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TIME DELAYS
1. Time delayed logistic
 delays recognized by Volterra
 time delayed logstic introduced by Hutchinson
Let t = rt, n = N/K, \mathcal T = r T, yielding
which has equilibria n^{*} = 0 or n^{*} = 1.
To find the stability, let
where
Here a is a small undetermined constant and l is the growth rate of the perturbation.
We find



 (5)  

(1+a e^{lt})  (1+a e^{lt})(1+a e^{l(tT)}) 
 (6)  

a e^{lt} a e^{lt}a e^{l(tT)}a^{2} e^{l(tT)}e^{lt} 
 (7) 
 
 
Ignore the a^{2} term to get
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
Equating real and imaginary parts we get
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
References
* means required reading
*Gurney, W.S.C., Blythe, S.P. and Nisbet, R.M. 1980. Nicholson's blowflies revisited. Nature 287:1721.
*May, R.M. 1981. Models for single populations. in May, R.M., ed. Theoretical Ecology.
Turchin, P. 1990. Rarity of density dependence or population regulation with time lags? Nature 344:660663.
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