04 - SN Partial Differential Equations and Applications Webinar
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Join Yihong Du of University of New England as he considers the Fisher-KPP equation with free boundary and "nonlocal diffusion". He shows the problem is well-posed, and its long-time dynamical behavior is governed by a spreading-vanishing dichotomy. Moreover, he completely determines the spreading profile, which may have a finite spreading speed determined by a semi-wave problem, or have infinite spreading speed (accelerated spreading), according to whether a threshold condition on the kernel function is satisfied. Furthermore, for some typical kernel functions, he obtains sharp estimates of the spreading speed (whether finite or infinite). Don’t want to miss another webinar? Register here: https://go.sn.pub/lXgEaxHave an article on the Fisher-KPP equation? Submit Your Article: https://go.sn.pub/GT525VLearn more about publishing with SN Partial Differential Equations and Applications: https://go.sn.pub/UFulhi
00:09
To speak in these webinars series.
00:14
And I.
00:17
I think I'll just.
00:22
Have started with a brief outline of the talk,
00:27
so I'll start by giving very briefly review.
00:32
Uh, for results on free boundary problem with the local
00:37
diffusion.
00:38
That's what we started at 1st,
00:41
and then I'll just briefly discuss.
00:44
So what's the difference between local and nonlocal diffusion?
00:50
So I assume that some of the audience tonight or
00:53
for me is the night tonight.
00:55
But I know people in different places that might be
00:59
different time of the day.
01:02
So anyway, because some of the audience may not be
01:06
an expert in this area,
01:08
so I'll just mention some basic effects on the difference
01:13
between local and nonlocal diffusion,
01:17
and then I'll talk about some.
01:20
A recent model we worked on is a windy through
01:24
boundary model with the nonlocal diffusion and then explain some
01:29
of the main results.
01:30
We obtained the recently and then compare the results between
01:36
problem with the local diffusion and the problem with the
01:41
nonlocal diffusion here.
01:43
So now let's start with these brief review for local
01:48
diffusion problems.
01:50
Now let's start with this very classical one,
01:54
so this is usually called the Fisher KP equation.
01:59
When the nonlinear function FU takes a special type or
02:04
mention in the next slide.
02:07
So this is a very simple parabolic equation in one
02:13
space dimension.
02:15
And this is basically a koshy problem,
02:18
so we are giving an initial function.
02:21
We want to know the behavior of the solution U.
02:26
And it started with these.
02:29
Kind of application. And so motivation,
02:34
so here. So we need to started.
02:37
When Fisher used these model,
02:40
he used to these U the unknown function to represent
02:45
the population density.
02:49
Of some species that are carrying good genes?
02:54
And data.
02:56
Time T and space location XX.
02:59
Knowledge simplified to be in just R1.
03:07
And this initial function U0 represented the initial population,
03:12
so naturally is a non negative function.
03:16
And it becausw. So these are model was proposed to
03:20
describe the spatial spreading of the species,
03:24
and so it's natural to assume initially the population concentrate
03:30
on a bounded part of the space.
03:33
So this are so in mathematically say it has compact
03:38
support.
03:39
And the growth of function F of U.
03:43
And usually we assume you see one function and F
03:49
0 = 0.
03:50
So if we think of it as a Ragusan term,
03:53
so when the population density is 00 when the density
03:58
of a chemical species 0,
04:00
there's no reaction. So this is pretty natural.
04:05
And so these term du Double X.
04:10
So these represent the diffusion.
04:13
So in the model is assumed that the individuals in
04:18
the species they move in space randomly.
04:24
Now for these nonlinear function's official,
04:28
just used this one, you times 1 minus you.
04:31
These quadratic function very simple.
04:35
And so KPP stand for Kolmogorov,
04:38
Petrofsky and peace can off in the same year.
04:42
They also considered these same.
04:47
Equation.
04:49
And with a a nonlinear function more general than this
04:52
one,
04:52
but behaving similarly.
04:55
A striking feature of this equation,
04:58
one is of the following,
05:01
so you can prove there exist a constant C stop
05:05
positive so that so.
05:07
As T goes to Infinity,
05:09
the population, the density of the population is close to
05:16
1.
05:16
If these are set in this region,
05:18
so this is a Regent,
05:20
well in one space dimension,
05:22
this is just the interval with the writer and close
05:26
to see star times T.
05:30
And the left aimed at monasteries.
05:34
Older side of these interval the population is close close
05:38
to 0.
05:39
Now considering the interval is a changing with the time.
05:44
So in the right hand side,
05:46
so the rate of a change is roughly see star.
05:51
So because of that, people use that interpret this as.
05:57
Seeing that the population spread into new space with the
06:03
speed system.
06:04
Now of course we want to know how to determine
06:08
these assist are well,
06:10
this is determined by a criminal wave problem,
06:14
so that was how these things started with the Fisher
06:19
and called a cat pee pee.
06:21
When you look forward, special solution is at fault.
06:27
For this equation. Here with a few these particular form.
