Past exam of the topics in applied mathematics course of the University of Oxford / 2009 / A / 1 / b / ii

Apply the same vertical-then-horizontal cobweb construction to $H(u)=9u{e}^{-u}$. Since no numerical value of the large initial biological population is prescribed, an exact five-generation list is $u_1$, $H(u_1)$, $H^2(u_1)$, $H^3(u_1)$, $H^4(u_1)$. For the plotted example $u_1=5$, these values areThe initial slump is followed by rapid growth and an alternating overshoot about $\log 9\simeq 2.197225$, which is now unstable because its periodic-orbit multiplier $1-\log 9<-1$. A still larger initial biological population can remain extremely small for more than these four updates; the example illustrates the requested construction without imposing an unstated universal numerical starting value.