LessonPopulation models We have already encountered three models of population dynamics. The first model is the doubling function in Lesson 1 (figure 5), whereby the population simply doubles at each generation. The second model follows the Fibonacci number sequence, as exemplified by the multiplying rabbit pairs in Lesson 4 (figure 8). There, the rabbit population ultimately increases by the golden ratio
As emphasized in the function iteration diagram of figure 1, we can freely choose the parameter r in between 0 and 4. Note that the population function of Lesson 1 corresponds to the special case of r = 2. In the right-hand side of equation (1), the first term rx represents population gain and the second term -rx2 gives rises to population loss. Hence, the goal is to find out how the gain and loss terms balance out to give stable population for a given r. A word of caution is: since x takes a value in between 0 and 1, we must interpret it as a ratio of the actual population to maximum population that can be sustained by a given environment. For instance, with the maximum population of, say, 100 rabbits, x = 0.5 means 50 rabbits. Also, a negative x should be rejected, for it has no physical meaning.
Figure 1. Population function iteration |
1.618, in contrast to factor 2 of the first model. In any event, both models give a continuous population growth without an upper ceiling. Hence, they cannot predict a realistic population level of ecological equilibrium between the gain by birth and loss due to the food shortage and predator attacks. We, therefore, consider here a generalization of the population function first introduced in Lesson 1.
