Question

please complete the whole question

4. A population P of organisms dies at a constant rate of a organisms per unit time, and the growth rate is proportional to the population size with the proportionality constant k A. Assume the initial population P(0) = Po. Write a differential equation that models the size of the population P(t) at ay time t. B. Write the equation from part A in standard form, and solve. (The answer must contain the terms Po, a,k but not a constant C.) C. Find limo0 P(t) if Po > k

Answer 1

Grivlen by Popalabion Pof ans ds at a Constant rate of Ov2ans per unit time. rOath Tate is Popubatien propatonal to the and the Si 3e with the Constadt POpedatien plao dilfoestal Popatatoy A)The initral a equaiogthat wodds th ? Lt) at the Sige time t Populahon Si3e at time t. A Let plt) dlenote the sate attiue tis nde The The dath de is Coustad eqals &gauisus uuit time Per The sade af dhagftu poptaen ze is gver ths le Papalashop also knoh thd the Choyg SYoM bivths &slahs. So the rate ST3e s the bith k Sise We Comes changppaldin dP kP-a

The giken initral Popuatn s3e fo the inital Condition thes enuahan. tu wodd the two, Papalation 047 as dKP-a Plo) = Po Stadoud &nm BThe equhi Pat A to S to Solve th Lo us dt kp-q eation standand forn, We Write this dP -kp dt faNad as e talt tt The Can be Cultply tu ea kt -ae kt dlp e. kt kpe alt Re wrik the eaA -Kt ae

3 bath sides twith veped We Can to t to a -kt Pe kt -e + C -K kt tu expressiug Pt +Cekt th given inital Conditioo PCo) -P, to find the Constant C k to S.lbetitatng faeives usthu P Lt) + Co-t k

CC) asympatre bahaiou tu Ce look otthe Solahion P.Po-0 a K k(o The 1s texm in th Sotchio is a Consta and term has an exponertal &grauth nate he term. Sina it is the Ppatiouality Coustart kt Positive. So thu te e ast- k is is also multipled byy This exporential ten constant K Positive, (Po- kt Constad is Sound each tein in the th behaviouY We as -ta0 . Sdation lim P)lin kt Ka +lin(Po-et lim K Ktoo Lim PLE t