Big is beautiful - bioenergetics again (long)

[Martin Baeker <martin@raptor.ifw.ing.tu-bs.de>]

Thanks to everyone who answered my earlier question on bioenergetics. 

The reason behind it was that I tried to do a
back-of-the-envelope-calculation on why dinosaurs became so big. 

After having read those of the sources recommended to me that were
available, I came up with some nice figures, and, surprisingly, an idea
that I never found mentioned anywhere else (which probably only proves
that my knowledge of the literature is more than poor).

So here is what I did, and I would be grateful for any comment on this.

First of all, I tried to estimate the food requirements of big vs. small
animals. 

Taking the equation for the metabolic rate found in The Complete Dinosaur,
valid for mammals:

metabolic rate = 3.75 * M**0.75 

I get a specific metabolic rate (i.e. rate/mass)

sp. m.r. = 3.75 * M**(-0.25) 

This is in Watt/kg.

So with this, the spec. matabolic rates of animals of different sizes
and the total energy requirements are:


1 kg:     3.75 W/kg  3.75 W
100 kg:   1.18 W/kg  118 W
10000 kg: 0.375W/kg  3750 W

Now Watt is not a handy unit, so I convert everything to kcal/day,
something everyone who ever has seen a diet plan can relate to.
Fortunately, you just multiply by a factor of 20 (not very exact, but
these are estimate anyway). So, how much food do you need to get this
energy? As it is always said that being big gives you the advantage of
being able to use food with low nutritious value, I assume food with a
value of 100kcal/kg. This is more or less what tomatoes have for a human -
you will not get fat on this for sure!

So our three animals need per day:

1kg:     750 g
100kg:  23.6 kg
10000kg: 750 kg

These numbers look reasonable, as they would mean that an elephant would
have to consume a few hundred kg of low-value food per day.

As it is not practical to have more than perhaps a few percent of your own
mass in your guts (would make you VERY slow), this shows that it is not
possible for a small mammal to live on this kind of low-level food,
whereas a big one can easily afford to do so.

Now here comes the first surprise: What if dinosaurs had reptile metabolic
rates? In this case, every figure above has to be divided roughly by a
factor of 10, so even the small 1kg-reptile can easily exploit the
low-level food source. And the turtle I kept as a child was indeed happy
with a few leaves of lettuce, a tomatoe or a carrot, and it did not have
to eat huge amounts of those.

So does this show that being big is energetically favourable only for an
animal with high metabolic rate? I never heard this argument for dinos
being hot-blooded before. Am I totally in the left field here?

After this calculation I tried to understand WHY the metabolic rate scales
as it does. It was pointed out to me that this has probably to do with
energy costs of locomotion, something I now also read in the book of
McGowan.  The standard argument is as follows:

Two animals of different sizes move at the same equivalent speed if their
Froude numbers Fr are the same:

Fr = v**2 / l * g  where v is speed, l length (e.g. hip height) and g is
gravitational acceleration. 

So the equivalent speed scales as v ~ sqrt(l)
and the stride frequency then has to scale as
str. freq. ~ 1 / sqrt(l)

So to run at the same equivalent speed the small animal needs more strides
per second than the large one and, as the cost of moving one unit of mass
through one stride is more or less constant, has to have a higher
metabolic rate.

However, something I did not find adressed anywhere is the following
question: Why cannot the smaller animal run at the same speed in relation
to its own body size. I.e., if the large animal can do 1000 body lengths
per hour, why cannot the small animal function at the same relative speed?
It will cover the same area as does the large animal, relative to its
size, so there is no scaling problem here. 

I found two possible answers to this question:

First of all, it does not help the mouse, "running" at 0.1 m/s, to tell
the cat: "Hey, don't catch me, I am as fast as you are, relatively
speaking." 

But, perhaps more important than that, there is another constant coming
in: What I call the "reaction time". If you are not running on very smooth
ground, you will have to adjust your movements to the stumbling stones you
encounter, otherwise you will frequently land on your nose. Now the time
you have to recover from stumbling or to adjust to uneven ground, is
obviously proportional to the time you have until you belly hits
the ground, i.e. until your center of mass has fallen a certain distance.
This time scales like 

t ~ sqrt(l/g)

so it gets shorter, the smaller you are. So our slow-going mouse will have
to react VERY quickly whenever the ground is uneven, even if it would like
to be slow. (If you ever hiked in the mountain, you have perhaps noticed
that sometimes it is indeed easier to go more quickly on uneven ground, so
as not to have to make quick adjustments during slow moving). So I argue
that for efficient locomotion your stride frequency must be not much
slower than the reaction time you need. Now this argument I also never saw 
before
- is it again something only a stupid physicist can cook up, with no
relation to the real world?

If not, it has another interesting aspect: It has been argued that a T rex
cannot have run because it would severly injure itself on falling.
However, as it is so big, it would have a lot of time to catch itself,
when stumbling. So perhaps, the T rex simply did not fall, because for it
the reaction time was quite large. 

So, here is what I cooked up during the weekend. Any comments are more
than welcome.

Martin.



                   Dr. Martin Baeker
                   Institut fuer Werkstoffe
                   Langer Kamp 8
                   38106 Braunschweig
                   Germany
                   Tel.: 00-49-531-391-3073                      
                   Fax   00-49-531-391-3058
                   e-mail <martin.baeker@tu-bs.de>