06:33
Any confined, so here, so these are,
06:35
say, is a parameter. You look for solution in that
06:39
form,
06:39
you change the partial differential equation to odie.
06:45
And because you are looking for well now we call
06:48
to traveling wave solutions.
06:50
If you have such a solution,
06:52
you say the solution is a traveling wave with speed
06:56
C.
06:56
Well, you can show that foresee gruetli equal to these
07:01
sister well with a few in this special form you
07:05
can calculate this study to Formula Two Square Root D.
07:10
Force recruiters are equal to the deceased are.
07:13
You can find such a solution before.
07:15
See lesson 6 dot you don't.
07:18
So for that original $6 called the minimal wave speed
07:22
of traveling waves to these problems.
07:25
Now, these earlier slide tells us that.
07:29
Well, this model tells us that.
07:31
So the population spread with the speed sister and the
07:36
stock can be calculated in this way to square root
07:41
AD.
07:41
So this is a pretty nice.
07:45
So you use a very simple model,
07:47
then it tells you a spreading speed,
07:50
which is a very important notion in biology.
07:55
Now of course this is not that perfect,
07:58
and in many situations you want to know well if
08:02
the population is spread waves at the very front,
08:06
geometrically. Well, if we follow.
08:10
Meaning of the.
08:13
Solution you use the population density,
08:16
then the population range at Time TA Little time is
08:21
given by data set,
08:22
so all the X values in R such that U
08:26
is positive.
08:27
So you change TV's set to change.
08:30
Well actually this set doesn't change because of the Marks
08:35
and principle leader said once T is greater than zero
08:40
is always the entire R.
08:42
Though you start with the.
08:44
The initial function initial population concentrate on a bounded set.
08:51
Well, this does not spoil the thing for application because
08:56
we can nominate a small Delta and consider these set.
09:01
For any Delta greater than zero,
09:04
small leader said, we call it only got dirty depends
09:08
on that.
09:08
However, this time, no matter how small is T as
09:12
is Delta for any positive T.
09:14
This set of Omega Delta is a foreigner to say.
09:19
For any T positive.
09:22
And then we can use these to approximate the population
09:27
range and then the boundary of this said can be
09:30
regarded as the spreading front geometrical front.
09:34
And then using this. Rigorously prove the result.
09:41
You can say this front.
09:44
So you look at the writer.
09:46
Most are point on the.
09:49
Population range now this is a setting are and then
09:53
you look at the leftmost point on the range.
09:57
Well, we go to Infinity as T goes to Infinity
10:01
at little speed.
10:02
See star.
10:04
So now this indeed gives.
10:08
A clear meaning for these.
10:10
See stop. This is the spreading speed.
10:13
You follow these front, you go to Infinity when T
10:17
goes to Infinity at these speeds,
10:19
say stop.
10:21
Well, so about 10 years ago in the joint work
10:25
with professor Jacqueline of Yangzhou University in China,
10:30
we charged, too.
10:33
Reformulate, modify the original classical model.
10:39
These costly problem by incorporating these exact front.
10:45
Geometrical front in the spreadings process.
10:50
So we just use the same semonin equation semolina parabolic
10:56
equation.
10:57
But we only assume. Is a certified over these changing
11:03
interval.
11:03
So now the writer side and the left side of
11:07
the interval changed with time.
11:10
And enter the bond or so out of the two
11:13
aimed less.
11:14
Or do we try to represent as the geometrical front
11:17
over the spreading,
11:19
the population density is zero and outside will assume there's
11:24
no such population.
11:26
And then we also.
11:29
Put the initial function here.
11:32
So initial population at it equals zero,
11:35
and then of course mathematically,
11:37
if you. Make the domain change you have to.
11:44
Tell how is the change.
11:46
So now here we borrowed this a so called Steven
11:50
condition here.
11:52
So these equation together with the equals zero,
11:56
determines how these two end evolve with time.
12:00
So now these two is a well posed problem.
12:04
So we can show.
12:06
It has a unique solution is defined for OT and
12:11
then the was left is to analyze its longtime behavior.
12:19
And all that.
12:22
Would large new modified model give us a behavior that
12:27
Rep that mimic the spreading phenomenon?
12:31
Well, it turns out that.
12:33
Indeed it is.
12:38
So we can show. So for these problem and two
12:42
we have these properties.
12:44
So this time we have a dichotomy for the long
12:48
time behavior.
12:49
So one possibility is the Specious Spreader,
12:53
successfully we call it a spreading.
12:56
So this moving interval the population range when T goes
13:01
to Infinity.
13:02
Goes to our. And the population density goes to one
13:08
locally uniformly in space.
13:11
The other possibility is what we call vanish.
13:14
The species vanish is eventually.
13:18
So look population range converts to a Farneti Interval.
13:23
And the population density converge to zero uniformly.
13:28
So we can show actually both cases can happen.
13:33
And the other result is about a spreading speed.
13:37
So when we modify the model and we still want
13:41
to keep the good features of the original one,
13:45
Fortunately we still have a spring speed when spreading is
13:50
successful.
