Problem:
A population of rabbits on an island is growing at a rapid rate. The population is
currently 100 rabbits and is doubling every month due to a lack of predators and
abundant resources. However, the island's resources are finite, and the carrying
capacity for rabbits is estimated to be 1000 rabbits. The island's resources can only
support up to this number of rabbits.
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Create a mathematical model to describe the population of rabbits on the island
as a function of time (in months).
Use your model to predict when the rabbit population will reach its carrying
capacity.
How long will it take for the rabbit population to reach 80% of the carrying
Discuss the limitations of your model and any real-world factors that it doesn't
account for.
Note: The problem involves exponential growth and logistic growth, which are common
concepts in mathematical modeling when dealing with populations and limited
resources.
The population growth of the rabbits on the island can be modeled using
exponential growth, which can be described by the following equation:
P(t)=P0×ert
Where:
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P(t) is the population of rabbits at time t.
P0 is the initial population of rabbits (100 in this case).
e is the base of the natural logarithm (approximately 2.71828).
r is the growth rate.
t is time in months.
Given that the population is doubling every month, the growth rate �r can
be calculated as follows:
r=1ln(2)
Now, the carrying capacity K for the island is 1000 rabbits. To account for this,
we can modify the exponential growth equation to include the carrying
capacity:
�(�)=�×�0×����0+(�−�0)×���P(t)=P0+(K−P0)×ertK×P0×ert
Where:
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�K is the carrying capacity (1000 rabbits).
This modified equation ensures that as the population approaches the
carrying capacity, the growth rate slows down. The population growth will be
rapid initially, but it will gradually approach the carrying capacity over time.
(1000*100*e^(ln(2)*x))/(100+(1000-100)*e^(ln(2)*x))
y=100×2^x