The Fate of a Stocked Lake Population Under Fishing Pressure

[Shackleford]

Consider a lake that is stocked with walleye pike and that 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. Suppose 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.

P' = 0.1P(1- P/10) - 0.1

(b) Find and classify the equilibrium points for your model.

Equilibrium points: 1.12702 (asymptotically stable), 8.87298 (unstable)

(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 having 2000 fish?

With 1000 fish, the population will decrease to zero.

With 2000 fish, the population will increase to the 8.8K fish EQ point.

I hope I did this correctly.

 

[Billy Bob]

I like your model and your EQ pts, but how did you decide stable vs. unstable?

 

[Shackleford]

I like your model and your EQ pts, but how did you decide stable vs. unstable?

Hey, Bob.

Well, I drew a phase line and tested the sign of P' in each of the three intervals. I just redid it and got a different answer.

Each of the intervals has a negative sign, so they are all unstable.

 

[Billy Bob]

Each of the intervals has a negative sign

That's still not what I got.

 

[Shackleford]

That's still not what I got.

Oops. The 1.2 < t < 8.8 interval is positive. The other two are negative. So, the 8.8 EQ point is stable. I think I had that originally.

 

[Billy Bob]

Oops. The 1.2 < t < 8.8 interval is positive. The other two are negative. So, the 8.8 EQ point is stable. I think I had that originally.

That's what I got.

 

[Shackleford]

That's what I got.

Great. So, my answer to (c) should be correct.