TIME DELAYS

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) = ae^lt.

(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 e^lt

(1+a e^lt) - (1+a e^lt)(1+a e^l(t-T))

(6)

a e^lt- a e^lt-a e^l(t-T)-a² e^l(t-T)e^lt

(7)

Ignore the a² term to get

l = - e^lT

(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

-e^-iqT

(9)

-cos( qT ) + isin( qT)

(10)

Equating real and imaginary parts we get

cos( qT )

(11)

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

References

* 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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