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Lesson 7.52 Logistic Equations

Exponential growth modeled by y = Cekt assumesunrealistic for most population growth.

More typically the growth rate decreases as the populat'on \nqXeaBQS Qs thepopulation 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 theequation 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 horizontalasymptote?

\{A Q Q=0 a Q=\00a*\\.r$

trkuAhon+ a

X?il,"hti eSpqq,.r a\eWhat happeris'if the starting point is above theasymptote?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 lt40\

\.lu \

lrhi,l

eu1Ieo/

/tCI/

I7ilrr

111111\1\\\\\11\\\\\\\\\\\\\\\\\\\\\\\\\::::::::

t/t/l{

tdtltlttII/II

decreatet r\fi{1\ r\ \ep\oac\',et \kref 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)

{t .. lf the park begins with ten bears, sketch a graph of P(r) on the slope field.

d. Solve the differential equation to find P(t) with this initial condition. _\t \\ --*A'{-P.*dQ N(\\D(rnn-n\ -L tf=Aa+t + Ari.,\tt ,t+=tQ' n,)\s

b=to Jte-\eDtP' " I a- A, r_L / \oot^ ^ -!tot

*-mr = * n' Gs\ fr ='bb\ p (uu -p) a rrrl:,)r*f:, :.-: . I 1 ="Grq\';i';:o'"""' .1:'::' ;- ; \ ':t '# \+= c"g5\=N S'ke-\aGv \ ,--,,-q-, \ n I v--rou

b0(\\qQ--'o*s(t+t0-k*) * \!!*^ * \\0.rv-, n^"!i

\.. Lq q- to'= _\ {

e. lnstead of solving the differential equation, use the general form of a logisti. * = * ] ^equation P : #to find the same solution. ^ \u\J

0 = \\o \N - -S9- \N.. \', = \Iq\, \* tfloorr.u(1.it t \\ = l* c ^ _ 9.0^o A

\sa / t=\ \ -"J3Io.\\\ =\t((ffi\ -r v ' ,o__$: \.r\- ^^t\qY )'\o'

f . Find lim P(t) - 1gg

X H ov \ eon\ a\ at$r$-?\ o \ ea\ Q*\00"

ob' When will the bear population reach 50? \-L \dto-= \ \=-\\\n\q)\ {_-\u'\ . \n(\) 1 ., 1_\ srr_h t\ = \*.' (\i) tiq) \_**\1-t

:L\r.

., {-|)\\\q a

5\ \\\Vr,ax Q P =5S

5Se-i l

.s."-(h. When is the bear population growing

*(SJ = &(n.w -u.rr',?*) a# ='s\ * " u'toLa o+

)q = Bto.\ -a t\Lv) -/d\' dt '-

the fastest.