Concept explainers
The size of an undisturbed fish population has been modeled by the formula
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 {pn} is convergent, then the only possible values for its Limit are 0 and b − a.
(b) Show that Pn+1 < (b/a)pn.
(c) Use part (b) to show that if a > b, then limn→∞ pn = 0; in other words, the population dies out.
(d) Now assume that a < b. Show that if P0 < b − a, then {pn} is increasing and 0 < pn < b − a. Show also that if P0 > b − a, then {pn} is decreasing and pn > b − a. Deduce that if a < b, then limn→∞ pn = b − a.
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Essential Calculus: Early Transcendentals
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