LOGISTIC EQUATIONS

Sect. 5-L(6-3 in book)

Logistic EquationsExponential growth modeled by assumes unlimited growth and is unrealistic for most population growth. More typically the growth rate decreases as the population grows and there is a maximum population M called the carrying capacity. This is modeled by the logistic differential equation .

ktCey

)( PMkPdtdP

Logistic EquationsFor The solution equation is of the form

Note: Unlike in the exponential growth equation, C is not the initial amount.(Proof page 429: book uses L for upper limit or carrying capacity)

)( PMkPdtdP

kMtCeMP

1

Logistic Equations1) A national park is capable of supporting

no more than 100 grizzly bears. We model the equation with a logistic differential equation with

a. Write the differential equation.

001.0k

b. The slope field for this differential equation is shown.

Where does there appear to be a horizontal asymptote?

What happens if the starting point is above this asymptote?

What happens if the starting point is below this asymptote?

c. If the park begins with ten bears, sketch a graph of P(t) on the slope field

d. Solve the differential equation to find P(t) with this initial condition.

e. Now use the general form of the logistic equation to find the same solution

f. Find

kMtCeMP

1

)(lim tPt

g. When will the bear population reach 50?

h. When is the bear population growing the fastest?

2) A 200 gallon tank can support no more than 150 guppies. Six guppies are introduced into the tank. Assume that the rate of growth of the population is

where t is in weeks.

a. Find a formula for the guppy population in terms of t.

)150(0015. PPdtdp

2) b. How long will it take for the guppy population to be 100?

c. How long will it take for the guppy population to be 125?

HOME WORK Worksheet 5-L and page 432 # 71-75, 77 a,b,d, 79, and 80