Equilibrium points for 3 nonlinear ODE

*To*: mathgroup at smc.vnet.net*Subject*: [mg70135] Equilibrium points for 3 nonlinear ODE*From*: Virgil Stokes <vs at it.uu.se>*Date*: Thu, 5 Oct 2006 03:32:28 -0400 (EDT)

I am trying to find the equilibrium points for this system of 3 nonlinear ODEs: vars = {x[t], y[t], z[t]}; bi = r1*x[t]*(1 - x[t]/K1) - Î·*x[t]*y[ t] - a1*x[t]*z[t]/(1 + Î±*x[t] + Î²*y[t] + Î³*z[t]); ra = r2*y[ t]*(1 - y[t]/K2) - a2*y[t]*z[t]/(1 + Î±*x[ t] + Î²*y[t] + Î³*z[t]) - Î¼r*y[t]; ca = -d*z[t] + ( b1*x[t]*z[t] + b2*y[t]*z[t])/(1 + Î±*x[t] + Î²*y[t] + Î³*z[t]) - Î¼c*z[t]; eqns = {x'[t] == bi, y'[t] == ra, z'[t] == ca} I then give, equi = Solve[{bi == 0, ra == 0, ca == 0}, vars] which gives: \!\({x[t] -> 0, y[t] -> 0, z[t] -> 0}, {x[t] -> 0, z[t] -> 0, y[t] -> \(K2 \(( r2 - Î¼r)\)\)\/r2}, \) with another very, very, ... very long result. These first two solutions are indeed correct; but, I do not get the solution {x[t]->K1,y[t]->0,z[t]->0}, which on examination of the first equation (bi) is clearly correct. And I believe that these are the only solutions for equilibrium. How can I "tweak" Mathematica to give me all three correct solutions? --Thanks, V. Stokes