L OGISTIC E QUATIONS Sect. 8-6 continued. Logistic Equations Exponential growth (or decay) is modeled by In many situations population growth levels off.

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Presentation transcript:

L OGISTIC E QUATIONS Sect. 8-6 continued

Logistic Equations Exponential growth (or decay) is modeled by In many situations population growth levels off and approaches a limiting number M called the carrying capacity. In this situation the rate of increase or decrease is modeled by the logistic growth equation.

Logistic Equations For The solution equation is of the form Note: Unlike in the exponential growth equation, C is not the initial amount.

Logistic Equations 3) The population of fish in a lake satisfies the logistic differential equation Where t is measured in years and a)

b) The logistic equations graph 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? What is the range of the solution curve?

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

d) Now use the general form of the logistic equation to find the same solution e) Find

f) For what values of P is the solution curve concave up? Concave down? g) Does the solution curve have an inflection point?

4) A population of animals is modeled by a function P that satisfies the logistic differential equation where t is years. a) If p(0)=20, solve for P as a function of t

b) Find P when t = 3 years c) How long will it take for the animal population to be 80 animals?

H OME W ORK Worksheet 8-6-B