I’d like you to imagine that your
paddling down the river
suddenly you spot this exotic fish
swimming right alongside the boat so
naturally you lean over to take a closer
look and as you do that boat tip
slightly you’re not concerned you feel
fairly stable but you don’t realize is
it now
your friend in the back seat has also
decided to lean over and this brings the
boat to a very precarious position
beyond which an irreversible transition
takes place you’ve probably worked out
the transition and referring to it
certainly gets you closer to that fish
but let’s be honest it could make or
break a friendship anyway I’ll let you
make up the rest of our story for
yourselves the point is this canoe
possesses something known as a tipping
point a point where the dynamic behavior
of a system suddenly changes resulting
in an abrupt shift to some contrasting
stable state and such a shift is known
as it critical transitions now critical
transitions can occur in systems far
more complex and that of this canoe let
me give you some examples financial
crashes epileptic seizures disease
outbreak species extinction despite
having widely different mechanisms each
of these events share one thing in
common
there are consequence of crossing a
tipping point so will it be fantastic if
we could anticipate these tipping points
before they arrive well in this talk
we’re going to see how these seemingly
unpredictable transitions are actually
more predictable than we think due to
some of the generic mathematical
features that these systems exhibit as
they approach a tipping point but before
I launch into how we might go about
anticipating these transitions we first
need a brief understanding as to what
drives them in the first place and the
key players here are feedback loops
we’ve all heard that painful squeal that
can occur when a microphone gets in
front of its own amplifier this is an
example of a positive feedback loop as
illustrated on the left of this slide
any input to the microphone gets
amplified
which in turn gets fed back into the
microphone which gets amplified and so
on creating this runaway effect until
the system reaches its painful limit so
in essence a positive feedback loop
refers to a process whereby change is
self enhancing causing a system to run
away with itself and lose control
negative feedback on the other hand acts
to counter change and so in general
promotes stability consider the
thermostat that regulates the
temperature of your house as the health
temperature drops below some threshold
value the thermostat kicks in and sends
more power to the heaters bringing the
house temperature up as that house
temperature then goes above that
prescribed threshold the thermostat
again reacts to this and reduces power
to the heaters bringing the temperature
back down
and so this thermostat is providing a
negative feedback loop that stabilizes
the temperature of your house well with
regards to a critical transition a
positive feedback loop is the key
ingredient as you might imagine these
negative feedback loops tend to create
these locally stable states that a
system can sit in and trick us into
thinking that a system is stable on a
global scale while in fact these
underlying positive feedback loops can
create tipping points nearby and when
the system crosses one of these tipping
points this stabilizing negative
feedback loop snaps leaving the system
to get swept away in this positive
feedback loop towards the squealing
microphone towards the upturned canoe
towards the epidemic outbreak etc well
the examples we’ve considered so far
we’ve been able to understand through a
relatively descriptive means and our
intuition so it tells us why these
changes come about however nature and
society possess some incredibly complex
systems that can no longer be understood
through basic intuition let me show you
the reaction scheme for the cell cycle
of budding yeast
I think we can all agree there’s quite a
lot going on here
this reaction scheme is comprised of a
network of positive and negative
feedback loops or interacting with each
other in various ways suddenly it
becomes very difficult to see how the
abundance of the SCF protein complex is
affected by a change in the bundles of
the cyclin 3 molecule for example the
only way to capture these feedbacks in a
rigorous way and so understand the
qualitative behavior of the system as a
whole is through the lens of mathematics
using a mathematical framework allows us
to map out the stability structure of
this system and therefore see where the
critical transitions are lacking however
there is a caveat in order to accurately
predict critical transitions in this way
one would require a perfect knowledge of
the system parameter values characterize
these interactions and in reality this
simply isn’t the case due to small
uncertainties in these parameter values
the exact location of these tipping
points is seldom known so how do we go
about predicting a critical transition
if we don’t know where the tipping
points are the answer to this question
lies in complementing this model-based
approach with a data based approach that
directly uses time series data of the
system evolving to assess the system’s
resilience but what do we mean by
resilience do you think you could remove
one of these Jenga bricks without the
entire Tower collapsing on a more
serious note if we remove a particular
species from an ecosystem how well does
that ecosystem sustain itself if at all
if MMR vaccination rates across Canada
drop by 5% how safe is that population
from a widespread measles outbreak
