Population Dynamics: Answer to Example1
Example: Let P(t) be the population of a certain animal species.
Assume that P(t) satisfies the logistic growth equation
- Is the above differential equation separable?
- Is the differential equation autonomous?
- Is the differential equation linear?
- Without solving the differential equation, give a sketch of
the graph of P(t).
- What is the long-term behavior of the population P(t)?
- Show that the solution is of the form
Find A and B.
Hint: Use the initial condition and the result of 5.
- Where is the solution's inflection point?
Hint: This can be done without using the answer of 6.
- What is special about the growth rate of the population
P(t) at the inflection point (found in 7)?
- This differential equation is autonomous, i.e. the variable t is missing. Therefore, this equation is indeed separable.
- The answer is Yes! (see 1.) Every autonomous differential equation is separable.
- The equation is not linear because of the presence of .
- The graph of the solution P(t) is as follows:
- Clearly, because of the initial condition (see the graph of the solution below), we have
200 being the carrying capacity.
- Let us solve this equation (use the technique for solving separable equations). First, we look for the constant solutions (equilibrium points or critical points). We have
Then, the non-constant solutions can be generated by separating the variables
and the integration
Next, the left hand-side can be handled by using the technique of integration of rational functions. We get
Hence, we have
Easy algebraic manipulations give
Therefore, all the solutions are
where C is a constant parameter.
Remark: We may rewrite the non-constant solutions as
where a and B are two parameters. If we use the conditions
we will be able to get the desired solution. Indeed, we have
Thus, and consequently .
- From the graph (see 4.), the graph of P(t) has an inflection point between 0 and 200. Let us find it. Since we must have P''(t) = 0, we get
where we used the chain rule and the fact that . Since our solution is not one of the two constant solutions we are only left with the equation
This simplifies to
which gives P=100 (half-way between 0 and 200).
Remark: You still need to convince yourself that t=100 is indeed the inflection point, that is the second derivative changes sign at that point.
- The growth rate at t=100 is maximal.
S.O.S MATHematics home page
Do you need more help? Please post your question on our
S.O.S. Mathematics CyberBoard.
Copyright © 1999-2017 MathMedics, LLC. All rights reserved.
Math Medics, LLC. - P.O. Box 12395 - El Paso TX 79913 - USA
users online during the last hour