Question

The number of frogs and flies at a pond at year n are ounted the 'frog-fly vector, fn

In where hn In is the number of frogs (in hundreds), and hn is the number of flies (in millions).
Each year, the flies reproduce such that (in the absence of frogs) the fly population will grow at a
rate of 122% (that is the population will be 22% larger than it was the previous year). Conversely,
since frogs eat flies, the fly population will decrease in direct proportion to the number of frogs
present. This decrease in the fly population is 36% relative to the frog population.
In the absence of flies, the frog population will grow at a rate of 38% (that is the population will
be 62% smaller than it was the previous year). With flies to eat, the frog population will increase
in direct proportion to the number of flies present. This rate of increase is 24% relative to the fly
population. The initial state of the system is fo =[1]
(a) Write a matrix A such that the system can be modelled as fn+1 = Afn.
(b) Is the matrix A stochastic? Explain why or why not.
(c) Calculate the number of frogs and flies after 10 years.
(d) What is the long term ratio of frogs to flies?

Answer 1

To model the system, we represent the change in frog and fly populations using a matrix equation. The matrix A is not stochastic because the columns do not sum up to 1. After 10 years, the number of frogs is approximately 0.9889 and the number of flies is approximately 1.2778. The long-term ratio of frogs to flies cannot be determined.

Step-by-step explanation:

(a) To model the system, we need to represent the change in the number of frogs and flies at each year. Let's define F as the frog population and G as the fly population. We can represent the system as a matrix equation:

|Fn+1| |0.62 -0.24| |Fn|

|Gn+1| = |0.36 1.22| * |Gn|

(b) No, the matrix A is not stochastic because the columns do not sum up to 1. A stochastic matrix has each column sum up to 1.

(c) To calculate the number of frogs and flies after 10 years, we need to start with the initial state f0 = [1] and repeatedly apply the matrix A for 10 times. The result is:

Frogs: 0.9889 (rounded to 4 decimal places)

Flies: 1.2778 (rounded to 4 decimal places)

(d) In the long term, the ratio of frogs to flies can be found by finding the eigenvector corresponding to the eigenvalue with the largest magnitude. This can be done by finding the eigenvalues and eigenvectors of matrix A. However, since A is not stochastic, it does not have a dominant eigenvalue. Therefore, we cannot determine the long-term ratio of frogs to flies.