The size of an undisturbed fish population has been modeled by the formula pn11 bpn a 1

Chapter 11, Problem 93

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The size of an undisturbed fish population has been modeled by the formula pn11 bpn a 1 pn where pn is the fish population after n years and a and b are positive constants that depend on the species and its environment. Suppose that the population in year 0 is p0 . 0. (a) Show that if h pnj is convergent, then the only possible values for its limit are 0 and b 2 a. (b) Show that pn11 , sbyadpn. (c) Use part (b) to show that if a . b, then limnl` pn 0; in other words, the population dies out. (d) Now assume that a , b. Show that if p0 , b 2 a, then h pnj is increasing and 0 , pn , b 2 a. Show also that if p0 . b 2 a, then h pnj is decreasing and pn . b 2 a. Deduce that if a , b, then limnl` pn b 2 a.