13:51
So we can show that when this happens,
13:54
so the front. So now the front,
13:56
the right front is edge of TX equals HT,
14:00
left front is X equals qifti the front of the
14:04
population range.
14:06
So these both goes to Infinity Lokasie 0 *
14:10
T will. Now we have a new positive constant.
14:16
So all explain how to determine these constant in a
14:21
moment.
14:21
But now let me say so we can determine away
14:25
in these two cases happen exactly by using them You
14:29
as a parameter,
14:30
the meal now appears. Here is in the statement condition,
14:35
so what's the meaning of this statement condition in this
14:40
context?
14:41
Well, in this joint paper with Doctor Bunting and Doctor
14:46
Krakowsky and we.
14:47
MA. We made assumption. Say if the species.
14:54
Near the front.
14:56
Had some loss. So there's a K units of loss
15:01
per unit of volume out of the front,
15:04
and then them You would be D /
15:07
K. So these care unit slots and the deals are
15:11
the diffusion rate.
15:16
So if we use this immune as a parameter,
15:19
then we can show we have a threshold and you
15:23
start positive so that so spread happens exactly when you
15:27
grew on me is greater than mustar.
15:36
Now this.
15:38
Come back to these speed C0.
15:41
Now it depends on this private meal.
15:44
And these are C0. These C zero here depends on
15:48
you is also determined by Audi basically.
15:52
Similarly we look for a criminal waiver solution.
15:56
We end up with these.
15:58
This problem now because we have this definite condition.
16:03
At zero we have an extra condition.
16:06
So this mu appears in the Steven condition.
16:10
So here we call it a semi wave.
16:12
Becausw is defined on.
16:15
On the half reola. Change the original problem,
16:21
so these are sistar is determined by the problem on
16:25
the real line full real.
16:27
So now is on half real line,
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so we call this solution a semi wave.
16:33
And here is not a minimal.
16:35
Speed is the unique speed you give me a new
16:39
positive I can show I have a unique C0.
16:43
And C0 is related to the classical seized are in
16:47
that way as a meal goes to Infinity.
16:50
So these C zero would increase to this system.
16:55
Actually this entire problem.
16:59
These two and the classical wines are related when you
17:03
look at these prime to mute when you goes to
17:07
Infinity under the classical problem,
17:09
why actually can be viewed as the limiting problem of
17:14
these knew problem too?
17:17
So we can show that.
17:18
So if we now to stress the dependence on meal,
17:22
we denote the solution of a 2.
17:24
The free boundary problem. Well,
17:27
for that problem you Anna both you and the interval
17:31
Omega T are unknowns.
17:33
So they we mark them as U mu and Omega
17:38
Mu.
17:38
To stretch the dependence on mute,
17:41
how could we can show when Mu goes to Infinity?
17:44
So these. Goes to R.
17:48
And these are solution goes to capital you capital use
17:52
of the unique solution of one.
17:54
Of course we have to assume they have the same
17:58
initial value.
17:59
So therefore so this one.
18:01
The classical problem can be viewed as a limiting problem
18:05
for two.
18:06
If you let the prime time you go to Infinity.
18:09
Well, in in that interpretation of care goes to 0,
18:13
there's no loss near the boundary.
18:16
Then you go back to the classical model.
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Well, this is just a very brief account for the
18:27
very simple,
18:29
most simple cases of 1 under two.
18:32
And as we know.
18:35
So there are many many more advances on these models
18:40
in multiple directions.
18:43
So now let me just mention that the result.
18:48
The result full one I mentioned earlier is actually from
18:54
the paper by Aronson Weinberger.
18:57
And the result of O2,
18:59
the rubanda problem is from the joint work.
19:03
My joint work with the degree Lena younger University and
19:08
as onium Gore for Henan Normal University in China and
19:12
also with a bang down low.
19:15
Of Shanghai at Normal University in China.
19:18
So there are many more results of which have appeared.
19:23
Since these earlier ones add in many directions.
19:27
However, because today I want to concentrate on nonlocal problems
19:34
or just escape this.
19:36
And just to say so far for both one and
19:40
two.
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So we used these local diffusion or random diffusion to
19:46
describe the spatial dispersal over the population.
19:50
In which. As we know,
19:53
is not our deal. There are.
19:55
There are many situations, real situations for which a nonlocal
20:01
diffusion approximation would be better for the spatial dispersal.
20:07
And of course, for the classical problem or one less
20:11
problem without a free boundary,
20:14
and there have been many,
20:16
many research on this non local version of one and
20:20
for example is attained last 1020 years.
20:23
There are already many many deep works and there are
20:28
still many work on that in that direction,
20:31
so it's a faster progress.
20:33
But for the free boundary problem is a relatively new.
20:38
So we have adjusted started the recently.
20:41
Well, now let me just briefly explain why the need
20:48
to locate a nonlocal diffusion.
20:52
Instead of local diffusion, well,
20:56
one raison is that in many practical situations there's a
21:02
so called long range dispersal.
21:05
And that cannot be captured by using the random diffusion
21:10
operator.
21:11
So it's better to use a different one.
21:14
And so now I'll just mention a particular one.
