We took some time today to finish looking at the functions in question 5 of the Iterated Functions sheet, exploring different cases of what happens with the slope of a curve immediately at its fixed point is precisely 1.
From there, we moved to a different family of functions defined by the graph y = ax(1-x). This function is called a logistic map, and it represents the growth of a capped population where x represents the ratio of the current population to a maximum sustainable population (as defined by the carrying capacity), and a is known as a fecundity rate. Naturally, both x and y are bound within the interval (0,1), as a population that reaches 0 is extinct and a population that reaches 1 will necessarily exceed its carrying capacity and become doomed. Your homework tonight is to consider this equation and those limits and to derive the boundaries for a.