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