Cambrian Explosion and Population Growth
At the beginning of the Cambrian, more than 500 million years ago, there began a burst of speciation and adaptive radiation which resulted in an "explosion" of diversity of complex, marine animal life.
The development of complex animal life gave rise to organisms which could move over long distances (relative to the protists' very limited range), forage for food effectively, and actively protect themselves against predators.
As organisms become more complex, they must become larger so that they can fit in all the new goodies such as muscles, eyes, etc. If the organism is a sphere, then the volume increases as the cube of the radius but the surface area increases only as the square of the radius. So, the ratio of the surface, over which the food is ingested, to the volume, which is using the food, must decrease as the organism becomes larger. So, our larger, hypothetical spherical animal is able to absorb less energy per unit volume. This would be doubly serious, if it occurred, because the larger animal doesn't need just the same amount of energy per unit volume as the smaller animal, it needs more! The larger animal would likely be more complex and thus would need the additional energy per unit volume for new activities like powering muscles, not less energy! The solution to this problem was the evolution of structures which increase the surface area for absorption of nutrients and oxygen and removal of waste products, all without an increase in volume. This sort of requirement led first to the development of digestive pockets in organisms like the sponges and eventually to tubular digestive and respiratory structures. We will talk about these problems some more in a couple of lectures from now.
The trilobites are the penultimate example of the evolution of complex animal body plans during the Cambrian and were the first animals to have complex eyes, similar to the compound eyes of modern insects. The complex eyes enabled them to avoid predators and see their environment in detail not previously possible. (Even present day Cnidarians have primitive, multicellular eyes and other sensory structures so the trilobite eyes aren't something that just popped out of nowhere!) The trilobites had a complex, segmented body plan with the "joint-footed" system of locomotion characteristic of arthropods. The complexity of the metazoan animals enabled the space in the oceans and the vast food resources which had accumulated to be utilized in a really effective way for the first time.
Exponential and Logistic Population Growth
Exponential growth is the result of a process where the rate of growth of a population is proportional to the number of individuals in the population:
dn(t)/dt = r*n(t)
We don't have good mathematical models for the complex processes of speciation and adaptive radiation. But, we can get a hint of what may have happened by looking at the analogous process of population growth. A population is a group of animals of the same species. As Malthus and Darwin understood, in the absence of limitations on resources (space, food), populations will grow ever faster with time and have no limit. This sort of growth is exponential. The reproduction rate, r is the average number of offspring for each adult in the population per generation. The higher the reproductive rate, the faster population will increase. Look at population growth for three cases: r = 2, 5, and 10. Which curve corresponds to each of these numbers?
Logistic growth is the result of a process where the rate of growth of a population is proportional to the number of individuals in the population multiplied by a factor which limits population growth when the number of individuals is large:
dn(t)/dt = r*n(t)*(K-n(t))/K
However, if resources are limited, populations will begin to grow exponentially but, before long, will begin to "feel" the effects of the limited space or food. When this happens, growth will slow down to a rate below that of unconstrained exponential growth. Eventually, growth will asymptotically approach a limit. This limit on population size is called the carrying capacity and is given the symbol K. One mathematical model of such resource limited growth is called logistic growth. The three curves below correspond to K = 100 and r = 2, 5, and 10:
Notice that in all three cases, the number of individuals in the population will eventually rise very close to the limit of 100. However, depending on the reproductive constant, it will take a shorter or longer time to come close to the limit.
The Cambrian explosion may be no more than the consequence of exponential growth in diversity and complexity, followed by a leveling off of diversity as nitches are filled.
The s-shaped, sigmoid curve of logistic growth looks very similar to the relationship between diversity (measured as number of invertebrate genera) and time during the Cambrian. The "explosion" is perhaps nothing more than the fact that the development of complex body plans for the first time permitted radiation into a variety of adaptive nitches and a dramatic increase in diversity. This diversity was limited, just as population growth is, when all available nitches or resources are used. Read the the article by Stephen Jay Gould in the book Ever Since Darwin to learn some more about the Cambrian explosion.
The models of exponential and logistic growth don't have any time lag between generations. Development and maturation to sexual reproduction are all considered to be instantaneous.
There are several things wrong with the models of exponential and logistic growth which we have described. One of the more serious problems is the assumption that offspring are created continuously and instantaneously. The mathematics works instantly; the offspring is conceived, the embryo grows and becomes an reproductively active adult, all in an instant! Of course, real organisms don't work like this. Since there is a time lag, during which all the activities from fertilization to maturation of a reproductively active adult take place, the population can overshoot the carrying capacity. Many more offspring can be produced than the carrying capacity support. By the time these new organisms actually require the full resources of the environment, it is too late! There are already too many! So, some die off, bringing the population below the carrying capacity at which time the population then increases again. This sort of cycle can continue indefinitely. The results of a more realistic model, incorporating a time lag between generations, are shown in the table below. (The graphs show plots of the number of individuals in the population, scaled to a fraction of one because of the procedure used for calculation, vs. time in generations.)
Note that as the reproductive constant increases (more offspring per adult during each generation), the overshoot becomes more noticeable. Finally, for high values of the reproductive constant, the oscillations break down into chaos, a phenomenon which can look periodic but has a random component to it, making it impossible to exactly predict when the next peak or valley in population will be.
Why is chaos important for our discussion of the Cambrian explosion? Well, once populations increase to levels near the carrying capacity, the population numbers are especially subject to strange and unpredictable oscillations. To the extent that we can apply the ideas on population changes to speciation and adaptive radiation, we might expect that periodic extinctions or bursts of speciation, as in the model of punctuated equilibrium, might occur just as the result of simple interactions between individuals and groups and not necessarily the result of external forces.
Especially for those with a mathematical inclination: You can view an explanation and simulation of population growth and chaos by clicking on the Mathematica icon.
|r = 1.2
||This looks almost exactly like logistic growth!
|r = 1.5
||The population overshoots the carrying capacity, then drops slightly below it, then rises above it, continuing this pattern for a few small oscillations
|r = 1.8
||Bigger oscillations but the population still settles down near the carrying capacity.
|r = 2.1
||Oscillations build to a level which is sustained indefinitely.
|r = 2.4
||Oscillations build to a level which is sustained indefinitely
|r = 2.7
||The development of chaos. Notice that the pattern looks periodic. But, if we look closely, it almost repeats, but not exactly. There is an underlying general pattern, but the pattern does not repeat perfectly.
|r = 3.0
||Notice how the population drops to very low levels from time to time. Imagine that there was a minimum sustainable population. If so, the species might become extinct during one of these huge dips in population number.
In class, we will discuss how this cool "bifrucation diagram" relates to Chaos.