10 000 trout eggs, 1000 baby trout and 800 adult trout are placed in a pond to establish a trout population.
In establishing this population
-
- eggs (`E`) may die (`D`) or they may live and eventually become baby trout (`B`)
- baby trout (`B`) may die (`D`) or they may live and eventually become adult trout (`A`)
- adult trout (`A`) may die (`D`) or they may live for a period of time but will eventually die.
From year to year, this situation can be represented by the transition matrix `T`, where
`{:(qquadqquadqquadqquadqquadtext(this year)),((qquadqquadqquadE,quad\ B,quad\ A,\ D)),(T = [(0,0,0,0),(0.4,0,0,0),(0,0.25,0.5,0),(0.6,0.75,0.5,1)]):}{:(),(),(E),(B),(A),(D):}{:(),(),(qquadtext(next year)):}`
- Use the information in the transition matrix `T` to
The initial state matrix for this trout population, `S_0`, can be written as
`S_0 = [(10\ 000),(1000),(800),(0)]{:(E),(B),(A),(D):}`
Let `S_n` represent the state matrix describing the trout population after `n` years.
- Using the rule `S_n = T S_(n – 1)`, determine each of the following.
- `S_1` (1 mark)
- the number of adult trout predicted to be in the population after four years (1 mark)
- The transition matrix `T` predicts that, in the long term, all of the eggs, baby trout and adult trout will die.
- How many years will it take for all of the adult trout to die (that is, when the number of adult trout in the population is first predicted to be less than one)? (1 mark)
- What is the largest number of adult trout that is predicted to be in the pond in any one year? (1 mark)
- Determine the number of eggs, baby trout and adult trout that, if added to or removed from the pond at the end of each year, will ensure that the number of eggs, baby trout and adult trout in the population remains constant from year to year. (2 marks)
The rule `S_n = T S_(n – 1)` that was used to describe the development of the trout in this pond does not take into account new eggs added to the population when the adult trout begin to breed.
- To take breeding into account, assume that 50% of the adult trout lay 500 eggs each year.
The matrix describing the population after one year, `S_1`, is now given by the new rule
`S_1 = T S_0 + 500\ M\ S_0`
where `T=[(0,0,0,0),(0.40,0,0,0),(0,0.25,0.50,0),(0.60,0.75,0.50,1.0)], M=[(0,0,0.50,0),(0,0,0,0),(0,0,0,0),(0,0,0,0)]\ text(and)\ S_0=[(10\ 000),(1000),(800),(0)]`
- Use this new rule to determine `S_1`. (1 mark)
This pattern continues so that the matrix describing the population after `n` years, `S_n`, is given by the rule
`S_n = T\ S_(n – 1) + 500\ M\ S_(n – 1)`
- Use this rule to determine the number of eggs in the population after two years (2 marks)