Exercise 45 EM

Show that 1 has to satisfy the equation 1 = — re—1t and then explain why if r is sufficiently small there is a solution of this equation corresponding to decay towards the steady state (it may be easiest to sketch a graph with 1 on the x-axis and y = 1 or y = —re—1r on the y-axis and look for their intersections and note that if r = 0 then 1 = —r).

To further increase our intuition, note the following. Suppose we knew that n(t) = Acos(pt/2r), so that dn/dt = —(Ap/2r)sin(pt/2r). Furthermore, n(t — r) = Acos[(pt/2r) — (p/2)] and if we recall the angle addition formula from trigonometry, cos(a + b) = cos(a)cos(b) + sin(a)sin(b), we conclude that n(t — r) = —Asin(pt/2r) from which we conclude that in this specific case dn/dt = (p/2r)n(t — r). Thus, if we start with an oscillatory solution, we know that we can derive a differential equation similar to Eq. (4.27). This suggests that we might seek oscillatory solutions for the more general delay-differential equations.

An oscillatory solution would mean that we assume l = i + i!, where i is the amplitude of the oscillations and ! is the frequency of the oscillations. We take this and use it in Eq. (4.27) to obtain

1 + i! = —re—(l+i!)r = —re—¡r e—i!r = —re—¡r[cos(!r)— i sin(!r)] (4.28)

We now equate that the real and imaginary parts to obtain equations for 1 and !:

1 = —re—¡r cos(!r) ! = re—¡r sin(!r) (4.29)

We want to understand the conditions for which i < 0, which will mean that the dynamics are stable. From the first equation in (4.29), we conclude that one condition for i < 0 is that !r < p/2. Furthermore, we know that when r = 0, we have the solution i = — r, ! = 0, so that perturbations decay without oscillation. The classic result is that the steady state N(t) = K of Eq. (4.25) is stable if 0 < rr < p/2 (see Murray 2002, p. 19; I have not been able to find a better way to explain the derivation, so simply send you there). If the condition is violated, then perturbations from the steady state will exhibit oscillatory behavior. Although the logistic equation with a delay seems to be highly simplistic, it both provides insight for us and, in some cases, leads to good fits between theory and data. In Figure 4.11, I show the fit obtained by May (1974) to the data of Nicholson (1954) on the Australian sheep-blowfly

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