Welcome to Calculus? I'm Professor Greist.

We're about to begin Lecture 24, Bonus Material.

Let us consider a differential equation model for population dynamics.

This time, a model that's a bit more sophisticated.

The [UNKNOWN] model of exponential growth without bound.

This is the so-called Logistic Model and it states that the rate of change of a

population size, P, is proportional to P, by some constant, r, the reproduction

rate. But there's another term on the right

hand side as well. This is negative c times P squared.

That is, there is a death rate c that operates in front of a quadratic term in

P. Now, if we were to factor this right hand

side, pulling out c times P and having left over a constant kappa minus P, then

of course, this kappa would be the ratio of the reproduction rate to the death

rate, r over c. Factoring in this way is going to allow

us to apply the method partial fractions. For when we do separation we get on the

left dPP over P times kappa minus P and on the right, c.

Integrating both sides and applying the method of partial fractions to the left

hand side, we get a decomposition of 1 over p times kappa minus p.

As 1 over kappa, times 1 over p, plus 1 over kappa times 1 over kappa minus P.

Now, what happens when we perform the integral?

Well, on the right hand side we have the interval of cdt.

Let's move the kappa over there and multiplying both sides of the equation by

that constant, and then reducing c times kappa to r.

So that on the right hand side we get rt plus constant, and on the left hand side,

we get log of P minus log of kappa minus P.

Now, when we go to solve this, we're going to have to do a bit of algebra.

Combining those logarithms together, and then exponentiating both sides gives P

over kappa minus P equals e to the rt plus a constant.

We can, by the usual trick, pull that constant out in front.

From which we see that the constant is equal to the initial population P not

over kappa minus P not. That's what we get evaluating at t equals

zero. Now, the only thing that's left to do is

a bit of algebra to solve for P. When we do that multiplying through and

expanding out the multiplication, we collect terms on the left.

And with a little bit of effort, we see that P is Kappa times C e to the rt over

1 plus C e to the rt. Substituting in our value for the

constant C and simplifying gives a final answer of kappa P not over quantity kappa

minus P not times e to the negative rt plus P not.

That looks a bit complicated. Let's think about what is happening.

Let's let t go to infinity and see what happens to the population size.

Does it grow without bound. At what rate, does it grow?

Well, it's pretty obvious that this has a finite limit.

In fact, if we set the right hand side of the differential equation equal to 0 and

solve for the equilibria, we see that there's an equilibrium at 0.

And an equilibrium at kappa. And indeed, taking the limit of our

explicit solution, we can see that as long as your initial condition is

non-zero, the solution tends to kappa. We can see this easily, if we plot P dot

versus P. This is a quadratic, with the coefficient

in front of the quadratic term, being negative.

It definitely passes through the origin and it crosses the horizontal axis at

Kappa. Therefore, we see that kappa is a stable

equilibrium, and 0 is an unstable equilibrium.