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

displaymath23

1.
Is the above differential equation separable?
2.
Is the differential equation autonomous?
3.
Is the differential equation linear?
4.
Without solving the differential equation, give a sketch of the graph of P(t).
5.
What is the long-term behavior of the population P(t)?
6.
Show that the solution is of the form

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Find A and B.
Hint: Use the initial condition and the result of 5.

7.
Where is the solution's inflection point?
Hint: This can be done without using the answer of 6.
8.
What is special about the growth rate of the population P(t) at the inflection point (found in 7)?

Answer:

1.
This differential equation is autonomous, i.e. the variable t is missing. Therefore, this equation is indeed separable.
2.
The answer is Yes! (see 1.) Every autonomous differential equation is separable.
3.
The equation is not linear because of the presence of tex2html_wrap_inline92 .
4.
The graph of the solution P(t) is as follows:


5.
Clearly, because of the initial condition (see the graph of the solution below), we have

displaymath96,

200 being the carrying capacity.

6.
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

displaymath98

Then, the non-constant solutions can be generated by separating the variables

displaymath100,

and the integration

displaymath102.

Next, the left hand-side can be handled by using the technique of integration of rational functions. We get

displaymath104,

which gives

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Hence, we have

displaymath108.

Easy algebraic manipulations give

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where tex2html_wrap_inline112 .
Therefore, all the solutions are

displaymath114

where C is a constant parameter.

Remark: We may rewrite the non-constant solutions as

displaymath118,

where a and B are two parameters. If we use the conditions

displaymath124

we will be able to get the desired solution. Indeed, we have

displaymath126

Thus, tex2html_wrap_inline128 and consequently tex2html_wrap_inline130 .

7.
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

displaymath136,

where we used the chain rule and the fact that tex2html_wrap_inline138 . Since our solution is not one of the two constant solutions we are only left with the equation

displaymath140

This simplifies to

displaymath142,

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.

8.
The growth rate at t=100 is maximal.

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Author: Helmut Knaust

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