# Modeling with first order Diff Eq.

## Homework Statement

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:

## Homework Equations

dC/dT = Rate In - Rate Out

## The Attempt at a Solution

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.

I probably started it off wrong, but

dC/dT = mC (where m represents the mass, since it is proportional to the time)

=>
dC/C = mdt

=>

ln|c| = mt+c

=>

Ke^mt = C

Probably started off on the wrong foot, the only difficult part of these problems are actually setting up the equation, the differentiation after isn't a problem. Some help being guided in the right direction would be a great help!

dm(t)/dt = (riCi -roCo)

ri is rate in Ci concentraion initial

ro rate out and Co concentrain out

this give mass as a function of time, i found this way really helpful.

in most question you are probly missing the Ci orCi

then you Ci or Ci = m(t)/volume

hope this helps

P.S i did not read you question in detail...