Math Forum: Lee: Lesson 1 - Do it by Functions, part 4

Population function

By doubling numbers one can express the initial population growth by a parent rabbit pair multiplying to 2 rabbit pairs of first generation, 4 rabbit pairs of second generation, and so on. Although the rabbit population may grow initially by doubling, it cannot grow forever in such a simple doubling fashion. For one thing, there might not be enough food supply to support an ever-doubling population growth. Moreover, the larger the population, the greater the threat of predator attacks. Therefore, for a realistic population model it is necessary to account for population depletion by including a loss term of simple form -2x² in function (1)

(3)

This is a population function. As we shall see in Lesson 9, it is a special case of the population growth model

, where r is a number (parameter) chosen in the range (0, 4). We must restrict the value of x between 0 and 1, because outside that range x becomes a negative number, which is not acceptable for population. Also, since being in between 0 and 1, we must interpret x as a ratio of actual population to a certain maximum population that environment can support.

Figure 6.Population function

We now iterate population function (3) by the function diagram as shown in figure 6. In table 2, beginning from initial population x₀= 0.3, the first iterate is x₁= 0.42. Then, entering x₁ as input for the next round of iteration, we get the second iterate x₂ = 0.4872, the third iterate x₃ = 0.49967, the fourth iterate x₄ = 0.49999, and so on, as shown in table 2. After one more iteration on a pocket calculator, you get x₅ = 0.5.

Table 2. Iteration of function (3)

Once you have x₅= 0.5, all further iterates are frozen at 0.5. This is because

So, x = 0.5 is a special value for function (3). That is, the input value is the same as the output, and such a value of x is called a fixed point. In fact function (3) has another fixed point; it is x = 0 as indicated by

Summing up, there are two fixed points, one is at x = 0 and the other at x = 0.5. There is however something very special about the fixed point at x = 0.5, and we shall come back to discuss it shortly. Now, at the fixed point x = 0 both the input and output have zero value. That is, if you start with zero population we expect the population to remain at zero, and this is the meaning of fixed point at x = 0. Now, at the upper end point value x = 1, we see that the first iterate is zero

so that further iterations get stuck at x = 0 forever. In other words, if you initially populate at the maximum level, function (3) will lead to total extinction due to overpopulation.

We now discuss the fixed point at x = 0.5. It is to this fixed point that population function (3) iterates any initial input in between 0 and 1, except for two end point values at 0 and 1. As a typical initial input value larger than 0.5, we take x₀= 0.7 for instance, and the first iterate is x₁= 0.42, as shown

Since x₁ = 0.42 from x₀ = 0.7 is the same as the first iterate from the initial input x₀ = 0.3 in table 2, population function (3) will approach the same final iterate x = 0.5 from the initial x₀ = 0.3 or 0.7, as summarized in table 3. What is the physical meaning of this symmetry about x = 0.5? Let us suppose the maximum allowable rabbit population is 100 pairs. Then, x₀ = 0.3 and 0.7 correspond to the initial populations of 30 and 70 rabbit pairs, respectively. According to population model (3), the initial 30 rabbit pairs will increase up to the sustainable population level of 50 rabbit pairs. On the other hand, if we initially overpopulate above the sustainable level, some 20 rabbit pairs will eventually die off due to the lack of food supply or predator attacks. In this way, the fixed point x = 0.5 corresponds to the stable population level of 50 rabbit pairs.

Table 3. Iteration result

x₀	x₁	x₂	x₃	x₄	x₅
0.7	0.42	0.4872	0.49967	0.49999	0.5