21:19
But before I mentioned that one,
21:22
let me explain.
21:25
In very simple terms, in this one space dimension well,
21:30
why you say these?
21:33
UW robot chief represented local diffusion random diffusion.
21:39
OK, now let's try to understand these from the modeling
21:44
point of view.
21:46
So we discretize space and time.
21:50
So divide our one into.
21:53
Small blocks or small intervals of equal length Delta X
21:59
and divide time into steps of lens Delta T.
22:04
And then one random Walker basically says an individual from.
22:09
The current time tied to the next step,
22:13
T Plus Delta T, so it oughta move.
22:17
From his current position, X to either the left neighboring
22:22
block.
22:23
So now remember we divided R into equal intervals infinitely
22:29
many equal length the intervals.
22:32
So now my individual is 1 block.
22:36
Would move.
22:39
From the current time to the next time step,
22:42
either to the left neighboring block,
22:45
all the right neighborhood blog with the equal proper probability
22:51
half.
22:51
It has to move. It cannot not move,
22:54
it must move other to the left or to the
22:57
right.
22:58
But with probability half to both.
23:02
And if you tried to pass to the limit to
23:07
get a continue continuous continuum model.
23:11
I think you these process would give you this term
23:16
here in this modeling.
23:18
So. So you would arrive out of these term here.
23:23
So these are random movement of individuals would entertained give
23:29
you this,
23:30
and this D determines how fast you move.
23:33
So this is the ratio between space and time.
23:37
But so anyone doing numerical analysis would know this is
23:42
not a linear.
23:43
You say it's like this Delta X ^2 /
23:46
2 Delta D converted to D.
23:48
Delta T converted the anyway,
23:51
so this is how these local diffusion operate arrives.
23:57
Now for nonlocal diffusion.
24:01
So we do the same kind of.
24:04
Division of R1 and division of time into equal lengths
24:10
of the blocks.
24:11
However, this time you assume that an individual from its
24:16
current location X can jump.
24:19
In the next time, at the next timestep to any
24:23
other location,
24:24
why not only the neighboring?
24:28
Block can be anywhere, but with the probability given by
24:33
a probability function J.
24:36
So here you can make.
24:37
These are more general is a function of two variable,
24:42
but usually.
24:44
So in these in this talk I would just concentrate
24:48
on these simpler kind of a probability function.
24:52
It only depends on the distance between X&Y.
24:57
So now if you use this in the.
25:01
Modeling process then, instead of this term local diffusion term,
25:06
it would arrive at this term.
25:09
So these integration with these convolution.
25:14
Colonel Jay, so these are probability function J becomes a
25:19
convolution kernel.
25:20
So you can say so.
25:22
So these these times are that integrated with the.
25:28
We live, why for why it goes over.
25:32
Our basically tells you how many individuals comes to from.
25:38
Elsewhere 2X and then this morning is.
25:41
This integral tells you how many individuals moved away from
25:47
X,
25:47
so the difference would just tell you in the next
25:51
time a moment how many individuals are are at X.
25:55
So that's how you arrive at least.
25:58
So then the classical Model 1 would become that.
26:02
Basically you does the replace these local diffusion terms with
26:07
that term.
26:08
Involving these probability Colonel.
26:12
So everything else is the same.
26:14
You have the same nonlinear function.
26:16
You have the same initial function,
26:18
it's just these. His term for diffusion has changed from
26:23
a local diffusion to what we call nonlocal diffusion.
26:29
Under the usual kernel, well,
26:31
there are many many choices.
26:34
So now basically we are free to chose.
26:38
So in particular for tinnitus talk,
26:41
mostly concentrate on kernels of these type,
26:45
continuous, nonnegative even and normalize.
26:51
And for these kernels too well,
26:54
of course you change the kernel.
26:58
The dynamical behavior of three with the change and but
27:02
one class of a kernel is singled out is called
27:06
a thin tailed,
27:07
so if these are function now we concentrate on continuous
27:12
functions.
27:13
Now, negative even normalized. If you have this property.
27:18
So if it turns a twosome,
27:20
Lambda X is integrable. Because it's a.
27:24
Cement is even just conceded integration from zero to Infinity.
27:31
So if this is integral for some land,
27:34
not every just for some lamp.
27:36
Think we would call this thing killed.
27:39
Otherwise we see the kernels affected tail.
27:43
It turns out that forcing tailed kernels and the dynamical
27:48
behavior of three share many similar properties is the one
27:53
for with the local diffusion one.
27:56
Through the work of many,
27:58
many people here, just to mention a few who started
28:03
early and about now.
28:04
Many, many people working in this area.
28:12
So one.
28:14
Very striking difference. Between. Three,
28:20
the nonlocal model and one.
28:23
The local model happens with the fat tailed kernels.
28:29
And it in the, in the view of spreading,
28:32
and for these kind of Kernels,
28:35
excellerated spreading happens. So we still try to use three
28:40
to explain.
28:41
So now we all see three are also mean.
28:44
We take a few, the official Kappa P1.
