Math 152 Project 1

Discrete Population Models

Choose any computer in the lab in Ayres 324. Open Matlab by going to start - programs- Matlab-Matlab5.3.
A window for the Matlab program will pop up. At the Matlab prompt type cd F:\Applications\MATH\MATH151\mfiles.
Type the name of the mfile you wish to run, without the .m extension. This will run the program in Matlab.
For more information about the program you are running, you can go to F:\Applications\MATH\MATH151\mfiles and doubleclick on any file name. This will open the file in an editor window so that you can see detailed information about the program and see the program code itself.

The three files you need to run are:

exponen.m: Simulates exponential growth (also geometric growth) according to:

where a is the growth rate and b is the net immigration rate.

analog.m: Simulates logistic growth (analog of continuous logistic growth) according to:

where r is the growth rate.

logistic.m: Another discrete version of logistic growth is

where r is the growth rate and here x_nis beween 0 and 1.

Notice that all three of these models are first order difference equations. Thus you will need to input an initial condition x₀. You must also input the number of generations the model must run for.

For exponen.m run the model with values of a = 0.2, 0.9, 1.5 and values of b = -0.5, 0 0.5. Use the initial conditions x₀ = 1, 10 and run the models for at least 100 generations. Summarize your results by compari ng the behavior of the models (e.g. increasing or decreasing population) and rank them in order of which is fastest growing to the slowest growing (see table 1). There are nine different models you will run for each value of x_0.Do your resu lts seem reasonable? Why or why not? Print out a graph from the top, middle and bottom of your list.

For analog.m run the model with values of r = 0.6, 1.2, 2.1 and use the initial conditions x₀= 0.01, 1, 2, 10. Determine the long-term behavior of the model by using at least 100 generations. Does the pop ulation size approach a limit for each value of r? If so, what is the limit for each r? Does this limit depend on the value of x₀? (i.e. does the limit change when r stays the same but x₀ changes?) Print out only two or three grap hs to support your answers. (See table 2)

For logistic.m run the model with values of r = 0.9,2.3,3.2, 3.7 and use the initial conditions x₀ = 0.1, 0.5, 0.9. Again, determine the long-term behavior of the model by using at least 100 generations. Does the population size approach a limit for each value of r? If so, what is the limit for each r? Does this limit depend on the value of x₀? (i.e. does the limit change when r stays the same but x₀ changes?) Print out only two or three graphs to support your answers. (See table 3)

Table 1

x₀=1 , T = 100 (at least)

a	b	Growth/Decay/Constant	Rank
0.2	-0.5
0.2	0
0.2	0.5
0.9	-0.5
0.9	0
0.9	0.5
1.5	-0.5
1.5	0
1.5	0.5

x₀=10 , T = 100 (at least)

a	b	Growth/Decay/Constant	Rank
0.2	-0.5
0.2	0
0.2	0.5
0.9	-0.5
0.9	0
0.9	0.5
1.5	-0.5
1.5	0
1.5	0.5

Table 2

b = 100 (at least)

x₀	r	Is there a limit?	Limit	Does the limit depend on x₀?
0.01	0.6
1	0.6
2	0.6
10	0.6
0.01	1.2
1	1.2
2	1.2
10	1.2
0.01	2.1
1	2.1
2	2.1
10	2.1

Table 3

B = 100 (at least)

x₀	r	Is there a limit?	Limit	Does the limit depend on x₀?
0.1	0.9
0.5	0.9
0.9	0.9
0.1	2.3
0.5	2.3
0.9	2.3
0.1	3.2
0.5	3.2
0.9	3.2
0.1	3.7
0.5	3.7
0.9	3.7