american bison

The Leslie matrix for the American bison is given by

A =(0, 0, 0.42;

0.60, 0, 0;

0, 0.75, 0.95)

The population is divided into calves, yearlings, and adults (age two years or more).

We would like to see how the whole population and the individual classes (calves,

yearlings, and adults) changes over time and how those results vary when we change

the various parameters in the Leslie matrix for the bison population. Thus, if the

matrix

x(0) =

(x1;

x2;

x3)

represents the American bison population at time t = 0, where x1 is the number of

calves, x2 is the number of yearlings, and x3 is the number of adult bison, then

x(t) = A^(t)x(0)

gives the bison population at time t. Suppose, at t = 0, there are 42 calves, 0

yearlings, and 95 adults.

Tasks

1. Using the Matlab skills you have learned in this Unit write an m-file that

(a) Find the number of calves, yearlings, and adults at times

t = {1, 2, 3, 4, 5, 10, 20, 25, 50, 75}

(b) Make a graph that has times t = 0 to 100 on the horizontal axis and

population size along the vertical axis. Plot the calf, yearling, and adult

population sizes for every time step between 0 and 100.

(c) Change the adult bison fecundity to 0.2, 0.42, 1.0, and 1.4. How do these

changes aect the bison population equilibrium structure? Create one

graph that has adult fecundity along the horizontal axis, and equilibrium

population structure along the vertical axis with a data set plotted for each

population class (calves, yearlings, and adults).

(d) Change the calf survival rate to 0.3, 0.5, 0.6, 0.7, and 0.85. How do these

changes aect the the bison population equilibrium structure? Create one

graph that has calf survival rate along the horizontal axis, and equilibrium

population structure along the vertical axis with a data set plotted for each

population class (calves, yearlings, and adults).

(e) Change the adult survival rate to 0, 0.3, 0.5, 0.75, 0.95, and 0.99. How do

these changes aect the bison population equilibrium structure? Create

one graph that has adult survival rate along the horizontal axis, and equi-

librium population structure along the vertical axis with a data set plotted

for each population class (calves, yearlings, and adults).

2. Using Matlab, try to discover any parameter changes that cause the population

to die out by t = 200. You do not need to write an m-le for this exploration.

What conclusions can you make about the possibility of the extinction of the

American bison based on this exploration?

Matlab and Mathematica

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