Question

A population of bison can increase its numbers by 10% per year. Let Bn be the population count after n years and assume that h bison are removed from the herd just before the end of each year. (a) Relate the number of bison from one year to the next year by finding an equation for Bn+1 in terms of Bn: (b) Find B, if Bo = 1000 and h = 20. (c) Is this removal strategy of 20 bison per year sustainable in the long term without driving the bison population to extinction? Why? (d) Again assuming that Bo = 1000, find the largest h so that B10 > 1500.

Answer 1

@ Bn = population count after in' years for nt year the Bison population increases by 104. of Bn and h bisons are removed hance Buti= Bn+ 10% of Bnth Buti= Bn+ 0.1 Bnth Buti= 1.1 Bn-h 6 from part @ we can write Bn=1.1 Bnrith Bn=1.1 (1.1 Bn-zh-h Bn=1.12 BM-27h (1+1.1) Bn=1.12 (1.1 Bn5h) -h(1+1.1) Bn = 1.13 By-s-h (1+1.1 +1.12) expanding similely we get Br = 1.,M Bo-h (1+1+1+12+...+171-1) Br = 1.1" Bo-(1.7P-) (using geometric Selis) 0.1

OO and 20 علم hance for Bo=1000 and h= 20 we have Br = ln" (1000) – 20 (1-1) Br = (1.1)" (1000) – 200 (1-1) Bm= 800 (1-1)"+200) ③ we have B = 800 (1.1)9 +200 when we remove 20 Bison per pel Ch=20) obseve that Bn is increasing with n hence Bn will not become sero hence this removal Strategy is fustainable in Both (1.8M-1) (d) we have in general Bn=11" Born Bn = (1000) (1-1)-10h((1-1)-1) Bio Z 1500 = 1000 (1:1)'Oloh (1.- 1) 1500 2593.742 - 15.93742h 2 1500 - h = 68.62 . Largest h for Bio 21500 is [h=68] 0.1