Question

Consider a lake that is stocked with walleye pike and where population of pike is governed by the logistic equation P'= 0.1P(1-10 where time is measured in days and P in thousands of fish. Suppose that fishing is started in this lake and that 1o0 fish are removed each day. (a) Modify the logistic model to accurately account for the fishing. b) Find and classify the equilibrium points for your model. (c) Use qualitative analysis to discuss the fate of the fish population with this model, in particular if P(O) 1 and if P(0) 2 (again in thousands of fish)

Answer 1

a) we need to modify the logistic model to account for fishing if the population is governed by the logistic equation p' = 0.1p(1- p/10) p thousand of fire h = 0.1 therefore p1 = 0.1(1-p/10) - 0.1 b) we need to find the equilibrium point for the model p' = 0.1p(1-p/10)-0.1 The equilibrium point is obtained by p' = 0 0.1p(1- p/10) -0.1 = 0 0.1(p- p^2 /10)-0.1 = 0 0.1(p-p^2 /10 -1) = 0 p^2 -10p + 10 = 0 The roots of the above equations are p = 10 plusminus squareroot 100-40 /2 = 5 plusminus squareroot 15 p' = 5- squareroot 15 one equilibrium point and other point is 5 + squareroot 15 c) we need to use the equilibrium use the quantitative analysis to completely discuss the fate of the fish population and the effect on the fish population if the initial population is 1000 and 2000 respectively we have the modified logistic equation p' = 0.1p(1-