Question

Here you will be plotting results and commenting in written form. Hence, you must show your work in the form of some numbers, some plots, and some simple statements. You may write your answers out on a sheet of paper (pen/pencil & paper), do this in an electronic spreadsheet or word processing document. You may submit your work and answers electronically or turn it in physically before end of class (1:15 pm) on February 27, 2018. In discrete time, the logistic equation can be written (see the hint on Problem Set HW 2, question 3): where N is the population, our time is represented by “t” (let’s use days), K is the “carrying capacity”, r is the growth rate. Consider the growth of a population of bacteria. At initial time, N(0) = 0.2 bacteria. The net growth rate has a value, r = 1. Carrying capacity, K = 1. Don’t worry about the units here (you can think of them as fractions of 1000 individuals). You are going to plot out the population to the 30th day, starting at day 0. You can use a spreadsheet or your calculator and paper/pencil, or software like matlab (if you use that). There are many graphical software packages that can manage plotting this. a) What is the population on each day - plot this. b) Change the net growth rate to, r = 2. What is the population on each day – plot this. Is a pattern emerging? What would you call this pattern? c) Change the net growth rate to, r = 3. What is the population on each day – plot this. Is this pattern similar to b or different? What is happening to this population? Is it reaching carrying capacity, K? d) change the net growth rate to, r = 3.001 (a tiny change). Now what happens?

Answer 1

Logistic Growth equation:

a). Based on the formulae, the following table was made to show per day growth at r=1

time (day)	Pop.        N	r	Carrying Capacity K	dN/dt	N+dN/dt
1	0.2	1	1	0.16	0.36
2	0.36	1	1	0.2304	0.5904
3	0.5904	1	1	0.241828	0.832228
4	0.832228	1	1	0.139625	0.971853
5	0.971853	1	1	0.027355	0.999208
6	0.999208	1	1	0.000792	0.999999
7	0.999999	1	1	6.28E-07	1
8	1	1	1	3.94E-13	1
9	1	1	1	0	1
10	1	1	1	0	1
11	1	1	1	0	1
12	1	1	1	0	1
13	1	1	1	0	1
14	1	1	1	0	1
15	1	1	1	0	1
16	1	1	1	0	1
17	1	1	1	0	1
18	1	1	1	0	1
19	1	1	1	0	1
20	1	1	1	0	1
21	1	1	1	0	1
22	1	1	1	0	1
23	1	1	1	0	1
24	1	1	1	0	1
25	1	1	1	0	1
26	1	1	1	0	1
27	1	1	1	0	1
28	1	1	1	0	1
29	1	1	1	0	1
30	1	1	1	0	1

b). The population growth with the r=2

time (day)	Pop.        N	r	Carrying Capacity K	dN/dt	N+dN/dt
1	0.2	2	1	0.32	0.52
2	0.52	2	1	0.4992	1.0192
3	1.0192	2	1	-0.03914	0.980063
4	0.980063	2	1	0.03908	1.019142
5	1.019142	2	1	-0.03902	0.980125
6	0.980125	2	1	0.03896	1.019085
7	1.019085	2	1	-0.0389	0.980186
8	0.980186	2	1	0.038842	1.019028
9	1.019028	2	1	-0.03878	0.980247
10	0.980247	2	1	0.038725	1.018972
11	1.018972	2	1	-0.03866	0.980308
12	0.980308	2	1	0.038609	1.018917
13	1.018917	2	1	-0.03855	0.980368
14	0.980368	2	1	0.038494	1.018861
15	1.018861	2	1	-0.03843	0.980427
16	0.980427	2	1	0.03838	1.018807
17	1.018807	2	1	-0.03832	0.980486
18	0.980486	2	1	0.038267	1.018753
19	1.018753	2	1	-0.03821	0.980544
20	0.980544	2	1	0.038155	1.018699
21	1.018699	2	1	-0.0381	0.980602
22	0.980602	2	1	0.038044	1.018645
23	1.018645	2	1	-0.03799	0.980659
24	0.980659	2	1	0.037933	1.018593
25	1.018593	2	1	-0.03788	0.980716
26	0.980716	2	1	0.037824	1.01854
27	1.01854	2	1	-0.03777	0.980772
28	0.980772	2	1	0.037716	1.018488
29	1.018488	2	1	-0.03766	0.980828
30	0.980828	2	1	0.037609	1.018437

Is the pattern emerging?

No.

c).Growth rate with r=3

Is the pattern similar to b? What is happening?

No, the pattern is different than that of b. The rapid growth of population causes using up of the resources leading to the decline in the population below the carrying capacity. The population lesser than the carrying capacity again grows. This led to the zig-zag pattern of the growth curve.

d)

Now what happens???

The pattern of the growth curve obtained weight the r=3.001 is almost similar to r=3.0.