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MathGroup Archive 2006

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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) - η*x[t]*y[
t] - a1*x[t]*z[t]/(1 + α*x[t] + β*y[t] + γ*z[t]);
ra = r2*y[
t]*(1 - y[t]/K2) - a2*y[t]*z[t]/(1 + α*x[
t] + β*y[t] + γ*z[t]) - μr*y[t];
ca = -d*z[t] + ( b1*x[t]*z[t] +
b2*y[t]*z[t])/(1 + α*x[t] + β*y[t] + γ*z[t]) - μc*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 \((
r2 - μr)\)\)\/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


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