Entry #: Lesson 7.52 Logistic Equations Exponential growth modeled by y = Cekt assumes unrealistic for most population growth. More typically the growth rate decreases as the populat'on \nqXeaBQS Qs the population growslnd there is a maximum population. This is modeled by the \OSilS\rg differential equation # = kp(U - P). rhe sotution equation is or the rorm p = =f "tt} Ja"v\'..st\:o"ci\) Note: Unlike in exponential growth equation, C is not the initial amount. Example: A national park is capable of supporting no more than 1 00 grizzly bears. We model the equation with a logistic differential equation with k : 0.001. a. Write a differential equation. *=,oo\a(\oo-a) b. The slope field for this differential equation is shown. Where does there appear to be a horizontal asymptote? \{A Q Q=0 a Q=\00 a*\\.r$ trkuAhon+ a X?il,"hti eSpqq,.r a\e What happeris'if the starting point is above the asymptote? r\ \he r\avhrrq D$\r.* \t akm.le \e a\\m$\s\e", \he SoSu\a\-r*,c, Un\irn\\ ed growth and is __ t' rju l t 40\ \ .lu \ lrh i,l eu1 I eo/ / tCI/ I 7ilrr 111111\1 \\\\\11\ \\\\\\\\ \\\\\\\\ \\\\\\\\ :::::::: t/ t/ l{ td tl tl tt I I / I I decreatet r\fi{1\ r\ \ep\oac\',et \kre f A'( \ \$\\fl\ f n\.)frr^ r i r, '..I{'re \', ,\: "rbi\ftet) What happensTf the starting pbint is below the asymptote? Lq \he' t'\ at \nq ?Crrr,*; \a \t'1,':',,,1 i.."i.,{.' .-,. r'., ,. . 5' , \(ret Q0pr,r\1.\.(,'i'\ .. (., f t\'*( \\'i\\ r\ app\ e'ac\'{:, \\re C'6\ \ \rt.t c h\-,'rr. r i "i g\nen, t\abr\ite-s)