Delay DE's

• To: mathgroup at smc.vnet.net
• Subject: [mg48180] Delay DE's
• From: Virgil Stokes <virgil.stokes at neuro.ki.se>
• Date: Mon, 17 May 2004 03:21:48 -0400 (EDT)
• Organization: Karolinska Institute
• Sender: owner-wri-mathgroup at wolfram.com

```This is a followup of my earlier query on delay differential equations.

The following is code that I have used to solve a simple 1st order
non-linear delay differential equation:

k0 = 1; k1 = 1.7; tau = 1;
k1 tau
f = k1 x[t](1 - x[t - tau]/k0)
tlimit = 14.1;
NDelayDSolve[{x'[t] == f}, {x -> (0.01 &)}, {t, 0, tlimit}]

Indeed Mathematica 5.0.0.0 will find the correct solution (using the
NDelayDSolve package). This is the Hutchinson-Wright equation which is
often used in biological models. The solution when plotted with,

int = x /. %[[1]]
Plot[int[t], {t, 0, tlimit}];

shows that x[t] approaches a periodic limit cycle (as it should).
Hoppensteadt (Analysis and Simulation of Chaotic Systems,
Springer-Verlag, 1993) has described an interesting method for
integrating delay-differential equations and the complete solution (for
all non-negative t) can easily be obtained with this method.
Unfortunately, when I use NDelayDSolve to verify this result, it fails
for values of t  > 14.1.

Finally, to my question --- What can be done to "fix" NDelayDSolve to
give a more complete solution to this problem (e.g. from t = 0 to t = 50)?

--V. Stokes

```

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