Some results we obtained recently in collaboration with Alberto Bressano, who was my postdoc mentor,

Najima Salehi, postdoc and Penn State with me at the same time, and Stefano Bianchini

from CESA.

And we've been working on the control of spread of invasive biological species.

So let's start from the very beginning.

So for those of you who are not familiar with invasive species, what is it?

What is this species?

It's an energetic organ, an active organism, which spreads for artificial or natural reasons

in an environment and creates trouble to human health, economy, or simply the environment.

And this is something that we experience in our everyday life.

If we think, for instance, to the Bible episodes telling about the Egypt plague, or how many

countries are still facing the malaria, which spreads through mosquitoes.

But the idea behind the project, which is very close to my heart, is a possible application

for the eradication of Xylella fastidiosa.

Xylella fastidiosa is a plant disease that is spreading in southern Italy, and it's infecting

olive trees and really undermining the oil production in my land.

So this is not simply a particular picture, it's the skyline in Puglia almost everywhere.

So the natural choice, the question is, what is the type of equation involved in this modeling?

Well the natural choice is for a reaction-diffusion equation in which the unknown fraction is

our u here and represents the population density.

And then we have a term here, f of u, representing the rate of reproduction, a diffusion term,

and then we have this piece here, which depends on u, the population, and half, which is going

to be our control term in the control problem.

So since I come from hyperbolic conservation laws, I may be tempted in future during the

talk to call this rate of reproduction flux.

I apologize, no, it's not a flux, it's just a rate of reproduction.

So which is the possible choice for this g alpha?

So first of all, g alpha is going to represent the amount of population that we eradicate

through our optimal strategy, through our control alpha.

So alpha can be imagined, for instance, in the case of the dedication of a plant disease

as a pesticide.

And the choice would be, for instance, g equal to alpha, which means that simply the amount

of population we are eliminating is proportional to the amount of pesticides that we are using.

Or a more natural choice is alpha u, which means simply that we are putting more pesticides

where we have a larger density of population.

And about the rate of reproduction, the two possible choices, the two cases that we studied

are monostable case in which f is concave, increasing in 0 and decreasing in 1, and by

stable in which we have f of u that is convex between 0 and star, concave but not really

between u star and 1, and then it's decreasing in 0, increasing in u star, and decreasing

again in 1.

So why between 0 and 1?

Well, because in the end of the story, we care about 0, which means free from contamination,

and 1, which means a full contamination.

So all this assumption on the sign, on the increasing and decreasing function f will

play an essential role in the classification of the equilibrium point.

So the same equation, so the same reaction diffusion equation with this control piece

may be used for different type of modeling.

For instance, in the harvesting problem, you can be imaging as the population of fish,

and we are seeking for a strategy alpha in order to maximize the amount of harvest biomass

minus the cost for harvesting.