[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.