1.  Time delayed logistic

• delays recognized by Volterra
• time delayed logstic introduced by Hutchinson

 dN/dt = rN(1-N(t-T)/K)
(1)
Let t = rt, n = N/K, \mathcal T = r T, yielding
 dn/dt = n(1-n(t-T))
(2)
which has equilibria n* = 0 or n* = 1.

To find the stability, let

 n(t) = n* + dn(t)
(3)
where
 dn(t) = aelt.
(4)
Here a is a small undetermined constant and l is the growth rate of the perturbation.

We find

 dn/dt = d (dn)/dt
 =
 n(t)-n(t)n(t- T)
(5)
 la elt
 =
 (1+a elt) - (1+a elt)(1+a el(t-T))
(6)
 =
 a elt- a elt-a el(t-T)-a2 el(t-T)elt
(7)

Ignore the a2 term to get

 l = - elT
(8)

We wish to find values of T for which all solutions l of (8) have negative real part. We first see that l = 0 is never a solution, so we ask what happens if l = iq. In this case (8) becomes

 iq
 =
 -e-iqT
(9)
 =
 -cos( qT ) + isin( qT)
(10)
Equating real and imaginary parts we get
 0
 =
 cos( qT )
(11)
 q
 =
 sin( qT)
(12)

Therefore q\mathcal T = p/2, etc., which implies q = sin(p/2), etc. Choosing the value p/2, we see that q = 1, so \mathcal T = p/2.

We thus have an upper limit to the length of delay that allows stability.

2.  Interpertation

• role of nondimensional parameter
• Nicholson's blowflies
• lemmings