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Lecture

# How Populations Grow.docx

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University of Toronto St. George

Biology

BIO120H1

Paul Thompson

Fall

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How Populations Grow: The Exponential and Logistic Equations
Introduction
Mitosis: 2x the population from the last time step
In the article, the species Paramecium is being observed in the case of population growth
Exponential Equation
Exponential equation is a model to be used for the growth of a single population
Best used for single celled organisms which reproduce asexually by cell division.
Examples: a bacterium or a ciliate
N(today)= 2N(yesterday) N(t) = 2N(t-1) N(t) = 2tN(0)
o N(t) = 2tN(0) in words: From however large the initial population is at t=0, the
population now (or the time that we’re calculating for) is simply the initial
population times 2 times the steps of growth it has gone through.
o This equation works for the situation: if one splits into two and two into four
N(t) = RtN(0) is the general equation where R is the multiple factor by which the
population increases.
o Ex. If an organism divides two times (4 individuals) and one dies every time, then
the equation will be N(t) = 3t(0)
R = “finite rate of population increase”
o Can be expressed in any way including 2.71828… = Euler’s Constant = e
rt
o When R = e, equation N(t) = e representing continual growth (no definite
time steps)
Gives you the growing population with a given time
Knowing that ln(e ) = x, lawn both side to change the equation to ln[N(t)] = rt
The constant “r” = intrinsic rate of natural increase
Very close to many populations’ growth model
This equation is used to calculate the time needed to reach a certain
population in exponential growth
o Note: if it’s (^) hard to understand, check out
http://betterexplained.com/articles/demystifying-the-natural-logarithm-ln/
Reminder:
Function Derivative
( )
( ) ( ) ( ) ( )
Differential Equation
o Solves for rate of growing in respect of t
( )
o rate of change is constant (“intrinsic rate of increase”)
( )
o Per capita = per person, /individual
How does the average change in numbers for an individual.
Exponential equation is useful in predicting simple population in a relatively short time
Example: Crops
o Threats to population: herbivores, pests
o Methods to prevent: pesticides
But is realized to be harmful for human
Set a limit to a threshold to the population of the pests at which the
farmers can start using pesticides to control the pest population
o Knowing the R of the population growth would have been really useful to
estimate when to start.
Density Dependence Modifies the Exponential Equation
Simple population is not that prevalent, there is always factors in the nature which controls
the population growth.
Below are the factors which limit population growth:
Intraspecific Competition

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