One of the few areas where mathematics has been applied to biology. Many of the concepts have been expressed in terms of algebra; (and also many by computer simulations) Interesting combinations of going out into the woods or onto coral reefs and counting animals and plants of different species, and then making lots of graphs and equations to make sense of the patterns of variation. Very relevant to such practical matters as fishing and hunting and game management, and conservation of wild species. Some of the concepts: #1) Mark-recapture studies: A method for finding out the total number of individuals in a given species in the some particular territory in the wild: Catch and mark as many as you can; let's say 100 individuals Then go back again and catch as many as you can, and notice what fraction of the ones you catch this second time had already been marked. Lets say this fraction might be 1/20th (The percentage already marked among the ones you caught the second time = 5%) What would that tell you about the total number in the wild? 20 * 100 = 2000 total in the wild. Is this significantly easier / more practical / cheaper than it would have been to catch ALL the ones in the wild? (Compare the total number caught, relative to total population) Is it more or less accurate than catching them all? How could this accuracy be influenced by differences is "catchability" between individuals of the species? Or what if being caught the first time makes them more wary, and therefore less likely to be caught again? What if the life span were very short: how would that influence your results, or perhaps change how you do the experiment? #3) Rates of population growth: Exponential growth (what seem like small differences in reproductive success can result in huge differences in longer-term results. In contrast to "density dependent growth" (classic S-shaped curve used to be called the "logistic equation") "r" is the reproductive growth rate of a population "K" has to do with the carrying capacity of environment. This gave rise to the concept (or jargon) distinction between species that are "r-selected" (lay very many eggs, etc.) Evolutionary "strategy" of many offspring, and/or fast growing offspring; capable of rapid expansion of populations where & when there is space. Rabbits, frogs, insects, etc. In contrast to those species that are "K-selected" (such as humans, elephants: & many others which are specialized to maximize their occupation of the capacity of the environment, and also tend to age slowly and to care for young) #3) Age-specific mortality: Figure 48:13 in textbook Type I: Of which an example are humans; & many small plants (tend to survive for a certain life span; then all die) Type II: Songbirds (in captivity can live 10x usual lifespan) (death probability remains more or less constant during life) Type III: Many plants, especially trees (Many die young; But the few survivors have long life-spans) #4) Turtles and alligators and some fish, just keep getting bigger and bigger, and older and older, as if with no limit (instead of reaching a set size). "Indeterminate growth" is correlated to longer life-spans! (but it is difficult to know which is cause and which is effect) #5) The cause of aging is not known (although there are many good theories) Aging is almost certainly NOT caused simply by wearing out! (except for a few specific structures; including teeth) Many parts of the body get constantly replaced during life, until the replacement process becomes ineffective late in life. Aging may be "programmed in" to help reproduction of offspring by not competing with them as much for food and space. (Hamilton wrote a famous paper arguing this shouldn't happen )<P> #6) Notice that "life expectancy" is misleading; It is the age at which 50% of a population has died. Thus in a society with high infant and childhood mortality, life expectancy may be only 30 or 40 years, in comparison with 70 to 80 years in modern countries; But that does NOT mean people there "get old" at 35! in the sense that people senesce in their 70s or 80s. (That is a common misconception, however) (It fits in with the other misconception, which is that aging results from wearing out, like a car wears out) In ancient times, and in poor countries, people didn't get old until their 70s, just like now! That hasn't changed, or at least not much. But debilitating heart arterial problems are now much more treatable. And dental care has improved enormously. To some extent, aging actually occurs slightly later in people in poor countries, rather than sooner! (you might try to guess a reason why; but nobody knows) Unfortunately, those stories about mountain villages where everybody lives to be 120+ have been convincingly disproved. (partly by simply asking how many of the people are 70 or 80!) #7) Some of the major theories about the cause of aging. i) Programmed self-destruction. (of various possible kinds) ii) Loss of ends of chromosomes. ("telomeres") iii) Accumulation of somatic mutations (in DNA of body cells) (with some of the mutations causing increased mutation rates) iii) Increased covalent binding between collagen molecules. iv) Chemical reactions caused by oxidizing chemicals. v) Attack of the immune system against body tissues. vi) Weakening of the immune system (reverse of the previous!) #8) Insurance companies keep very good data: & have exact equations; going back as far as the later 1700s. And they have found very clear and constant patterns:P> Compare graphs of Log % of survivors as a function of age. When you graph log death rate as a function of age; It increases in a straight line from age 35 to age 85. On the other hand: #9) In many experiments on tolerance of older animals to stress: How much blood can you drain out of them before 50% die? How long can you put them in the freezer until 50% die? Rats, mice, etc. Graph lethal stress as a function of age not a logarithmic graph, but