28:47
So prototype is U times one month you?
28:55
For these kind of Kernels,
28:57
we look at a three and well,
28:59
you can still show similar behavior.
29:02
So you started with a similar kind of initial function,
29:06
non negative with a compact support,
29:09
not identically zero. Then the solution exist for all time
29:14
and when he goes to Infinity it converts to one
29:18
locally uniformly in Spanish.
29:20
And then you want to know whether you have a
29:23
spreading speed.
29:24
How does the spreading happen and you do the similar
29:27
thing.
29:28
Use these level set. This is the boundary of these.
29:33
Is nominated. A population range with a small Delta.
29:39
You look at that. You look at the rightmost point
29:43
on these,
29:43
set the leftmost Point. Well,
29:46
you found these go to Infinity.
29:49
When T goes to Infinity,
29:51
but if Jay is a theme tailed,
29:54
it has a fondness Abate.
29:56
If Jay is affected, held no.
29:58
Is the infinite SB. This infinite speed is something completely
30:05
new and is called accelerated spreading.
30:09
And for these official Cappi model and these thresholds is.
30:15
Determined by these condition, whether it's a thing tailed determined
30:20
by the kernel.
30:21
And a half in this case,
30:23
when it goes to Infinity,
30:26
this speed is infinite, and how fast can it go?
30:30
So if we look at these that we can show
30:34
that so depend.
30:35
So if your kernel function at Infinity behaves like eggs
30:40
to the Sigma symbol lesson amonst too,
30:43
and for that case is affected,
30:46
tilled and then R.
30:49
So these would go to.
30:51
These would go to Infinity faster than T,
30:54
it's like E to the offerte exponentially.
30:58
Offer positive. Brief JX behaves like that near Infinity that's
31:03
still fat tailed,
31:05
then it would grow like a teacher.
31:08
The better, better group someone so the better comes from
31:12
here.
31:17
And also there are other kind of Kernels.
31:20
For example if you look out of the fractional laplacian
31:24
then the kernel function would be like this.
31:28
So here I talk about ainda mention this capital enemies
31:32
and dimension.
31:34
Well, these kernel is not continuous has a singularity at
31:39
0.
31:39
It is not normalized well,
31:41
it's done, and the integrals are Infinity.
31:45
Anyway, and you can still use it as a kernel
31:49
and you understand the convolution in that way.
31:54
Align so you replace.
32:01
So you replace these term in one the local different
32:05
by the fractional diffusion and you can show you have.
32:10
Accelerated the spreading and actually so you have by the
32:16
result of Korean rock,
32:18
Joffrey and they proved that so.
32:24
So so these.
32:28
Basically it says so.
32:32
So you hide dimension if you do a similar thing.
32:35
These ghosts Infinity like that.
32:41
OK, now, so let's. Come back to the free boundary
32:47
problem so.
32:48
So let's locate these.
32:52
Local diffusion one.
32:55
So that's the one with the local diffusion.
32:59
So now we want to do the similar thing,
33:02
or replace these by the nonlocal diffusion term.
33:05
Then we have.
33:11
We have this so just to replace that term by
33:15
this one.
33:16
And everything else to the ship.
33:18
And again, at the end point of the population density,
33:22
we assume it's zero, but.
33:26
So the the natural. Through boundary condition is no longer
33:31
these Steven kind of a condition here.
33:38
Not at these ones. And the natural one.
33:42
Is a.
33:44
Is the following. So we we integrate,
33:47
we use integrations involving the kernel function.
33:51
So while these integrals, well,
33:54
if we follow the same kind of biological assumption,
33:58
so now diffusion is been changed to this nonlocal diffusion
34:02
so so similar kind of reasoning would give you this.
34:11
So now the initial function we assume is continuous at
34:16
the boundary of this initial interval is zero and inside
34:21
is a positive and for the kernel function.
34:24
As I said we assume is continuous at zero.
34:28
Is a positive normalized even and is a bounded.
34:33
So this is in my talk today.
34:35
This is the standard assumption for the kernel and for
34:40
the nonlinear function we keep using the facial Kappa P
34:44
classical one.
34:45
You times one month U.
34:50
So the meaning of the free boundary condition.
34:54
So you look at these term actually to tells you
34:59
are.
35:00
The population that went up that passed across these right
35:06
boundary per unit time.
35:09
So this is the population there zalost.
35:11
And as in the local diffusion model,
35:14
we assume so these.
35:18
These movement over the through boundary is proportional.
35:23
So the the ratio of the moment is the derivative.
35:26
These proportional to these quantities and and similar from.
35:32
The population loss from the left.
35:36
Boundary is given by that a quantity,
35:39
so we assume that so the movement,
35:42
the change ratio over the boundary is proportional to the
35:47
zalost quantity.
35:49
Awesome constant mute well. Me over day because we have
35:53
a T here.
35:54
Then we would arrive at that condition.
35:57
So these are biological, meaning is similar to the one
36:01
with the log or diffusion.
36:05
And at least through bonder condition,
36:08
actually was using independently.
36:12
In these two papers were written in the papers.
