Understanding the Impact of Fishing on Fish Population Growth

[Gleveniel]

Hey everyone, I'm new here and wanted to see if I've gotten this question right, thanks in advance.

1. Consider a lake that is stocked with walleye pike and the population of pike is governed by the logistic equation:
P' = 0.1P(1-P/10)
where time is measured in days and P in thousands of fish. Supposed that fishing is started in this lake and that 100 fish are removed each day.

a) Modify the logistic model to account for the fishing.
b) Find and classify the equilibrium points for your model.
c) Use qualitative analysis to completely discuss the fate of the fish population with this model. In particular, if the initial fish population is 1000, what happens to the fish as time passes? What will happen to an initial population of 2000 fish?

2. The attempt at a solution
a) I got: P' = 0.1P(1-P/10) - .1
b) I set P' equal to 0 and solved, I then got 1.12702 and 8.87298.
c) Drawing a phase line, I see that any number less than 1.12702 results in a negative rate in population, therefore an initial population of 1000 would die off and eventually reach 0. As for the initial population of 2000, the phase line shows a negative growth until the equilibrium point, therefore the initial 2000 fish will die off until they reach that equilibrium point of 1.12702.

Is what I did correct? I'm not sure, and was hoping to get insight.

 

[mighty2000]

Hello and welcome to the forum! Your attempt at solving the problem is a good start, but there are a few things that can be improved upon.

a) You are correct in modifying the logistic model to account for fishing, but it should be P' = 0.1P(1-P/10) - 0.1, as this represents the removal of 100 fish each day.

b) Your equilibrium points are correct, but it would be helpful to classify them as stable or unstable. The equilibrium point at P = 1.12702 is stable, meaning that if the initial population is less than this, it will approach this value over time. The equilibrium point at P = 8.87298 is unstable, meaning that if the initial population is greater than this, it will eventually decrease and approach the stable equilibrium point.

c) Your qualitative analysis is correct, but it would be helpful to explain why the initial population of 1000 fish will eventually reach 0. This is because the negative rate of population growth (indicated by the negative slope on the phase line) will continuously decrease the population until it reaches 0. As for the initial population of 2000 fish, it will decrease until it reaches the stable equilibrium point of 1.12702, where it will then remain at that level.

Overall, your approach and understanding of the problem is good. Keep up the good work and don't be afraid to ask for clarification or help if needed. Best of luck with your studies!