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Welcome again to this third week of the class,

Simulation and modelling of natural processes.

In this module we would talk about an example of continuous dynamical system.

This example is the growth of a population.

So, what is the starting point?

We have population of any kind,

it can be bacteria, people, what you want.

And this population, we just count the number of individuals.

It will be described by P(t), which will be just the number of these individuals.

At some starting point, t = 0,

we have a given number of individuals, which is P(0).

We also assume that we have only two processes affecting

this population, which is the births and the deaths.

So the variation of P can be simply expressed as a differential equation, so

the variation of P is simply the births minus the deaths.

So, stated in a differential equation is that

time derivative of P is simply B minus D.

So now the thing we must to do is to give a model for

how B and D will evolve in time to be able to solve this problem.

So the most basic assumption we can

make is that the number of births is simply proportional to the population,

and the rate of births is a constant, which is given by rb.

So this is on the bottom of the slide, the left equation.

The number of deaths can be modeled exactly the same way, but

instead of having a coefficient which is rb, we choose another one which is rd,

so rd, which will be the rate of deaths.

So, if we put these two equations back in our

differential equation for the evolution of the population, what do we get?

We get the variation of the population, P dot, is equal to r,

which is the difference between rb and

rd times our population.

This differential equation is very easy to solve analytically.

And with P(0), the initial condition of this differential equation,

we end up with P(t), which is equal to P(0) times the exponential of r times t.

Of course, this model is not very accurate.

It has mainly three possible behaviors.

So the first one is that r, so the rate of variation of r populations,

is bigger than 0, so you have more births than deaths,

then your population will simply increase exponentially fast.

If, in turn, r is smaller than 0, which means that there are more deaths than

births, you see that your population will decrease exponentially fast to 0.

Finally, we have the case where the rate of births is

equal to the rate of deaths, and then your population will simply stay constant.

Now, we can make this model a bit more realistic because it's not very accurate.

You can imagine that you have a limited amount of food,

that the disease will spread faster also.

So if you

had only a very limited amount of people and

plenty of food then you can imagine they will reproduce very fast.

In turn, if you have a lot of people with a very low amount of food,

they will also reduce very fast and

reach some balance, some target population.

So, here, we model this target population with the number,

C, which can depend on many parameters, the amount of food available or,

I don't know, what kind of disease you can have in your system.

It can depend really on a lot of things.

The idea is that our population rate,

variation r, will go to 0 as P goes to C.

So we will replace our constant, r, by

our first equations here, which is v, a constant, times 1 minus P over C.

So here you see that when P is equal to C then r is equal to 0.

So now we simply replace this equation here in the model for

the evolution of our populations, and we have that the variation of P, P dot,

is equal to v times 1 minus P over C times P.

This differential equation has an analytical solution,

also, which is a logistic function, and you have the formula on the left.

I will not read it for you.

But on the right-hand side of the slide what you see is

different initial values for

a specified set of parameters of our population evolution.

So here at the target, population is set to 50.

And we start our model with either P(0) is equal to 10,

so smaller than C, or with P(0) equal to 19,

which is bigger than the target population.

So, in red you have the P(0) equal 90,

and in blue the P(0) equal 10.

What you see is that when you are above the target population,

then your population will decrease toward our target population.

On the other hand, when you are below, your population will grow and

reach, synthetically, our target population.

So, this is summarized in this slide.

What is interesting to note is that there are two points which are fixed points,

which means where if you are close to these points you will not move at all.

So if you are on C or on 0 then your population will not evolve with time.

It will just stay at this value.

Nevertheless, if you move a little bit around these points,

the behavior is very different.

So if your start very close to the fixed point, 0,

no matter what you do your population will

vary fast and reach the target population.

On the other hand, if you move a bit from the value C

with your population then you will be immediately attracted back towards C.

So these two fixed points have two very different behaviors.

I will not talk more about fixed point theory here, but these are keywords

you may want to remember if you want to study a bit deeper these kind of systems.

So with this I end this module, and in the next module we will try to

generalize a bit what we saw here and talk about balance equations.

Thank you for your attention.

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Growth of a population

Simulation and modeling of natural processes

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