answers to questions like these are
directly dependent upon the resilience
of the system in question my resilience
is to find Loosli as the ability for a
system to recover following some
disturbance so clearly in the early
stages of this Jenga game when the tower
is small it’s relatively easy to draw a
brick the tower is fairly resilient
however if the game increases the tower
becomes more susceptible to these
external forces and more liable to a
critical transition and so it’s
resilient decreases as the height of the
tower increases however these more
complex systems that we care about like
the population and the ecosystem it’s
much harder to assess this loss in
resilience by AI but fortunately there
is a universal property of complex
systems as they approach tipping points
and that is that they take longer to
recover following a disturbance so their
natural tendency to wobble begins to
slow down and there’s a very technical
term in the mathematical literature for
this phenomena that as a complex system
approaches a tipping point it slows down
and it’s called critical slowing down
and critical slowing down is at the
heart of many of the early warning
signals that are being developed today
to anticipate these critical transitions
let me show you how this works in an
example for a model of a fish population
that is subjected to harvesting this
model was proposed actually way back in
the 70s to explain how ecosystems can
suddenly collapse when subjected to
critical levels of harvesting on the
left here this diagram simply shows us
the stability structure of the system
the solid black lines are the stable
states that the fish population can sit
in for some given harvesting rate so for
a relatively low harvesting rate where
that Green Line is we can see that the
fish population are able to sustain
themselves at a high abundance this is a
good popular good state for the fish
population
the fishing community however if you
follow that black line down you’ll see
that as we increase the harvesting rate
there is a critical point where that
stable state ceases to exist this is a
tipping point this is a point where that
positive feedback loop kicks in and the
fish population collapses to this low
abundance undesirable state and this
transition can come as quite a shock to
the fishing community because near to
this tipping point
it only takes a very small increase in
the harvesting rate for the fish
population to suddenly collapse now let
me show you how this would evolve in
time on the right here we’ve got a time
series of our fish population
fluctuating about that desirable steady
state but now the harvesting rates
increasing well from the time series we
can’t tell that approaching a tipping
point at the moment anyway and you’ll
see that suddenly as that harvesting
rate across this a tipping point the
fish population collapses and it’s this
that we want to avoid this is that
critical transition okay but that’s a
big problem here we don’t know exactly
where that tipping point is as I said
earlier those parameter values I’m not
exactly known so how do we go about
anticipating this kind of transition
well we move to that data based approach
let me run that same time series again
but this time we’re going to compute two
statistical indicators namely variance
and auto correlation now simultaneous
increases in the variance and auto
correlation correspond to a decrease in
the resilience of the system they’re
picking up on that critical slowing down
that tells us when a system is
approaching a tipping point so here the
same time series running again the fish
population fluctuating about that state
but now we’re computing these indicators
directly from the time series of the
system in question and that increase is
telling us we need to intervene there’s
something going on here the resilience
of this system has gone too far down and
you’ll see that right before the
transition takes place we get that final
increase in those indicators and we need
to do something but in this simulation
we didn’t do anything the harvesting
rate went across
in point and we get this scenario here
something I find absolutely fascinating
about tipping points it’s the fact that
they can occur on such vastly different
scales empirical studies have found
tipping points right down on the
microbial scale of yeast all the way up
to the planetary scale of our entire
climate system and in each of these
studies researchers found the tell-tale
signs of an upcoming tipping point in
the time series data right before the
critical transition took place so this
is very promising work however I’ll add
that there’s still important progress to
be made before we use these indicators
universally assessing the reliability of
these indicators in different structures
of complex systems is currently an
active area of research in the complex
systems community and something that our
team at the University of Waterloo are
heavily involved in we live in an age
where humanity continues to apply
increasing levels of stress to natural
systems all over the globe potentially
bringing them to fatal tipping points
however at the same time we are now more
equipped to obtain data at a higher
frequency and at a higher resolution
making any early warning signals that we
develop more efficient more reliable so
I strongly believe that the
collaboration of mathematics with the
natural and Social Sciences will play a
key part in making sure that we as a
society do not collectively tip the
canoes of mother nature into
catastrophic States thank you very much
for attention
you
you
