Logistic Function and Population

7

Project ll

One of the most common models of population growth is the exponential model. These models use functions of the torm p(t) : po€rt, wherep6 is the initial population and r > 0 is the rate constant. Because exponential models describe unbounded growth, they are unrealistic over long periods of time. Due to shortages of space and resources, all populations must eventually have decreasing grovtrth rates. Logistic growth models allow for exponential growth when the population is small. However, as the population approaches a critical size called the carrying capacity, the growth rate approaches zero. The result is a self-regulating population.

- Logistic Growth

We letp(r) be the population of a community or species at time t > 0, which means thatp'(r) is the population growth rate. The population growth rate for pure exponential grOwth satisfiesp'(t) ,p, where r > 0 is the rate cOnStant; that is, the growth rate is proportionalto the population size. By contrast, the growth for logistic growth is given by p' (t) = rp(l where r is the rate constant and K is the carrying capacity.

:

*),

1. The growth

rate for a logistic model,

p'(t)

:

function ofp in Figure 1 using

r :0.1

and

K:

rp(l -

{.),

Ir graphed as a

500.

!r, :1x: /'r'ltrrt j

a. For what population is the growth zero? b. For what population is the grovuth mte a maximum? c. Does the population ever decrease in size? d. What does the population approach 6s l --r m?
2.
The goal is to find the actual population functionp that has a groMh rate

given

bY

p'Q)

:

rp(l

constants. This task requires solving a differential equation, which in this case means evaluating integrals. We first separate the dependent variable from the independent variable. Dividing both sides of the equation by gives

- {'),*n"re

we assume that r and Kare specified

*, \r-Tl

i@-''
Next, integrate both sides of the above equation with respect to r .