Growth Model - Leslie Matrices

Fish Farm Data | |||

Size Class | % Growing To Next Class per Week | % Dying | % Reproducing |

< 1 oz. | 50 | 20 | 0 |

1 oz. - 1 lb. | 30 | 10 | 15 |

> 1 lb | 0 | 5 | 35 |

From this we form the following transition matrix:

1-0.50-0.20 0.15 0.35 0.30 0.15 0.35 G = 0.50 1-0.30-0.10 0 = 0.50 0.60 0.0 0 0.30 1-0.05 0.0 0.30 0.95Note, in contrast to the Markov transition matrix, the sums of the columns aren't necessarily equal to 1. This is because there is possibly increase or decrease of the total population.

Let x^{(n)} = (x1^{(n)}, x2^{(n)},
x3^{(n)})^{T} be the number of fish in the 3 classes after
n weeks. Then we have the simple relationship

x^{(n+1)} = G x^{(n)}

Suppose when we start we have 1000 fish in class 1, i.e. x^{(0)} =
(1000,0,0)^{T}, then

x^{(1)} = G x^{(0)} = (300, 500, 0)^{T}

x^{(2)} = G x^{(1)} = (165, 450, 150)^{T}

x^{(3)} = G x^{(2)} = (169.5,352.5,277.5)^{T}

x^{(10)} = G^{10} x^{(0)} ~ (407,402,767)^{T}

Thus at first the population declines but as soon as we get enough of the larger fish the population begins to grow.

n = 10; for i = 1:n x = G*x; endAt the end we have a vector x of results after n weeks.MATLAB has some very power plotting routines. We can use them to visualize our results. Basically we just need to create a matrix of values we want to plot. Set up G and the initial x again and run this code:

n = 10; results(:,1) = x; for i = 1:n x = G*x; results(:,i+1) = x; endIf you typedisp(results)you'll get a 3 by 11 matrix of values where each column represents the state at a certain week. We can plot these results easily byplot(results')(Note the ')

## Exercises

1. For the fish farm model above, we had a total of over 1500 fish after 10 weeks starting with only 1000 fish. Adjust the death rate for the class of large fish to get the total after 10 weeks to be as close to 1000 as you can. This means adjust the (3,3) value of G down or up and then compute the result after 10 weeks. When you get the total to be close to 1000, record the death rate. Note that the death rate is not just the value of G(3,3) but is related to it.2. From the following data for University of ABC

Yearly Progress & Retention Data | |||

Class | % Moving to Next Class | % Droping out | % Graduating |

Freshman | 50 | 40 | 0 |

Sophomore | 60 | 20 | 0 |

Junior | 70 | 10 | 0 |

Senior | 15 | 10 | 70 |

5th-Yr Senior | 90 | 5 | 0 |

Graduated | 0 | 0 | 100 |

Use this data and an incoming class of 4200 to predict how many will have graduated after 4, 5 and 6 years.

After 4 years will there be any students that are still considered Freshmen or Sophomores?