So, can you please accept the recording?

Yes, I can hear.

Please, can you make your microphone, please?

Okay.

So, welcome everyone.

Today we have Professor Laurent Deliet from the Université de Paris, and he will be speaking

about some aspect of cost diffusion equation.

We're looking forward to it.

Okay, so thanks a lot.

Thanks for this invitation.

It's always a pleasure to give a seminar, even if it's online.

And so let me just say one word about this strange name of Université de Paris.

Actually it's a new name and it comes from the merging of Paris 5 and Paris 7.

Okay, so today I will present a certain number of results which were...

So it's Paris 12.

Yeah, or 6 if you take the average.

I mean, I can decide.

So anyway, I will present a certain number of results

which were obtained in collaboration with colleagues.

So in France there was Thomas Le Poutre, Ayman Moussa and Ariane Trescassès.

Actually the UPMC for Ayman Moussa is Paris 6, but it is now called Sorbonne Université.

So as you can see, the name has changed two times since I first used those slides.

Anyway, there are also three German speaking colleagues who were involved.

So two in TU Wien, so Esther Daus and Ansgar Jungl.

And one who is Helge Dietert, who is currently in Leipzig,

but who has also a position in Université Paris-Diderot,

which is a former name of Paris 7.

And so it also changed two times since the slides were written.

Okay, so let me start with a brief historical description of some of the models

which are used in population dynamics.

And I would like to first say that it seems that the first work on population dynamics

is the one which is due to Malthus at the end of the 18th century.

And they are basically proposed that the typical growth of a population is exponential.

So this is represented by the red line here.

And of course, very soon people realized that the growth can be exponential

only on a certain scale, only for a certain amount of time.

And then at some point, the resources will not be sufficient for a population to grow forever.

And so one should introduce a term in the equation for the growth of the population,

which doesn't make it possible for the population to grow forever.

And so this was seemingly introduced first by Ferhulst in the mid-18th century.

And his idea was to replace the equation of Malthus, which would just be u prime equal r0 times u,

by an equation in which you also multiply by a quantity which is 1 minus u over k,

where k is sometimes called the carrying capacity.

And it means that this is a maximum population which can be sustained by the resources which are available.

And so here u is basically the number of individuals in the population.

So if you believe in this small ODE, you end up with a curve,

which is sometimes called the sigmoid or logistic curve, which is a curve which is in blue.

And as you can see on the slide, it first starts as an exponential.

So at the beginning, it's really close to the red curve.

And then after some point, the effect of the lack of resources can be felt.