paddling down the river
suddenly you spot this exotic fish
swimming right alongside the boat so
naturally you lean over to take a closer
look and as you do that boat tip
slightly you’re not concerned you feel
fairly stable but you don’t realize is
it now
your friend in the back seat has also
decided to lean over and this brings the
boat to a very precarious position
beyond which an irreversible transition
takes place you’ve probably worked out
the transition and referring to it
certainly gets you closer to that fish
but let’s be honest it could make or
break a friendship anyway I’ll let you
make up the rest of our story for
yourselves the point is this canoe
possesses something known as a tipping
point a point where the dynamic behavior
of a system suddenly changes resulting
in an abrupt shift to some contrasting
stable state and such a shift is known
as it critical transitions now critical
transitions can occur in systems far
more complex and that of this canoe let
me give you some examples financial
crashes epileptic seizures disease
outbreak species extinction despite
having widely different mechanisms each
of these events share one thing in
common
there are consequence of crossing a
tipping point so will it be fantastic if
we could anticipate these tipping points
before they arrive well in this talk
we’re going to see how these seemingly
unpredictable transitions are actually
more predictable than we think due to
some of the generic mathematical
features that these systems exhibit as
they approach a tipping point but before
I launch into how we might go about
anticipating these transitions we first
need a brief understanding as to what
drives them in the first place and the
key players here are feedback loops
we’ve all heard that painful squeal that
can occur when a microphone gets in
front of its own amplifier this is an
example of a positive feedback loop as
illustrated on the left of this slide
any input to the microphone gets
amplified
which in turn gets fed back into the
microphone which gets amplified and so
on creating this runaway effect until
the system reaches its painful limit so
in essence a positive feedback loop
refers to a process whereby change is
self enhancing causing a system to run
away with itself and lose control
negative feedback on the other hand acts
to counter change and so in general
promotes stability consider the
thermostat that regulates the
temperature of your house as the health
temperature drops below some threshold
value the thermostat kicks in and sends
more power to the heaters bringing the
house temperature up as that house
temperature then goes above that
prescribed threshold the thermostat
again reacts to this and reduces power
to the heaters bringing the temperature
back down
and so this thermostat is providing a
negative feedback loop that stabilizes
the temperature of your house well with
regards to a critical transition a
positive feedback loop is the key
ingredient as you might imagine these
negative feedback loops tend to create
these locally stable states that a
system can sit in and trick us into
thinking that a system is stable on a
global scale while in fact these
underlying positive feedback loops can
create tipping points nearby and when
the system crosses one of these tipping
points this stabilizing negative
feedback loop snaps leaving the system
to get swept away in this positive
feedback loop towards the squealing
microphone towards the upturned canoe
towards the epidemic outbreak etc well
the examples we’ve considered so far
we’ve been able to understand through a
relatively descriptive means and our
intuition so it tells us why these
changes come about however nature and
society possess some incredibly complex
systems that can no longer be understood
through basic intuition let me show you
the reaction scheme for the cell cycle
of budding yeast
I think we can all agree there’s quite a
lot going on here
this reaction scheme is comprised of a
network of positive and negative
feedback loops or interacting with each
other in various ways suddenly it
becomes very difficult to see how the
abundance of the SCF protein complex is
affected by a change in the bundles of
the cyclin 3 molecule for example the
only way to capture these feedbacks in a
rigorous way and so understand the
qualitative behavior of the system as a
whole is through the lens of mathematics
using a mathematical framework allows us
to map out the stability structure of
this system and therefore see where the
critical transitions are lacking however
there is a caveat in order to accurately
predict critical transitions in this way
one would require a perfect knowledge of
the system parameter values characterize
these interactions and in reality this
simply isn’t the case due to small
uncertainties in these parameter values
the exact location of these tipping
points is seldom known so how do we go
about predicting a critical transition
if we don’t know where the tipping
points are the answer to this question
lies in complementing this model-based
approach with a data based approach that
directly uses time series data of the
system evolving to assess the system’s
resilience but what do we mean by
resilience do you think you could remove
one of these Jenga bricks without the
entire Tower collapsing on a more
serious note if we remove a particular
species from an ecosystem how well does
that ecosystem sustain itself if at all
if MMR vaccination rates across Canada
drop by 5% how safe is that population
from a widespread measles outbreak
answers to questions like these are
