Differential equation- solve for k

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In summary, biologists stocked a lake with 143 fish and estimated the carrying capacity (the maximal population for the fish of that species in that lake) to be 8000. The number of fish doubled in the first year. Assuming that the size of the fish population satisfies the logistic equation, the constant was determined to be .7115 and the size of the population after years was found to be 286.
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Homework Statement

Biologists stocked a lake with 143 fish and estimated the carrying capacity (the maximal population for the fish of that species in that lake) to be 8000. The number of fish doubled in the first year.

(a) Assuming that the size of the fish population satisfies the logistic equation

dP/dt = kP(1-P/N)

determine the constant , and then solve the equation to find an expression for the size of the population after years.

Homework Equations

none in particular

The Attempt at a Solution

I didn't think that this problem would be difficult but the answer I am getting doesn't make any sense. Basically I try to solve for P to start with. So dP/(P-P^2/N) = kdt and integrate

ln(P/(P-N)) = kt+c

It would make the most sense to solve for k right away so

(ln(P/(P-N)) - c)/t = k

The problem though is that N = 8000 while P will always be less than this. This will cause there to be a negative number inside the natural log and you can't take the log of a negative number. Where am I going wrong?

Last edited:
I get ln(P/(N - P)) = kt + C. For the log term to be defined, 0 < P < N.

Mark44 said:
I get ln(P/(N - P)) = kt + C. For the log term to be defined, 0 < P < N.

Really? Let me try to integrate this again and see what happens... hold on

Ya you're right on that. I must have screwed up when breaking I broke it up into partial fractions. Another part that confuses me is do I solve for p first and then solve for k or do I head straight for k? My answers are still sketchy...

I think you want to solve for C first, and then k. I'm not sure you have an initial condition, P(0), but you know that P(1) = 2P0.

Woops. Sorry that it didn't copy over in my original post. Starting fish population is 143 fish. So P(0) = 143 and P(1) = 286.

Ah! k is equal to .7115 ! Nice! I also found c to equal -4.006.

1. What is a differential equation?

A differential equation is a mathematical equation that describes the relationship between a function and its derivatives. It is commonly used to model natural phenomena in physics, engineering, and other fields.

2. Why do we need to solve for k in a differential equation?

In many cases, the value of k represents a constant or parameter in the equation that affects the behavior of the system being modeled. By solving for k, we can better understand and predict the behavior of the system.

3. How do we solve for k in a differential equation?

The specific method for solving for k will depend on the form of the differential equation. In general, we can use techniques such as separation of variables, substitution, or integration to isolate and solve for k.

4. What are some applications of solving for k in a differential equation?

Solving for k in a differential equation is often used in real-world applications such as predicting population growth, modeling chemical reactions, and understanding the behavior of electrical circuits.

5. Can differential equations with multiple unknowns be solved for k?

Yes, it is possible to solve for k in a differential equation with multiple unknowns. However, this will require more advanced mathematical techniques and may result in a system of equations that need to be solved simultaneously.

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