September 25, 2017, Monday

# MBW:Evolution and Distribution of Species Body Size

## Executive Summary

In the article Evolutionary Model of Species Body Mass Diversification [1] Aaron Clauset and Sidney Redner take current theories of the mechanisms of the evolution of species body mass distribution of mammals and express them as a reaction-diffusion-advection PDE. The PDE can be solved using eigenfunction expansion. The steady state solution, atwo-parameter model, was fitted to extant mammalian body mass distributions. Then, the time-dependent solution was compared to fossil data for the body mass distributions of extinct mammalian species, with convincing results. The simplicity of this model, combined with its success in roughly modeling this aspect of evolution dynamics opens the door to further study of evolutionary biology through reaction-diffusion-advection equations. For more information on the transference of a set of genes through a population see Spatial Genetics, and for many more research projects involving reaction-diffusion equations, refer to the "Reaction/Diffusion" section of the main MBW:MathBioWiki page.

## Background to Body Size Evolution

### Cope's Rule

A good starting place for the evolution of body size is to consider Cope's Rule, which states that organisms in a population get larger over evolutionary time. That is, given a certain size distribution of animals within a species, over the course of generations, the average size of the animal is expected to increase.

Note that this applies to all animals that evolve—not just mammals. However, insects tend to be small. Mammals range in size from mice to men to mammoths. And birds come in all sizes as well. Therefore, it would be reasonable to assume that either Cope's rule is wrong, or other factors are at play.

Increased body size may be of evolutionary advantage due to factors such as:

• Better thermal efficiency—consider the ratio of body volume to body surface area. Larger creatures have larger ratios, and must therefore devote less body mass and metabolism to maintaining body temperature.
• Ease of fighting off predators and finding/capturing prey (for information on recent predator-prey models, please visit MBW:Three Species Food Chain Modeling)
• Better survival during times when food is scarce
• Better resistance during rapid climactic changes

### Balancing Cope's Rule

On the other hand, there must obviously be factors that weigh against large body size:

• Increased likelihood of extinction
• Need for oxygen diffusion into the body—consider the small insect, who cannot get so large that oxygen doesn't diffuse into its interior
• Physical limits—if birds get too large, they may have trouble flying.

Another important effect to consider is a species' ecological niche. If we consider an animal that preys only on small burrowing rodents by following them down their burrows, it would be of evolutionary disadvantage to be so large that maneuvering through the burrows was impossible. Thus, we expect the predator, occupying a niche, to remain the same size over generations, as those predators displaced from the niche would struggle to survive.

## Scope: Evolution of the distribution of land mammal size

Rather than attempt to model the evolution of one particular species, which would require extensive knowledge of that species' evolutionary history, Clauset and Redner attempt to model the evolution of the distribution of body size of a land mammals. This summary cannot capture the motivation better than the original paper[1]:

```Animals—both extant and extinct—exhibit an enormously
wide range of body sizes. Among extant terrestrial
mammals, the largest is the African savannah
elephant (Loxodonta africana africana) with a mass of
10^7 g, while the smallest is Remy’s pygmy shrew (Suncus
remyi) at a diminutive 1.8 g. Yet the most probable
mass is 40 g, roughly the size of the common Pacific rat
(Rattus exulans), is only a little larger than the smallest
mass. More generally, empirical surveys suggest that
such a broad but asymmetric distribution in the number
of species with adult body mass M typifies many animal
classes  including mammals, birds, fish,
insects, lizards and possibly dinosaurs.
```

Therefore, since there exists such an asymmetric distribution of body masses, a model may help to bring better understanding to the evolutionary reasons for an asymmetric and right-skewed distribution.

## Model

### Overview

The essential idea of the model is that over time, as species of various sizes reproduce, they do so according to some basic rules. These rules and their effects are shown in the figure below.[1] Note the log scales!

• Lower Boundary Effects: As mammals become smaller and smaller, they fail to be able to maintain a high enough body temperature to allow for metabolism. This limit appears to be around 2g.
• Long Term Risk of Extinction: Larger animals have a higher chance of becoming extinct.
• Diffusion: When an animal of mass M reproduces, its offspring are of size kM, where k is a random variable, very close to 1.
• Short Term Selective Advantages: See Cope's Rule

### Specifics

Assumptions:

1. Each species of mass M produces descendants with masses mM, with m a random variable. The sign of ln(<m>) denotes a bias toward larger or smaller descendants. <r> denotes "average of r." Thus, implementation of Cope's Rule for mammals suggests ln(<m>)>0.
2. Species extinctions are independent of each other. Probability of extinction increases gradually with mass.
3. No species can be smaller than some minimum mass (ca 2g).

Let c(x,t) be the number or density of species of log-mass x=ln(M) at time t.

Then, the reaction-diffusion-advection equation governing the system is [1]:

Terms in the equation:

• k is the growth rate constant of the number of species.
• A+Bx is the extinction rate constant of the species. It is linear, with B>0 so that extinction "reactions" occur more often with higher body mass. Note that the sign of this term is negative, showing loss of density due to extinction.
• v=<ln(m)>.
• D=<(ln(m))^2>

### Solution

Solution is via eigenfunction expansion, transforming the equation for each eigenfunction into the conventional Airy Equation (not shown). Of the two solutions, given that empirical data show no species of divergent body size, only the Ai(z) are kept, while the Bi(z) are discarded.

The Ai(z) form a complete set, since the Airy Differential Equation is a Sturm Liouville type.

Details of the calculations and PDE solution are left to any curious readers, interested in delving into the original Clauset and Redner paper. For this summary, results are presented below.

## Results

### Graphical Results

The steady state model, plotted against extant mammalian empirical data is shown below [1]:

The time-dependent model, plotted against fossil data for extinct mammalian species is shown below [1]:

Model parameters were calibrated by using the steady state equation and the extant mammalian empirical data. Then, with model parameters fixed, the model at various times was shown to be very close to swathes or bins of fossil data. HOWEVER, it is very interesting to note that model time does not scale linearly to real time. In other words, between each 5 million year bin, the model's time change varies. The implication is that other factors may have slowed (or nearly stopped) the body mass diversification processes.

## Important Conclusions

The most important conclusion of this model is that reaction-diffusion-advection type physics have the potential to be applied to the field of evolutionary biology, with interesting results. This particular model works well for extant and extinct mammal species.

The nonlinearity of model time as a function of real time presents some interesting questions as to what events may have occurred to induce nonlinearities and to highlight future potential developments of similar models.

With this model paving the way, there are many other types of questions that one could ask, followed by an effort to create like-minded models to attempt to answer those questions, e.g. including the actual populations of each species in calculations, rather than just densities of body mass according to species type.

## Paper Summary

### Mathematics Used

This model uses a PDE to model the mammalian evolution of species body mass distribution. The PDE is solved by eigenfuction expansion, which transforms the equation into the Airy Equation for each eigenfunction. Both the steady-state and time-dependent solutions are analyzed.

### Model Type

This model involves three ingredients: each species produces descendant species ("Cladogenesis"), the assumption that species become extinct independently, and lastly, that no species can be smaller than a certain mass. These three features enable the use of the Reaction-Diffusion-Advection PDE. The results for the steady state and time-dependent solutions are modeled against extant (still alive) mammalian empirical data and extinct mammalian species fossil data, respectively, both of which show species body mass versus density.

### Biological System

This paper studies the change in species body size over time, as well as their asymmetric distribution. Rather than specific species, Redner and Clauset utilize terrestrial mammal data (both empirical from extant populations and fossil from extinct populations) to create their model.