How to Model Fish Population Dynamics Using First Order Differential Equations?

[HerpaDerp]

The initial mass of fish in a lake was 7 thousand pounds on January 1st, 2001. Since the time, there was a 4-year moratorium on the harvesting on this specific type of fish. This species of fish reproduce at a rate proportional to the mass and by next year on the same date, there were 11.54 thousand pounds of fish.

After the moratorium (2005) ends, a certain company is given exclusive rights to harvest 24 thousand pounds of fish per year from the lake.


I need to set up a Diff Eq. modeling the mass of the species of fish in thousands at time T (T in years), and T=0 on Jan. 1 2001. Then a solution for the differential equation must be found.
The only way I know to start off is:

dC/dT = Rate In - Rate Out

the first reproduce at rate prop. to mass. and we model the fish using mass, so.

dM/dt = aM where M is the total mass of fish and a is some constant.

that was before the moratorium.

24000 lb fish harvested per year. so

dM/dt = aM - 24000 after the moratorium.

After this, I am rather confused and lost.

 

[HallsofIvy]

The initial mass of fish in a lake was 7 thousand pounds on January 1st, 2001. Since the time, there was a 4-year moratorium on the harvesting on this specific type of fish. This species of fish reproduce at a rate proportional to the mass and by next year on the same date, there were 11.54 thousand pounds of fish.

After the moratorium (2005) ends, a certain company is given exclusive rights to harvest 24 thousand pounds of fish per year from the lake.


I need to set up a Diff Eq. modeling the mass of the species of fish in thousands at time T (T in years), and T=0 on Jan. 1 2001. Then a solution for the differential equation must be found.
The only way I know to start off is:

dC/dT = Rate In - Rate Out

the first reproduce at rate prop. to mass. and we model the fish using mass, so.

dM/dt = aM where M is the total mass of fish and a is some constant.

that was before the moratorium.

Yes, that's right. Now you solve that equation for M in terms of both a and t and then can use the fact that M(0)= 7000 and M(1)= 11540 (as you said, t= 0 corresponds to 2001) to find both a and the constant of integration. Then find M(5).

24000 lb fish harvested per year. so

dM/dt = aM - 24000 after the moratorium.

After this, I am rather confused and lost.

By this time you know a and M(5) so you can solve this differential equation with M(5) equal to the value you just found to determine the constant of integration.