36:16
Both appeared last year.
36:19
And so this is a my joint worker with a
36:24
jovens,
36:24
how from.
36:27
From Lanzhou and a family doctor family from Guangzhou and
36:33
professor want only from Lanzhou University.
36:38
And so into the consider the case every is identically
36:43
0,
36:43
so therefore the behavior is a longtime behaviors are very
36:48
different,
36:49
so that one conceded nonlinearity of these thought.
36:55
So, well, we basically can show something parallel to the
37:00
local diffusion model,
37:02
so of course we have existence uniqueness.
37:06
We can prove it.
37:10
And we also have a spread and vanish in dichotomy.
37:14
So again, went Eagles Infinity.
37:16
We have two possibilities. Are the spreading successfully?
37:20
So the population range goes to R,
37:23
the population density goes to one and all vanished,
37:27
so population range converted to affinity interval.
37:30
Population density converted to 0.
37:33
And also we can.
37:36
Determine exactly when. Spreading orwin vanishing happens.
37:45
So, so this is a complete classification,
37:48
so no cases left, so we can determine that.
37:52
And of course if you look at here very carefully
37:56
you will observe some difference between these nonlocal model nonlocal
38:01
diffusion model and the local diffusion model.
38:05
For example there is no cross opponent for the first
38:09
case in the local diffusion model,
38:11
but of course so we want to take the spreading
38:15
speed.
38:15
Wing spreading happens, and that's where we will see the
38:21
striking difference.
38:22
So about a spreading speed and we also have a
38:26
threshold of condition on the kernel function.
38:30
So for the classical nonlocal.
38:34
Diffusion model without a free bound?
38:37
Or we know the theme tailed condition is the thresholded
38:42
one,
38:42
now for nonlocal diffusion.
38:46
With a free bhanderi. And this is a threshold and
38:50
condition.
38:51
So J X * X is integrable over the positive
38:56
half,
38:56
X is.
38:58
So our result this is based on a recent joint
39:03
work with the family that's the same family from Guangzhou
39:08
and Doctor Moline Joe.
39:11
Who worked in Australia for several years and has recently
39:17
moved back to China to non care University in Tianjin?
39:23
And we proved that well.
39:25
If Jaehwan holds, then we have fallen into spreading speed.
39:31
If J1 does not hold,
39:33
then we have infinite spreading speed or accelerate the spray
39:39
so so we assume the standard.
39:41
Assumption J so the standing assumption.
39:46
Here.
39:48
And then all that standing assumption.
39:51
So this J one is the threshold condition to determine
39:56
whether we have accelerated the spread.
39:59
So if this does not hold,
40:01
then we have accelerated the spread.
40:06
Now of course there are now two questions we want
40:10
to answer.
40:11
When there's a linear spreading or formative spring speed,
40:15
we want to determine C0.
40:18
And when you. Accelerate the spreading.
40:22
We also want to say how fast these HT and
40:27
GT go to Infinity.
40:29
First, it is determined the C0 and the leader C
40:32
Zero is determined by considering what we call center with
40:37
problem.
40:38
This is basically the corresponding one for the local diffusion
40:43
problem with the free boundary.
40:45
So here we again you look for travel with solutions
40:49
in the nonlocal diffusion setting and you would arrive at
40:54
this and then the free boundary condition would.
40:58
Give you another condition. So now to determine the DC
41:03
zero we solve these system 5 simultaneously.
41:07
Now we can show on the condition J.
41:10
This has a unique solution pair if and only if
41:15
Jay one holds.
41:17
And when Joanne holds or we can determine a unique
41:22
30 positive and unique 50 which is decreasing.
41:27
And these are C0 is.
41:30
Is these one so we have to solve these?
41:34
Completely well, so again, now this jaehwan determines whether this
41:39
has a solution.
41:44
Until then, we want to know.
41:48
Weather we can.
41:51
Estimated this. So when J1 does not hold,
41:56
can we estimate how fast HT goes to Infinity?
42:00
Now is faster than India is accelerated spreading?
42:05
For these purpose we look at A.
42:09
A special type of kernel function.
42:13
So we look at these kind of kernel function which
42:17
at Infinity it behaves like X to.
42:19
The models are for well for this kind of Kernels,
42:23
let's check. So the standing assumption J means Arthur has
42:28
to be greater than one because the Jay has to
42:32
be integrable itself.
42:33
So there's equivalent to offer greater than one for this
42:38
country and J1 hold.
42:40
Is the equivalent to Arthur grew to that 2?
42:44
So if it. Offer is greater than one but less
42:48
than or equal to two.
42:50
Then J1 does not hold.
42:54
Well. By these earlier result,
42:57
it means we would have accelerated spreading.
43:02
But for our for greater than two,
43:04
we have.
43:07
Farnet spreading speed here.
43:11
Well, in this case, actually we can estimate the difference
43:17
with C0 minus HT.
43:18
So if Alpha is greater than three,
43:21
well of course J1 holds.
43:23
We can show this difference is bounded.
43:27
So we use that to mean this,
43:30
so this is order one.
