The fish population P(t) in a lake is suddenly attacked by adisease that render the fish unable to reproduce. Assuming that thedeath rate of the fish is inversely proportional to the square rootof the fish population (i.e. k1 = Î²/â(P) where Î² is aconstant), perform the following tasks:

1. Write a differential equation that models the population offish as a function of time. Assume that the fish are infected bythe disease at t = 0 and define your constants.

2. Solve the differential equation with the initial condition thatP(0) = P0.

3. What is the half-life of the fish population?

4. If there were 900 fish initially, and 441 are left after 6weeks, how long does it take for all the fish to die? Assume thatthe disease was caused by radioactive waste dumped in the lake. Itis discovered that another population of fish, S(t), spontaneouslymutate into the original fish at a rate of k2 (per week) and neverdie. The population S(t) are born at a rate k3.

5. Draw a diagram that describes the life cycle of these twofish populations including your rate constants.

6. Write the differential equations describing the populationsof P(t) and S(t).