Chapter 3: Life's ups and downs

This chapter looks at fluctuations in nature, and the work of population biologist as they started incorporating chaos as a factor in changing wildlife populations.  It also looks how hard it is to come up with an equation complex enough to represent real phenomena.

First when analyzing a population a few factors pollution biology looks at are connected to the history of life, looking at how predators and prey interact, and how the spread of a disease is affected by population density. 

 Another way of analyzing a population over time is through modeling the data with an appropriate function. The ideal function to model the growth of a population over time would give way to many factors and allowthe population to settle into its long-term behaviour. One equation that was derived to represent population growth is X_{next=rx(1-x). The parameter r represents the rate of growth, (1-x) keeps the growth within reasonable bounds, since rises, (1-x) falls. Population is then expressed as a fraction between one and zerp. Zero representing extinction, and one representing the largest possible population in the particular environment. After calculating for high parameters, biologists began to notice a higher fluctuation of populations. Many scientists at this time just thought this chaotic behavior was because of malfunctioning calculators, as they noticed that populations tend to rise sharply and fall dramatically before reaching an equilibrium. But, this fluctuation of results could have been attributed to the oversimplified equations that were not capable to take into account all the factors of nature that influence a system. It would require nonlinear equations that would have the necessary variables, but would be extremely difficult to solve. In secondary school, students are only taught how to solve linear systems that are solvable. Non-linear systems with real chaos are rarely understood, as people are constantly trying to make sense of the world and not account for all the disorder amongst them. Robert May, a biologist, further reasoned for the necessities of students to learn about chaos. May believed proclaimed, “ The mathematical intuition so developed ill equips the students to confront the bizarre behaviour exhibited by the simplest of discrete nonlinear systems.” One example May looked at that explained the need for chaos to be understood by scientists in all fields was the debate of population change.  Some ecologists argued that populations are regulated and steady, while others argued that populations fluctuate erratically. May believed that the answer to this argument could be explained though chaos theory. The message that simple models could produce what looked like random behaviour, but was in fact behaviour with fine structure. It was like seeing disorder in an ordered system.}