43:32
But if our phase three,
43:34
so this difference is no longer order one.
43:39
Is order is laugherty.
43:42
If Alpha is between 3:00 and 2:00,
43:45
so that's still in that range we still have.
43:49
Farmers spread is to be the C zero,
43:52
but so HT the difference now.
43:56
Is tied to the three monitor is not bounded.
44:00
Though use a lower order than T.
44:03
We have these orders.
44:06
Knuffle offer less than or equal to two and greater
44:11
than one.
44:12
Then we move to the range.
44:14
Will jail, 1 does not hold.
44:16
We know now as T and GT.
44:19
Goes to Infinity faster than.
44:23
Nina, well, if Arthur equals two so from Google,
44:27
and if it's too late we know is T times
44:31
log T order.
44:32
So in the Romanian case between 2:00 and 1:00.
44:37
So cannot be below one or equals 1.
44:39
Otherwise the Jays not set for these does not.
44:43
Is not integrable anymore.
44:46
Showing that a range so this goes to Infinity,
44:50
like T2, that apparel these power of course now is
44:53
greater than one.
44:54
So faster than linear. But in that order exactly in
44:58
that order.
44:58
So so for these kind of Special Kernels,
45:01
so we know much more for the spreading speed.
45:05
So this is from very recent joint work with Doctor
45:10
Wingene,
45:11
who is now doing postdoc with me in Australia.
45:19
So now summarizing.
45:22
So we found. So under the standing condition J.
45:29
So look at the model 3,
45:31
the classical model with nonlocal diffusion,
45:34
but no free boundary for.
45:37
The modified model for three nonlocal different Wheeler through boundary,
45:43
so we know for these special official Kappa P nonlinearity
45:48
accelerated spreading for three the classical.
45:51
Nonlocal diffusion model is equivalent to say the kernel function
45:56
does not set it.
45:58
For J thin is not a thing thing tailed.
46:02
And here for full with the free boundary is the
46:06
equivalent to say.
46:08
The kernel does not satisfy J1.
46:11
So in this particular case,
46:13
so that's to say so are values between one and
46:17
two.
46:18
In this range.
46:22
Now the huge difference compared to the random diffusion problems.
46:27
While under two I mentioned earlier in the talk.
46:32
Is that in these models?
46:35
Accelerated the spread and never happened.
46:40
So in these models, three and four with the nonlocal
46:43
diffusion definitely accelerated,
46:45
spreading can happen. And it depends on.
46:49
How you choose your kernel and we know the threat
46:53
hold conditions on the kernel for these official Kappa P
46:58
nonlinearity.
46:59
Now let's compare these two conditions.
47:03
Thin tailed and Jaehwan well seeing tailed employees j1 'cause
47:08
if this is integrable you multiply by X of this
47:12
integrable because these eight to the Lambda X Lambda positive.
47:18
For some lamb deposit, but of course.
47:21
This does not imply that,
47:23
for example, these ones here.
47:29
So. So all these are not known not thin tailed.
47:39
So now basically these condition means so accelerators spreading happens
47:46
less often in the free boundary model.
47:50
Then the one without a free bundle.
47:54
And actually mean it happens.
47:56
So here we at least full.
47:59
Well for these kind of Kernels,
48:01
there's no exponential growth. Of the front.
48:06
So only like a power of T,
48:10
but it seems the. Low in the nonlocal diffusion problem
48:16
without a full bound.
48:19
In three we know exponential groups can happen to the
48:24
front.
48:29
So finally I just want to mention because this is
48:32
all very new.
48:33
There are some related works on free boundary problem with
48:38
the nonlocal diffusion.
48:39
Some of these.
48:42
Most of these are not got time to mention,
48:46
so these are all relatively new recent joint work that
48:51
I have been involved in there apparently also work about
48:57
other people.
48:58
Not a lot of familiar with.
49:03
I think I'll just stop here.
49:05
Thank you.
49:08
OK thanks, prefer to for his very interesting talk.
49:12
And when I talk. Any questions and comments?
49:21
If you have any questions you can write in the
49:25
chat platform.
49:34
You can post your questions in the text.
49:41
I have 3 questions prepared to and for the.
49:48
Do the other dimension affect your result?
49:53
You do some. Equations in one dimension and the info
49:58
in higher dimension.
50:03
Yeah, that's a very good question.
50:05
We are actually now working on the hard dimension case.
50:09
We've reduced symmetry. The easy in high dimension case,
50:14
but unlike the local diffusion case,
50:17
even with the radio symmetry,
50:19
the extension is.
50:22
Less easy compared to the local diffusion case.
50:26
Yeah yeah yeah yeah.
50:30
We are working on that right now.
50:33
I'm working on that. We win journey.
50:38
I I we are rather positive,
50:40
I think we are able to extend the results to
50:43
the high dimension pace.
50:45
Of course we will modifications.
50:50
I'm ready. OK, not now you do on radio case
50:53
maybe?
50:53
Yeah, we start with the radio case.
50:56
We need to get the feeling.
50:58
To say, yeah.
