2:08

In this second interval for the whole interval the graph must be always increasing.

But depending on this small interval or that small interval, concavity may changes.

If your initial population is between these two,

then it should be always a concave down and the increasing.

So you get graph of this type;

Increasing and the concave graph.

If your initial population is between 0 and r over 2a,

then up to a certain time

your graph is always increasing but your concavity may changes, somewhere in between.

So, increasing concave up and the changes concavity,

still increasing by concave down.

That's the shape of the general logistic curve.

You can read out some others things too from this graph.

For example, any logistic curves starting from the initial point

Po which is greater than zero approaches the equilibrium solution why?

the P is equal r over a but it

never reaches the value r over a at the finite time by Picard's theorem.

Can you figure it out?

why can touch the line P is equal to r over a In any finite time?

Here I claim the following.

If we started from any point,

starting from any point

which Po is a positive and look at the red curves and blue curves,

both of them they approach is to this equilibrium position,

equilibrium state as t turns to infinity.

But I claim that,

it never reaches the value r of a by the Picard's theorem.

Why is it then? Lets assume that for example,

here is the P axis and here is the time axis.

Here's the equilibrium solution,

which is the line p is equal to r over a.

That's the equilibrium solution.

Let's take any one of them for example this one.

Lets assume that solutions starting from somewhere from there,

this is a Po.

If it touches the value r over a as a finite time at a finite time say To.

If it can touch it and keep going away,

consider the following the initial value problem,

p prime is equal to (r-nP)P and p of

To that is equal to r over a. P of To is equal to exactly this equilibrium value.

Consider this initial value problem.

It satisfies all the conditions required by the Picard's theorem. What does that mean?

By the Picard's Theorem,

this initial value problem should have

a unique solution defined [inaudible].

There should be only one solution satisfying this initial value problem.

But as I assume if you have a solution starting from

this point and it touches the point To,

r over a at the finite time To then,

In any n value for To you have two distant solution.

This is the solution,

equilibrium solution, another solution given by this curve.

Can you see it right?

You have at least two different solution.

They cannot coincide because one

represents a straight line and the other one is a genuine curve.

That's a contradiction, because

for the given initial value problem you have two distinct solution,

which is impossible by the Picard's Theorem which says,

certain initial value problem should have one should have only one solution.

That's why there is such a solution starting from any positive initial value

approaches to this equilibrium solution as t times to infinity and never touches it.

Moreover, if your initial point Po

is between 0 and r over

a, so moving here.

If your initial population is between these two,

then your solution, your integral curve is these blue curves.

Their approach is to the value r over a as to test infinity but never touches

it and always should lie below this equilibrium line.

This horizontal line.

So, this value is P is equal to r over a is

a kind of upper limit for all those solutions.

So we call this number r over a.

It's very reasonable to call this number r over

a the carrying capacity or the saturation level of this environment.

So far we get all those informations about

the logistic curves without solving the given logistic differential equation explicitly.

We get all those informations on the logistic curves by the qualitative analysis.

But in fact in this case,

it's easy enough to solve this logistic equation.

The logistic equations say p prime is equal to r minus ap

times p. Because as I said before this is a separable differential equation.

So, why not separate the variables?

Divide it through by r minus ap times p and the multiply dt on both sides,

you will get this one.

One over one minus ap times the pdp that is equal to dt.

Let's take a partial fraction decomposition, one over one mines ap, times p,

which is one over r over p,

plus a over r over r minus ap.

That's a partial fraction decomposition of this quotient here.

We know what to the next right?

Take integration of both sides,

you are going to get,

from the left you will get,

one of r times log absolute value of p,

plus one of r times log,

negative one over r times

the log absolute value of a over r minus ap.

So by the log rule,

you are going to have the left hand side,

one over r log of the absolute value of p over minus aP,

that should be equal to integral of dt.

That is the T plus some integral constants of one.

So, we can get to the general solution of

the given logistic differential equation very simply through these manipulations.

The general solution of this logistic equation is given by p of t is equal to r times

C over AC plus exponential negative r of t. C is arbitrary constant now.

The initial condition P of zero is called a Po,

which I assume to be non-negative.

You can compute this C in terms of Po,

that is C is equal to Po over r minus a times Po.

Plugin this quantity for C into this expression and simplify,

then you will get P of t is equal to r times Po over a times Po,

plus r minus a times Po,

times e to the negative rt.

This is the general solution of this logistic differential equation.

The function given in here.

The equation number five,

we called it as the logistic function.

Using this explicit expression of p of t you can

confirm our previous consequences obtained by

qualitative analysis of the differential equation easily.

You can check all the things that we get through the table,

by using this expressive form of the solution.