Transcript Slide 1

Modeling with Differential Equations
1- Models of Population Growth
One model for the growth of a population is based on the assumption that the
population grows at a rate proportional to the size of the population. That is a
reasonable assumption for a population of bacteria or animals under ideal
conditions.
Let’s identify and name the variables in this model:
t = time (the independent variable)
P = the number of individuals in the population (the dependent variable)
The rate of growth of the population is the derivative
So our assumption that the rate of growth of the population is proportional to
the population size is written as the equation
where k is the proportionality constant.
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Example:
A population of protozoa develops with a constant relative growth rate of
0.7944 per member per day. On day zero the population consists of two
members. Find the population size after six days.
Solution:
The relative growth rate is
So,
Then:
Thus:
Example:
A bacteria culture starts with 500 bacteria and grows at a rate proportional
to its size. After 3 hours there are 8000 bacteria.
(a) Find an expression for the number of bacteria after hours.
(b) Find the number of bacteria after 4 hours.
(c) Find the rate of growth after 4 hours.
(d) When will the population reach 30,000?
Solution:
So,
Therefore,