directly dependent upon the resilience
of the system in question my resilience
is to find Loosli as the ability for a
system to recover following some
disturbance so clearly in the early
stages of this Jenga game when the tower
is small it’s relatively easy to draw a
brick the tower is fairly resilient
however if the game increases the tower
becomes more susceptible to these
external forces and more liable to a
critical transition and so it’s
resilient decreases as the height of the
tower increases however these more
complex systems that we care about like
the population and the ecosystem it’s
much harder to assess this loss in
resilience by AI but fortunately there
is a universal property of complex
systems as they approach tipping points
and that is that they take longer to
recover following a disturbance so their
natural tendency to wobble begins to
slow down and there’s a very technical
term in the mathematical literature for
this phenomena that as a complex system
approaches a tipping point it slows down
and it’s called critical slowing down
and critical slowing down is at the
heart of many of the early warning
signals that are being developed today
to anticipate these critical transitions
let me show you how this works in an
example for a model of a fish population
that is subjected to harvesting this
model was proposed actually way back in
the 70s to explain how ecosystems can
suddenly collapse when subjected to
critical levels of harvesting on the
left here this diagram simply shows us
the stability structure of the system
the solid black lines are the stable
states that the fish population can sit
in for some given harvesting rate so for
a relatively low harvesting rate where
that Green Line is we can see that the
fish population are able to sustain
themselves at a high abundance this is a
good popular good state for the fish
population
the fishing community however if you
follow that black line down you’ll see
that as we increase the harvesting rate
there is a critical point where that
stable state ceases to exist this is a
tipping point this is a point where that
positive feedback loop kicks in and the
fish population collapses to this low
abundance undesirable state and this
transition can come as quite a shock to
the fishing community because near to
this tipping point
it only takes a very small increase in
the harvesting rate for the fish
population to suddenly collapse now let
me show you how this would evolve in
time on the right here we’ve got a time
series of our fish population
fluctuating about that desirable steady
state but now the harvesting rates
increasing well from the time series we
can’t tell that approaching a tipping
point at the moment anyway and you’ll
see that suddenly as that harvesting
rate across this a tipping point the
fish population collapses and it’s this
that we want to avoid this is that
critical transition okay but that’s a
big problem here we don’t know exactly
where that tipping point is as I said
earlier those parameter values I’m not
exactly known so how do we go about
anticipating this kind of transition
well we move to that data based approach
let me run that same time series again
but this time we’re going to compute two
statistical indicators namely variance
and auto correlation now simultaneous
increases in the variance and auto
correlation correspond to a decrease in
the resilience of the system they’re
picking up on that critical slowing down
that tells us when a system is
approaching a tipping point so here the
same time series running again the fish
population fluctuating about that state
but now we’re computing these indicators
directly from the time series of the
system in question and that increase is
telling us we need to intervene there’s
something going on here the resilience
of this system has gone too far down and
you’ll see that right before the
transition takes place we get that final
increase in those indicators and we need
to do something but in this simulation
we didn’t do anything the harvesting
rate went across
in point and we get this scenario here
something I find absolutely fascinating
about tipping points it’s the fact that
they can occur on such vastly different
scales empirical studies have found
tipping points right down on the
microbial scale of yeast all the way up
to the planetary scale of our entire
climate system and in each of these
studies researchers found the tell-tale
signs of an upcoming tipping point in
the time series data right before the
critical transition took place so this
is very promising work however I’ll add
that there’s still important progress to
be made before we use these indicators
universally assessing the reliability of
these indicators in different structures
of complex systems is currently an
active area of research in the complex
systems community and something that our
team at the University of Waterloo are
heavily involved in we live in an age
where humanity continues to apply
increasing levels of stress to natural
systems all over the globe potentially
bringing them to fatal tipping points
however at the same time we are now more
equipped to obtain data at a higher
frequency and at a higher resolution
making any early warning signals that we
develop more efficient more reliable so
I strongly believe that the
collaboration of mathematics with the
natural and Social Sciences will play a
key part in making sure that we as a
society do not collectively tip the
canoes of mother nature into
catastrophic States thank you very much
for attention
you
you
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