linear!) The biggest riddle is to figure out a hypothesis by means of which the linear decline in stress-tolerance should produce the logarithmic increase in age-adjusted death rates! Logarithmic (=exponential) increases can indicate something autocatalytic about the process (element of positive feedback). Another cause of such patterns is that multiple independent events are required for aging & death. The patterns of increase of cancer rates with age are often interpreted to mean that about 6 independent mutations are required for a cell to become cancerous. (However heart disease, and most other causes of death have this same statistical pattern as cancer does! So what does that mean?) #10) Life table data: useful in conservation of animals: which patterns of age specific mortality lead to worst sensitivity to extinction; and maybe are a guide to finding the best way to save a species and restore populations. #11) Populations oscillations: especially in Arctic and other northern species: Snow-Shoe Hares and Lynxes (Canadian wild-cats) (data kept by fur companies for ~100 years Textbook example of Red Grouse population cycles; (Notice that the population graphs' Y axis is LOGARITHM of population. (so 999 out of 1000 were dying, apparently!) When birds were treated with anti-worm medicine, the die-offs were much less. But other oscillations may have entirely different causes. #12) Equations of Volterra (famous Italian pure mathematician) and Lotka (US actuary). These equations used to be a major part of the content of courses in ecology & population biology. They predict up and down oscillations in predator and prey populations. #13) 1980s and 90s: "Chaos theory" complex and seemingly random fluctuations can be produced by very simple equations that over-shoot. (the tell-tale finger print is fractal self-similarity) There are many books on this kind of chaos, which really should have been called "pseudochaos", because it only seems to have complex causes. Few physicists or mathematicians saw this coming; but it was accidentally revealed by computer simulations. ----- Questions that you should now be able to answer: a) Suppose that you needed a good estimate of the total number of fish in a given pond; and suppose you some kind of trap that catches fish randomly, then what else do you need to be able to do (NOT catching ALL the fish and just counting!), and give an example of some possible data and the conclusions that would be reached from them. *b) How could your conclusions be distorted (i) If some fish are more apt to go in the trap than others; (ii) If fish learn to avoid the traps, in the sense of being less likely to be trapped a second time; (iii) If the life span of the fish is short, relative to the time course of your experiment; (iv) if the population happens either to be growing or shrinking rapidly during your study? c) Draw an approximate graph of two common alternative patterns of growth of population size (on the Y axis) as a function of time (on the X axis). *d) Explain the idea of the contrast between "r-selected" organisms as compared with "K-selected" organisms. Which strategy would tend to be favored by unstable environments, such as ponds that frequently dry up; or droughts; or floods? e) Contrast (and sketch) the graphs of differences in age-specific death rates (% of those of different particular ages) that die per year, for (i) Trees, (ii) Humans; and (iii) Songbirds. *f) How well do these differences fit the categories of r-selected as contrasted to K-selected strategies? g) What is meant by indeterminate growth, how does it seem to be related to the process of aging, and what is an example of a kind of animal that has indeterminate growth? h) Is "life expectancy" really a measure of how soon aging occurs in a population, including human populations in different countries? [Hint: no it isn't; but you should be able to explain it] i) What are some of the kinds of theories about the causes of aging? [3 or 4] j) Is it analogous to the wearing out of a car, by cumulative wear and damage? [hint: the experts don't think so.] *k) In an animal (such as Dolly the sheep) that was produced by "cloning" = nuclear transplantation from an adult cell into an egg cell (whose own nucleus had been removed), then why might you expect this animal to age prematurely? (Perhaps at the time when aging occurs in the adult animal from which the nucleus had been transplanted?). Would some theories predict that? *l) If such an animal DID age prematurely, then which alternative theories of aging would that tend to support, and which would it not support? *m) Would you expect that "K-selected" species of animals would tend to age more slowly than equivalent "r-selected" species? [Hint: yes; but think about why.] n) Does death due to aging increase linearly with time, or what? [hint: "what"] o) What is the pattern of change in tolerance for stress, as a function of age? p) Do the numbers of particular species in the wild sometimes fluctuate up and down, with a regular wave-length of several years? q) When you find such fluctuations in two different species, one of which uses the other for food, then graphs of their numbers to be exactly the same? To have the same wave-length (length of time)? For the peaks (and valleys) of population levels of one species to come a little after those of the other? [hint: yes: but which one's peaks come before the other?] r) Have diseases sometimes been found to cause oscillations in population densities of particular animals? *s) Can simple equations sometimes produce wild oscillations that seem either to be random, or to have been produced by large numbers of causes. **t) What is the real point of all these best-selling books about "Chaos"? **u) What are fractals, and are fractal patterns ever related to chaos? **v) What might chaos and fractals have to do with population ecology?

 

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