Question

Please help with part c of ii - v

2 Snow Crab Fishing Suppose the snow crab population in the Gulf of Saint-Lawrence grows according to an instanta- neous logistic growth that is given by, F(x) X1- where X is the biomass (ie. quantity, in tons) of snow crabs, r = 0.1 is the intrinsic instantaneous (a) What are the two biomass levels X that define the two biological equilibria of the snow crab (b) What is the biomass that gives the maximum sustainable yield (MSY), i.e. what is XMSY? (c) Suppose that snow crabs are being harvested at no cost, and assume that harvest is equal to (i) There are two biomass levels X1 and X2, with Xi < X2, for which harvest is sustainable (ii) What will happen to the population of snow crabs over time if we start harvesting snow (iii) What will happen to the population of snow crabs over time if we start harvesting snow (iv) What will happen to the population of snow crabs over time if we start harvesting snow (v) Can we reach biomass level X1 over time if we start at any X f X1? Why or why not? growth rate of the biomass, and K = 5000 is the carrying capacity population assuming there is no harvesting? [0.25 point] What is the MSY, i.e. what is F(XMsY)? Show your work. [0.5 point] H 100 tons per period of time What are X1 and X2? Show your work. [0.75 point] crabs when X < X1? Explain. [0.25 point] crabs when Xi < X < X2? Explain. [0.25 point] crabs when X X2? Explain. [0.25 point] Explain. [0.25 point]

Answer 1

Snow crab population is given by:

F(X)= rX(1-X/K)

X=biomass

r=0.1, rate of growth of biomass

K= 5000 carrying capacity

c)

ii) if X< X₁

F(X)= .1X (1-X/5000)

= (500X - .1 X²)/5000

F(X₁)=(500X₁ - .1 X₁²)/5000

As we know X₁> X, F(X₁)>F(X), thus population of snow crabs will be less than that of X₁ biomass.

Since harvest is given as 100, and F(x) is less it will be less than 100.

iii) By the same explanation, since X2> X> X₁, thus. F(X2)>F(X)>F(X₁). Thus snow crab population will increase at a decreasing rate over time.

iv) When X> X2 , F(X)>F(X2)

Thus, snow crab population will increase at increasing rates as the biomass is higher than any other biomass.