Counting Rabbits: an Introduction to Population Dynamics and Ordinary Differential Equations

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Counting Rabbits an Introduction to Population
Dynamics and Ordinary Differential Equations

• Dr. Ken Mulzet
• Florida Community College at Jacksonville

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Question

• How can we effectively model the growth of a
  population over time?

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First Attempt

• Observe the population at a given time
• Wait a period of time then observe the population
  again
• Keep the process going, taking samples every set
  period of time
• Plot our observations and attempt to connect them
  with a regression curve

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Results of First Attempt

• It appears the population is increasing over
  time, at an approximately exponential rate

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Mathematical Analysis

• Let P(t) represent the population at time t.
  Here t is measured in months.
• P(0) 1000 represents the initial population.
• P(6) 1100, P(12) 1221, and P(18) 1431.

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Graph of P(t)
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Mathematical Analysis part 2

• P(t) can be written as an exponential function
• Now we differentiate

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Our First Differential Equation

• Therefore P(t) satisfies the following
  differential equation
• We can combine this with an initial condition

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Various Initial Conditions
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Benefits of Growth Equation

• As time increases so does the population
• The population increases at a fixed rate, i.e.
  22 per year
• The model fits the data reasonably well
• The function satisfies the initial value problem

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Drawbacks of Growth Equation

• The population can grow without limit as time
  goes on i.e. there is no upper limit on its size
• We could expect a sharp initial increase in the
  population size, to be tempered as time goes on,
  perhaps reaching a finite limit after a certain
  period of time

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Second Attempt

• We introduce the so-called logistic curve, which
  attempts to overcome the deficiencies of the
  previous model
• This curve solves the initial value problem

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Solution to the Logistic Problem

• The solution to this initial value problem can be
  written in the form
• K is referred to as the saturation level or
  environmental carrying capacity

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Graph of the Logistic Function

• The population grows rapidly at first, then
  levels off
• The limiting value appears to be 1500

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Different Initial Populations

• When the population starts at greater than 1500
  rabbits, we see a decline
• The population approaches 1500 as time goes on

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Properties of the Solution

• If the population starts above the saturation
  level, the population drops until it reaches this
  value
• If the population starts below the saturation
  level, the population grows until it reaches this
  value, regardless of the initial size of the
  population
• If the population starts at the saturation level,
  it remains constant for all time. This is
  referred to as an asymptotically stable
  equilibrium solution to the initial value problem.

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Threshold Values

• We introduce the concept of a threshold value,
  which requires the population to start at a
  minimum level in order to survive
• This value will be represented by T

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Threshold Value IVP

• We consider the following initial value problem

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Solution to the Threshold Problem

• This problem has the solution
• The graph of this function for various initial
  populations decreases when the population starts
  below T and increases when the population starts
  above T.

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Various Initial Populations

• The red graph exhibits an asymptote for a finite
  time value
• As time approaches this value the population
  grows without limit

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Drawback of Threshold Model

• If the population starts above the threshold then
  it will increase without limit, and if it starts
  at the threshold value it will neither increase
  nor decrease
• It seems that a combination of the threshold
  value and saturation level is necessary

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A Hybrid Model

• We consider the initial value problem
• using the threshold value of 750 and saturation
  level of 1500. Again r .016.

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Graph of Solutions

• This model exhibits well behaved solutions for
  any initial population

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Solution to the Hybrid Problem

• The graph of the solution has three equilibria
• y 0, y T and y K
• y 0 is a semistable equilibrium, y T is an
  unstable equilibrium, and y K is a stable
  equilibrium

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Advantages of the Hybrid Model

• This model successfully incorporates both the
  saturation level and threshold value discussed
  previously
• Regardless of the initial population, the final
  population is limited
• The maximum value of the population is the
  saturation level

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References

• William E. Boyce and Richard C. DiPrima (2005).
  Elementary Differential Equations and Boundary
  Value Problems (eighth edition), pp 78 88.