Consider a resource whose net growth in biomass (X) each year F(X) isgiven by a logistic growth...

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Consider a resource whose net growth in biomass (X) each year F(X) isgiven by a logistic growth function where r=.1 and K=1. (You can think of theunits of the biomass as being millions of animals so that the carrying capacity ofthe environment is one million animals.)F(X) ? rX(1? X / K) ? .1X(1? X)Suppose that harvest occurs according to the following relationship: Yield =Y=qEX, where E is “effort” pursuing the resource (number of boats for a fisheryor number of hunters for big game hunting) and q = .0025.
	
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SOS 325 
 
Homework #5: Renewable Resources, Common Property and Open Access 
 
Answer all questions on this sheet of paper and turn it in in-class on Thursday, 
December 6th.  If you cannot turn your assignment in in-class then you can place it in 
my mailbox (4th floor Wrigley Hall – you can leave your work with Theresa the 
receptionist) or submit it electronically via Blackboard.   Do NOT email it to me or the TA.  
Make your answers clear, legible and to the point.  
 
 XXXXXXXXXXpts.) Consider a resource whose net growth in biomass (X) each year F(X) is 
given by a logistic growth function where r=.1 and K=1.  (You can think of the 
units of the biomass as being millions of animals so that the carrying capacity of 
the environment is one million animals.)   
 
F(X)?rX(1?X /K)?.1X(1?X)  
 
Suppose that harvest occurs according to the following relationship: Yield = 
Y=qEX, where E is “effort” pursuing the resource (number of boats for a fishery 
or number of hunters for big game hunting) and q = .0025.   
 
a. (10 pts.) Graph the growth function of the biomass F(X), being clear to 
label maximum sustainable yield (MSY) harvest, biomass levels at MSY 
and carrying capacity on your graph.  What is the maximum amount of 
harvest that can be sustainably achieved in this system?  What level of 
biomass generates this amount of harvest?b. (15 pts.) Now, using the equation derived in class and in Keohane and 
Olmstead, graph the curve showing the sustainable harvest for any given 
amount of effort (the “harvest/effort” or “yield/effort” curve).  What 
level of effort exhausts the resource?   What amount of effort is 
consistent with the maximum sustainable harvest of the resource?

c. (15 pts.) Suppose ecologists discover that r=.05 rather than .1 so that the 
growth rate for any level of biomass is now much lower than thought.  
Graph the new yield/effort curve.  How does this change your...

Robert · Accepted Answer

Answer to 1: 
r  = 0.1 and K = 1 
Growth function F(X) = rX(1 – X/K) = 0.1X(1-X), where X = biomass 
Harvest/Yield (Y) = qEX, where E denote ‘level of effort’. 
a. 
F(X) = 0.1X(1-X) 
At maximum dF(X)/dX =0 i.e. 
0.1 -0.2X = 0, implies harvest maximizing level of biomass X* = 0.5 
So the maximum amount of harvest = 0.1*0.5*(1-0.5) = 0.025 
All these values have been represented in the figure1 below: 
Figure1:
b. 
Harvest (Y) = qEX = 0.0025EX   [given q = 0.0025] 
And Growth function F(X) = rX(1 – X/K) = 0.1X(1-X) 
At sustainable equilibrium, Harvest (Y) = F(X) i.e. 
0.0025EX = 0.1X(1-X), simplifying this we get steady state stock level as; 
XSS (ESS) = 1-0.025ESS 
Substituting this in harvest function: Y = qEX to get sustainable yield/effort curve as; 
YSS = 0.0025*ESS*(1-0.025ESS)  ……………….  (1) 
Level of effort that exhausts the resource is determined at a point where YSS = 0 i.e. 
0.0025*ESS*(1-0.025ESS) = 0 implies ESS = 40 
This implies level of effort that exhausts the resource is 40.
Further, at the maximum dYSS/dESS = 0 i.e. 
0.0025 – 0.000125ESS = 0, implying ESS*  = 20 
So the level of effort that maximizes the sustainable harvest is 20. 
These values have been shown in the figure2 below;  
[Note: the figure is yield/effort curve: YSS = 0.0025*ESS*(1-0.025ESS)] 
Figure2:
c. 
When r = 0.05 instead of 0.1, the growth function is given as; 
F(X) = 0.05X(1-X) 
And harvest (Y) = 0.0025EX 
At the sustainable equilibrium: F(X) = harvest (Y) i.e. 
0.05X(1-X) = 0.0025EX, simplifying this we get steady state stock level as; 
XSS (ESS) = 1-0.05ESS …………………..   (2) 
Substituting this in harvest function: Y = 0.

Consider a resource whose net growth in biomass (X) each year F(X) is given by a logistic growth function where r=.1 and K=1. (You can think of the units of the biomass as being millions of animals so...

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