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MathGroup Archive 2004

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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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