51:01
Out modifications is needed and then eventually we want to
51:06
look at the non radio case.
51:08
Yes yes yes.
51:12
Topic yes yes.
51:16
And any questions?
51:25
And another question I asked you the contents milk at
51:31
the for the first model water meaning about this in
51:38
the.
51:38
So you have content mu is if we know.
51:44
So if we assume there is a population loss of
51:48
care unit.
51:49
Per.
51:51
You need to volume. Out of the front.
51:54
And then you would be the diffusion rate D overpaid.
52:02
Of course this is a.
52:04
If we make these assumption where we are and we
52:08
can find a meaning of knew.
52:12
Some folding about all these info.
52:16
Biological problems very often. There's no guiding principle too.
52:22
Help you with the modeling you do you try to
52:26
make a reasonable assumption,
52:28
and when you work with your model and say whether
52:32
the model the phenomenon you derive from the model makes
52:37
sense.
52:40
Or cat yeah yeah.
52:45
DD is all.
52:49
Grading content rating. Content and care is.
52:55
From here and carry is drama assumption,
52:58
yeah?
53:00
My cell phone.
53:02
Yes.
53:04
Very easy.
53:07
So I.
53:10
Hello Joanne Yeah. A professor too nice to meet you
53:15
here.
53:15
Professor jump.
53:18
Today's talk, I learn a lot.
53:22
Give me a many informations.
53:25
So probably only also question there.
53:30
List the results are very nice for from local to
53:35
local for the single equations.
53:39
I wonder is there a summer results for four?
53:45
Like 2 species that have yeah no matter it is
53:49
a competition or cooperation.
53:52
Corporations we yes we did.
53:56
So some of the worker here.
53:58
OK, I thought these terms so this one with the
54:01
wind shear has just appeared.
54:03
These are the 4 two solutions for model with an
54:07
analog user and a free boundary.
54:12
For two suspicious competition model.
54:18
Yeah, these are awful. Two species,
54:22
yeah.
54:26
So because the equipment status result in this paper,
54:30
I just wonder if there are difference between single species
54:35
2 species I mean.
54:36
Mathematical point of view the result?
54:40
Yeah, definitely there should be many differences and for the
54:46
moment the system away worked on mostly behave like the
54:52
feature capability for this one.
54:55
One is a. Competition system,
54:58
but we we looked at a weaker competition we could
55:03
predict,
55:04
pray they behave rather differently.
55:08
However, our our results on these kind of models are
55:13
less complete.
55:14
OK.
55:16
So for these models, so these ones,
55:19
the Epidemical models and these West Nile virus model,
55:25
they basically behave like. A monostable model for these kind
55:31
of models.
55:32
We actually we are able to obtain.
55:36
Most of the results are mentioned tonight.
55:41
For these single species, of course,
55:45
is a complicated. OK. But for more full systems actually
55:54
we worked on A.
55:56
The equations system, but as long as they is cooperative,
56:01
they behave like a monostable.
56:03
OK, we can prove similar result.
56:08
OK, thank you very interesting here,
56:11
but look at this preprints on the archive.
56:15
There is something from there.
56:17
OK, thank you very much and professor too.
56:20
OK, thank you. Thank you for everyone and thank you
56:25
and thank you all and any questions or comments if
56:30
there is.
56:31
No, he sang perfect working and for his very nice
56:38
talk.
56:38
And their very intriguing topic.
56:42
OK, how old gotta? Ha yeah yes.
56:48
Then you're you're you're saying something?
56:56
Thank you you so really nice to talk to people
57:01
through this new medium so we can communicate across the
57:07
open.
57:10
Different time zones.
57:12
Yes yes yes yes.
57:17
Porn. Thanks a lot also to both of you to
57:19
all of you for joining and especially for professor do.
57:23
Of course for very nice talk,
57:24
think there's one more question actually in the chat.
57:27
Let me see there was a question.
57:30
If there's any work related to stochastic dynamics in your
57:33
work,
57:33
or if there's any relation,
57:35
I guess to stochastic dynamics in your work.
57:38
Yeah, so for these models are definitely stochastic.
57:43
Models are very, very meaningful,
57:46
but unfortunately have not worked in that direction.
57:51
I'm pretty sure there they are relations to stochastic models
57:57
which are very,
57:58
very meaningful, especially for problems arising in biology.
58:05
So these are spreading themes and actually,
58:09
uh, some some of some of these classical result.
58:13
I didn't get time to mention in this direction for
58:17
the local diffusions,
58:19
local diffusion model, the classical model,
58:23
some of these classical results will first approved by probabilistic
58:30
method.
58:31
And analytical method only came later.
58:33
Yes no. I also don't see any any questions anymore.
58:36
So thanks a lot again to everybody who joined the
58:40
professor do.
58:41
So I hope to see many of you again in
58:44
the coming weeks next week.
58:45
We have a talk, it's related to deep learning and
58:48
Piddies,
58:49
always a popular modern topic,
58:51
so I hope to see many of you again,
58:53
same time, same place.
58:56
Thanks a lot.