[Pablo Marquet:] "Scaling in ecology"

So, why is scaling so important
and why do you need to know about this?

Well, scalings provide a way to deal with
the diversity of scales

and also the... different type of organism

that exists and becomes
part of ecological systems.

Scaling... also makes apparent

the fundamental similarity
that underlies the diversity in nature

and how this has been molded
by the action of natural selection.

It's very important to realize that

scalings point out that
there are not many...

not many ways of actually having
a real functioning organism in nature

and provides you with the right way
of understanding the constraints

that are operating on diversity.

And, since most organisms
obey these constraints,

it points to the fundamental similarity
that they share.

So, the other important thing
about scaling is that

it provides us with a benchmark
against which we can compare

different species, populations
and ecosystems,

and you can actually measure
deviations from scaling relationships,

and those deviations mean
that there are some processes -

some important biological processes -

operating on these systems.

So, let's start by the most simple way
of characterizing scaling in ecology.

And, the characterization
is a mathematical one - it's very simple.

And usually, scaling relationships,
as you have already heard,

cannot be summarized with
a relationship like a variable "y"

or a trait of an organism.

It is proportional to a variable "M"
raised to a power.

In this case, I'm talking about M
as the mass of the organism

or the size of the system -
of the biological system.

So, you can also represent
this relationship as

y equals cM raised to alpha

which actually allows you to take
the logarithm of this relationship

and transform something
which is not linear

into a linear equation.

And, I'll show you an example of this.

In panel "A," you see
scaling relationships

that are in the nonlinearized way.

And then, to the right - in panel "B" -

you see the linear relationship
that results

after you take the logarithms
of this relationship.

That makes it easier to analyze.

So, examples of scaling abound in nature

and they affect the way organisms
are put together and evolved

through the action of natural selection.

You can see, for example,
relationships between

the size of the organisms
and weaning time,

or also, the size of the organisms
and longevity.

And, you can use this relationship -
as I show you here in this panel -

to compare different kinds of organisms.

For example, the gray dots represent
all mammalian species

and the red dots represent
one particular type of species,

which are the marsupials.

So, you can see the marsupials, they...

in general, they follow the trend's line
for longevity, for example,

but they deviate in maturation time,
for example.

So, these scaling relationships
allow you to compare

and you can use them as a benchmark
to try to understand what is going on

with this particular species
that they deviate.

Is this because of
their phylogenetic history?

Or, is it because of the environment
of the other kind of habits they have -

what they eat and so on.

So, you can answer
meaningful questions

using the scaling relationships.

In ecology,
these relationships also affect,

for example, things very fundamental,
like the average prey mass

that a particular carnivore species
will eventually eat,

or affect, as you can see in the panel,

the carbon turnover,
or the time it takes,

to replace one gram of carbon
in a particular ecosystem

and how that changes
with the average mass of the plants

in that ecosystem.

Or also, you can see scaling
relationship in terms of

the net primary productivity
of an ecosystem

and the total amount of
plant biomass in that ecosystem.

So, those are fundamental relationships

that tell you something very... general

about the way organisms
and ecosystem work.

So how do we understand
the origin of this relationship?

Well, let me tell you, the fundamental...

kind of relationship I would say is

the one that relates the organism
and its energy requirement to its size.

This is called "Kleiber's Law,"
and it points out that

the requirements of energy
of an organism, its metabolism,

which is the sum of
all the biochemical reactions

happening within the organism,

scale with the size of the system -
or mass -

raised to a three-quarters power.

And, you can see in the graph that

this relationship describes very well
how metabolism changes

as you increase the mass of an organism

from a mouse, for example,
to an elephant.

And, it tells you that somehow

an elephant is just a variation
on the same theme as a mouse is.

So, it's this relationship that points out

that there is a fundamental similarity
among these different kinds of organisms.

So, natural selection
is not acting randomly -

it's following some constraints or...
so, what are those constraints?

Well, in 1997, Geoffrey West,
Jim Brown and Brian Enquist...

they proposed a very simple
and elegant model

that points out to some
fundamental principles acting and

helping us to understand why
metabolism changes in the way it does

with the size of a system.

And, I'm not going to explain to you
all the details of the model.

I'm just going to give you
two major insights into it.

Well, the major thing is that
any organism that exists faces a problem

and the problem is that
you have to deliver energy -

the resources that you get
from the environment to live -

you have to deliver it to
all the different parts of your body.

