Mathematical modeling:

Predicting the size of the terror Bird with proportionality and geomatric similarity

Background on terror bird

  • The terror birds were giant,flightless,predatory birds.
  • Their ansistors lived in South America for almost 30 million years before migrating to Florida
  • It was feet hunter who would lie in amnush and attack from the tall grasses and kill with their beaks
  • Unique physical characterstic of these birds was that they had arms,not wings.

#Terror Bird

Using Mathematical Modeling to Infer the Body Weight

  • Since body weight of the Terror bird cannot be directly measured from its fossil bones, scientists need a way to infer the body weight from a quantity, which can be measured from fossils.Hence,“comparative anatomy” coupled with mathematics was used to determine the relationship between fossils and weight.
  • Comparative anatomy, the comparative study of the body structures of different species of animals in order to understand the adaptive changes they have undergone in the course of evolution from common ancestors.

Explicative Models: Proportionality and geomatric similarity Arguments

    • Proportionality: Two positive quantities x and y are said to be proportional if one quantity is a constant positive multiple of the other. We write \(y \alpha x\) to indicate that the quantity y is proportional to the quantity x. Thus \(y \alpha x\) if and only if y=kx for some k > 0

Geomatric similarity is the concept related to propertionality and is very useful in simplifying this portion of the mathematical modeling process The defination is as follows: two object are called geomatrically similar if there is a one to one corresponding points that is consatnt for all possible points

  • This allows us to make the comparison of the terror bird to similar species of which we have data

Modeling the Terror Bird

Problem Identification

Predict the weight of the terror bird as a function of the circumference of its femur

Assumptions and variable

  • Terror birds are geometrically similar to other birds of today or prehistoric dinosaurs from the past
  • Volume of bird is propertional to any characterstics dimension cubed

\(V \alpha l^3\)

  • Assume constant weight density
  • Volume displace an amount equal to its weight

V=W , \(W \alpha l^3\)

  • We assume that characterstic dimension be the circumference of the femur
  • Femur was choosen because it supports the body weight \(W=kl^3\) ; k>0

[Relative scaling of sizes is termed an “allometric relationship” and the mathematical form of an allometric function is \(y=Cx^p\), where y is the dependent variable, x is independent variable and c is any positive constant]

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Femur circumference and body weight of birds

Femur circumference (cm) Body weight (kg)
0.7943 0.0832
0.7079 0.0912
1.0000 0.1413
1.1220 0.1479
1.6982 0.2455
1.2023 0.2818
1.9953 0.7943
2.2387 2.5119
2.5119 1.4125
2.5119 0.8913
3.1623 1.9953
3.5481 4.2658
4.4668 6.3096
5.8884 11.2202
6.7608 19.9500
15.1360 141.2500
15.8500 158.4893

Modeling with Proportionality-Terror Bird example

Here we want to predict the weight of the terror bird using the circumference of the femur.We have already created the notional model for which we check further.

Scatter plot showing a curve trend.

Plot W versus \(l^3\) and obtain approximately a straight line through the origin

linear Regression out put

x
(Intercept) 1.3603268
Femur circumference (cm)^3 0.0399267

Hence our model will be \(W=0.0399*V\) where W represent weight and V represent volume

The model’s residuals shows some trends, so we might conclude that more work can be done in modeling. If we accept the model, having the circumference of the femur of terror bird ,we can precict that it has weight of 368.58 kg.

Refrence:

Fox,William P.,Richard West,Jack Robertson,Robert Chandler,Peter M.Jarvis and Robert Viav,“Determining the size of the Terror Bird”,ILAD,Project Intermath 2003