Modeling with First ODE

In summary: Then you have M(t) for t> 5.In summary, the initial mass of fish in a lake was 7 thousand pounds on January 1st, 2001. After a 4-year moratorium, the mass of fish increased to 11.54 thousand pounds on the same date in the following year. When the moratorium ended in 2005, a company was given the exclusive rights to harvest 24 thousand pounds of fish per year from the lake. To model the mass of fish, a differential equation was set up with the rate of reproduction proportional to the mass and the rate of harvest subtracted from the rate of growth. By solving this equation with initial conditions, the constant of integration was found and
  • #1
HerpaDerp
6
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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.
 
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  • #2
HerpaDerp said:
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.
 

1. What is a First Order Differential Equation (ODE)?

A First Order Differential Equation (ODE) is an equation that involves a function and its first derivative. It is commonly used in mathematical modeling to describe the relationship between a change in the dependent variable and the independent variable.

2. What are the steps for solving a First Order ODE?

The steps for solving a First Order ODE are as follows:

  1. Separate the variables by moving all the terms containing the dependent variable to one side of the equation.
  2. Integrate both sides of the equation with respect to the independent variable.
  3. Add the constant of integration to the solution.
  4. If necessary, use initial conditions or boundary conditions to solve for the constant of integration.

3. How is a First Order ODE used in modeling?

A First Order ODE is used in modeling to describe the relationship between a changing variable and its independent variable. It can be used to model various real-world phenomena such as population growth, chemical reactions, and heat transfer.

4. What is the difference between an explicit and implicit First Order ODE?

An explicit First Order ODE is one where the dependent variable is isolated on one side of the equation. This means that the solution can be easily obtained by solving for the dependent variable. In contrast, an implicit First Order ODE has the dependent variable on both sides of the equation and requires more advanced techniques to solve.

5. What are some common techniques for solving First Order ODEs?

Some common techniques for solving First Order ODEs include separation of variables, integrating factors, substitution, and using known solutions. Other numerical methods such as Euler's method and Runge-Kutta methods can also be used to approximate solutions.

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