In a multicellular organism,

this means that you have to deliver
this energy to all the cells in your body.

So, how do you do that?

Well, the way you do that is
you construct.

I mean natural selection has molded
the existence of networks

to deliver this energy towards
all the parts of your body.

So, those networks
actually generate constraints

that manifest in this fundamental
three-quarter scaling for metabolism.

Now, the way natural selection
acts on this is

by minimizing energy loss.

So, it generates networks
that minimize this energy -

so efficient networks -

when you take account of what
you consider these two major insights,

and they show
in a mathematical model

that this creates the three-quarter
scaling relationship that we see.

So, let me give you some example

of the implications of this relationship
in ecology.

One simple one is that you can ask
a very simple question like -

what is the maximum number
of individuals

that can be found in a given area?

So, to answer this question,

which I think is a very fundamental one,

you are just required to know
the amount of energy or resources

that are in a particular area,

and the requirements of those resources

by different kinds of individuals.

So... we know that the scale
with the size of the organisms

rises through three-quarters.

So, it's very simple then to compute

the maximum number of individuals
in a given area

just by dividing the amount of resources

over the requirements of each individual.

And... what that gives you is
another scaling relationship

that says that the maximum number
of individual scales with mass

raised to minus three quarters.

Now, let's see the empirical evidence.

The empirical evidence on this,
and I'm going to go back

to a paper published by
John Damuth in 1981

where he made a compilation
of the density

of different mammal species
around the world.

And, these happened to be
every [major] mammal...

and he computed the density
and size of each of these species,

and he plotted this relationship,
and, as you can see,

there's a negative slope and
this slope is minus three quarters.

So, right as predicted,
the maximum number of individuals

follow a minus three-quarter
scaling law.

The implications of this
are very interesting

because it means that mice actually...

they achieved larger densities
than elephants,

but somehow they are the same,
because they, as I will show you,

they use the same amount of energy
at the population level.

And, how we can calculate that?

Well, we call that
the "population energy use."

Population energy use - PEU -
is proportionate to

the number of individuals times
the requirements for energy.

So, that will be the total amount
of energy required by a population.

And, if we replace the known
scaling relationships

for number and for energy requirements,

we know that population energy use
will be proportional to

mass raised to minus three quarters
times mass raised to three quarters.

As you can see, these two exponents

they kind of annihilate themselves
and give you a population energy use

that is proportional to
mass raised to zero,

meaning that population energy use
tends to be invariant

regarding the size of the organisms,
meaning that elephants use

the same amount of energy
that mice use.

So, this is a very fundamental
invariant relationship.

There are many more scaling relationships

that show this kind of property
of invariance,

but it shows you that
things can be different,

but at the same,
time things can be equal

if you explore this relationship
between the scalings.

So, why is it fundamental?

Why is this very important too

is because it allows us to understand
something about ourselves too.

And here, I want to make the point that,
using this relationship,

you can understand that humans
are a hyper-dense species.

How do we know that?

Because, we know the scaling
relationship for all mammals.

We know how density changes as you
change the body size of the mammal.

We are mammals.

On average, a human being

you can consider weighs around 70 kilos.

And, you can try to plot which would be
the density that we should achieve

if we will kind of
follow this relationship.

And, our density will be around
2.12 individuals per square kilometer.

So, the realized density is

5.8 raised to 10 to 4 [fourth power]

meaning 58,000 times larger.

So, this is a very significant deviation
from the expected scaling relationship.

And, this is a very meaningful one
because

you can now answer how it comes -
what happens to humans

that they could shift to
such an extreme maximum density

as it does these days.

Well, let me tell you that

when we were in a different kind of
stage of development,

a social stage of development -
when we were hunter-gatherer -

we really fit into this relationship.

As we moved through time
and social complexity,

we were moving away
from this relationship

and now we are far away from it.

And, this has impacts in terms of
the amount of energy that we use

and the impact of that amount of energy
used by humans

on the rest of the ecosystem.

That is called, or is part of
what we call "global change."

We're not going to deal with this
in this class,

but it's important to keep it in mind.

That's why scaling relationships help you
to realize there are similarities,

there are fundamental principles
related to the way

organisms deliver energy
through networks

that explain the amount of energy
that different organisms require

and this constrains the density
and amount of energy

their populations use.

And... you can use them
to understand deviations,

like the humans' density extreme
or hyper-density of humans now.