Discrete Population Growth Lab Report Example | Topics and Well Written Essays - 500 words - 1

Discrete Population Growth - Lab Report Example y, during a period of an extremely populous generation resources are brought to scarcity and life and reproduction are made harder for individuals, leading to a decline in population numbers. These ideas are well reflected by modeling the situation with a function assuming low values on both ends of the range of argument, and high values in and around its center. The equation below is a simple example of this kind of model, using a variable parameter k to account for variable reproductive strength of different species and/or a given species in different environments. The equation predicts the size of (n+1)-th population pn+1 as a quantity considered dependent only on the size of n-th population pn: Using the program Grapher by R. Decker, the first 100 terms of the sequence generated by equation [1] were generated and plotted for several initial situations characterized by different combinations of p0 and k. Graphs in Figures 1 to 4 show four situations combining the values of 0.5 and 0.8 for p0 and 1.5 and 2.5 for k. The graphs make clear the sequence converges rather quickly under these conditions. The limit values are 0.33 for k = 1.5 and 0.6 for k = 2.5, regardless of p0. This behaviour is seen for k values between 1 and 3. The graph in Figure 5 is different in that there are two values between which the population numbers keep alternating. The graph has been produced for values k = 3.2, p0 = 0.5. Experimenting with the program proved such behaviour is characteristic for values of k between 3.0 and 3.4 (see also Figure 6). For values of k exceeding 3.4, the lines observed in Figure 5 become increasingly split or blurred (see Figures 7 and 8) until k = 3.6, where the alternating course of the sequence gives way to chaotic behaviour (see Figures 9, 10, 11 and 12). In this range, changing the value of k by only 0.001 has profound impact on the sequence terms (compare Figures 9 and 10, and Figures 11 and 12). Figure 12, produced for k = 4.0, reveals an