Exercise 43 MH

For Nicholson-Bailey dynamics show that p = 1 + [log(R)/(R — 1)] and that 7 = Rlog(R)/(R — 1). Then show that since R > 1, 1 + 7>p. However, also show that 1 + 7 > 2 by showing that 7 > 1 (to do this, consider the function g(R) = Rlog(R) — R + 1 for which g(1) = 0 and show that g'(R) > 0 for R > 1) thus violating the condition in Eq. (4.12), and thus conclude that the Nicholson-Bailey dynamics are always unstable.

What biological intuition underlies the instability of the Nicholson-Bailey model? There are two answers. First, the per capita search rate of the parasitoids is independent of population size of parasitoids (which are likely to experience interference when population is high). Second, there is no refuge for hosts at low density - the fraction of hosts killed depends only upon the parasitoids and is independent of the number of hosts. We now explore ways of stabilizing the Nicholson-Bailey model.

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