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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
Here a is a small undetermined constant and l is the growth rate of the perturbation.
(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
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.
- role of nondimensional parameter
- Nicholson's blowflies
* means required reading
*Gurney, W.S.C., Blythe, S.P. and Nisbet, R.M. 1980. Nicholson's blowflies revisited. Nature 287:17-21.
*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 344:660-663.
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