 0.8}, PlotStyle -> {Blue, Dashed, Dashing[{0.05}]}]1 D! ?9 T# m+ B; Y% E6 M9 x6 l, {
Plot[{Evaluate[D*Sa[t] /. Sol], & E" e4 [3 G" G5 u- t
Evaluate[(D*Sm[t] + (\[Alpha]*(gRate + \[Delta]))/(\[Rho] + ' u3 c! U6 t. [ gRate)) /. Sol], Evaluate[D*Ss[t] /. Sol]}, {t, 0, TT}, ; S# `6 I5 E8 m8 @/ ]% g AxesOrigin -> {0, 0}, PlotRange -> {0., 0.8}, " |" q0 _+ t' C4 f6 ~& q5 I PlotStyle -> {Blue, Dashed, Dashing[{0.05}]}] % c3 J0 }+ T7 X0 Q, k5 y7 e 7 W7 J8 A- M) q3 M4 z9 ]! i. u) Y7 s0 u, d4 D T3 J
# X4 y( T- f1 M: O* W
( A+ o% Q0 b. A$ l0 G3 NSet::wrsym: Symbol D is Protected.. I6 F6 S( q" A
$ g5 W7 q8 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.}.7 G2 o- _2 b8 d5 I. u; g
. X5 s" ?; J5 P8 e5 m0 | , u( }8 }, U# q" F$ F/ J" A! G \ O. O g6 N! K: u7 K " ]) E7 J+ h4 a P8 H) n" a, l