Thanks.

So last week I've been talking with you on numerical methods for quantum mechanics, and

indeed theoretical physics provided us, the applied mathematicians and computational mathematicians,

with problems and food for thought and challenges for the last hundred years.

However, more and more, greater and greater importance is given, and not always we're

paying enough attention to this, to problems originating in other areas, in particular

in biology, biochemistry, medicine, chemistry and so on.

And the subject matter of the talk today is problems originating from that domain, and

problems like this present different challenges.

Today will be an initial foray into these challenges, but definitely there is much more

there for all of us.

So it is positivity preserving methods, not only for population models, but for many other,

as we will see, models originating in chemistry and in biology.

So this is a joint talk with my colleagues and friends, Sergio Blanes from Politecnica

de Valencia and Shev McNamara from Sydney Institute of Technology at the other end of

the world.

So of course everything is done on Zoom very safely.

And the starting point really is that problems in these areas, many problems, probably most

of the problems can be formulated by a system of ODEs.

And we look at it and say, hallelujah, we know how to solve ODEs.

This is something that has been around since the 1950s, even earlier.

We have very robust methods, we have excellent software.

Why look again at this?

So we have a system of ODEs.

However, now this should not be zero, this should be Y0.

And the system has two features.

First of all, it has mass conservation.

The sum of the values of Y remains constant as the system evolves.

So you can think about each Yi as representing a mass of say, certain chemical species, species

undergo reaction, but the total mass remains constant.

There is another interpretation that we'll see soon, and another one is that positivity

is preserved.

So both the initial condition and the exact solution as the time evolves is always greater

than full zero.

And another example where this is very, very important is when the solution Y1 up to Yd,

the solution vector is a probability distribution, a discrete probability distribution.

And we want to see how this probability evolves under the action of the differential equation.

So being probability, it is always non-negative, it always leaves in zero, one, and the sum

is one.

And in these equations, there is a structure that is really behind both mass preservation

and positivity preservation.

Later, we'll see that we can go into more general structures, but this underlying structure

is the graph Laplacian structure.

In other words, the function f of t and y can be written as a matrix of functions, a

t of y times a vector y, where the matrix function a is a graph Laplacian.

And what I mean by graph Laplacian is that the diagonal elements of a are less equal

to zero, the off diagonal elements are greater equal to zero, and the sum of elements along

every column equals one, they say equals zero.

This is a graph Laplacian.

So remember, because we'll see it again and again, non-positive diagonal, non-negative