Regression

Data

length

width

Data

length

width

“fitting a curve to the data”

Data

length

width

length

widthlength

width

Pheidole dentataThe larvae of this species of ants are a favourite food of army ants. Pheidole dentata workers come in two different forms. The large headed “soldier” morph is adapted to fighting off army ants with its pincers.

Data

Problems:

• Reproducibility

• Which kind of curve?

Power Functionsy = axb

b = 1 b = 2 b = 1/2

Power Functionsy = axb

b = 1 b > 1 b < 1

b = 0.3? 3.1?

Allometry

Irish Elk

Stephen Jay Gould

Hadrosaurid

height

width

Power Functionswidth = a heightb

b = 1 b > 1 b < 1

height height height

width width width

width = a heightb

b = 1 b > 1 b < 1

height height height

width width width

Allometry

x

x

x

Cube of side x

Volume = x3

Surface Area = 6x2

2x

2x

2x

Cube of side 2x

Volume = 8x3

Surface Area = 24x2

= 4 x 6x2

rx

rx

rx

Cube of side rx

Volume = r3 x3

Surface Area = 6r2 x2

= r2 x 6x2

If an object is scaled up by a factor r in each direction, then its volume increases by r3 and its area increases by r2.

Scaling

Brown Recluse spider

Suppose a spider is 1 cm long and weighs 0.01 g.

Imagine scaling it up by a factor of 1000.

Now it is 1000 cm = 10 m long.

Its weight increases by a factor of 10003 = 1,000,000,000.

It now weighs 1,000,000,000 x 0.01 g = 10,000,000 g = 10,000 kg = 10 tonnes

Its legs are 1000 times as thick.

r

Their ability to support the weight above them depends on their cross-sectional area.

The cross-sectional area has increased by 10002 = 1,000,000.

The legs can support 1,000,000 times as much weight.

The spider weighs 1,000,000,000 times as much.

The stress on the legs is 1,000 times as much.

The spider cannot support its own weight.

The spider cannot support its own weight.

• hummingbirds eat their own weight each day

• mice can survive falling down a mineshaft

• horses cannot

• bigger animals tend to have thicker legs

• blue whales do not have to support their weight on legs

• hummingbirds eat their own weight each day; even a mouse eats nearly its own weight. Suppose a person is ten times as “long” as a rat. If they had the same shape, this would mean that the volume of the person would be 103 = 1000 times as much as the volume of a rat. Because we are made of similar material, this means the mass of a person is also about 1000 times the mass of a rat.However, the surface area of the person would be 102 = 100 times as much as the area of a rat. The ratio of the area to the mass is 10 times higher for the rat.

However, the surface area of the person would be 102 = 100 as much as the area of a rat. The ratio of the area to the mass is 10 times higher for the rat.The surface area of an organism is closely related to the rate at which it loses heat. A small animal, like a rat, loses heat much more rapidly than a human.

Maintaining its internal temperature requires a lot of energy, and this is a plausible explanation of the larger appetites of small animals.

• mice can survive falling down a mineshaft

This is for a related reason. The air resistance of a falling body is related to its surface area.

• horses cannot

A small animal experiences much greater air resistance relative to its mass, so it does not fall as fast. (Actually, it reaches a much lower terminal velocity.)

• bigger animals tend to have thicker legs

• blue whales do not have to support their weight on legs

The strength of an animal’s legs is related to the cross-sectional area of the legs. This must be proportionately larger for larger animals.

A blue whale is probably too big to be supported on legs.