Suppose that a particular endangered species of fish has three distinct stages of life, each lasting one year. Each fish begins life as a hatchling (H), then becomes a juvenile (J), then an adult (A). At year n we denote the number of fish in each stage by Hn, Jn and An.

• Each year, each adult produces 3 hatchlings and each juvenile produces 1 hatchling.
• Each year 50% of hatchlings survive to become juveniles.
• Each year 30% of juveniles survive to become adults.

a. Write a system of linear equations (in matrix form) to model this population.

b. Now assume that, each year, j juveniles and a adults are added to the population by hard-working conservationists.
Write a system of linear equations (again in matrix form)
to model the population with this additional condition. If H0 = 100, J0 = 70 and A0 = 20, find H2, J2 and A2 in terms of j and a.

c. Over time the population will tend to a steady state, with numbers that don’t change from year to year. You can find this steady state by setting Hn+1 = Hn = H∞, and similarly for J and A.
i. Find H∞, J∞ and A∞ if j = a = 0.
ii. Find H∞, J∞ and A∞ if j = a = 1.

d. Find values for j and a such that H∞ = 234, J∞ = 120, A∞ = 38.