1. According to the EF, is the human population living at, beyond or below the Earth’s natural biocapacity? For how long has this been the case? Is this sustainable? (4 marks) 2. According to the EF, is the Chinese population living at, beyond or below China’s natural biocapacity? For how long has this been the case? Is this sustainable? (4 marks) 3. If you assume that EF grows at a constant yearly rate, what is the approximate slope of the EF relationship with time for China? And what is its units? Give the equation of, and sketch this line, with EF on the vertical axis and year on the horizontal axis. (EF for 2012 is 3.4 and for 1961 was approximately 1). (4 marks) 4. If you assume that biocapacity grows at a constant yearly rate, what is the approximate slope of the biocapacity relationship with time for China? And what is its units? Give the equation of, and sketch this line, with biocapacity on the vertical axis and year on the horizontal axis. (biocapacity for 2012 is 0.9 and for 1961 was approximately 1). (4 marks) 5. If this trend continues, what will EF and biocapacity be in 2050? (Hint: use your equation from questions 3 and 4.) What will biocapacity minus EF be in 2050? Will there be a deficit or reserve? (4 marks) 6. Use the data presented in Figure 1 to suggest a policy to make land use sustainable. (4 marks)
The model describes the evolution of a population over time ??. ??(??) = ??0?? (??-??)?? (1) where ??0 is the population at time ?? = 0. 1. If the initial population is ??0 = 2000, the annual birth rate ?? = 0.03 (3% exponential birth rate) and annual death rate ?? = 0.02 (2% exponential death rate), what is the population after 1 year? Explain this result. (4 marks) 2. Find an expression for the growth rate (population change) of the population in terms of time and the parameters. Hint: differentiating with respect to time gives a rate of change. (4 marks) 3. If ??
The model in equation (1) in Part A describes a population untouched by humans: there is no harvesting. We can model what happens if we harvest ?? individuals from the population each period. The unlimited population growth with harvesting model is given by ??(??) = ???? (??-??)?? + ??-???? (??-??)?? (??-??) (2) where ?? is the birth rate, ?? is the death rate, and ?? is the number of individuals taken from the population in each period. 1. Confirm that the rate of population change is ????(??) ???? = (?? – ??)??(??) – ??. (3) (5 marks) 2. Give the meaning of the expression in equation (3). (4 marks) 3. Let ??0 = ??(0) = 2000, ?? = 0.05 and ?? = 0.02. At what rate can the population be harvested sustainably? Hint: The sustainable harvest, ??, will be the value that causes no change in the overall population, i.e. ????(??) ???? = 0. (4 marks) 4. Suppose an ecosystem is being sustainably harvested at exactly its replacement rate, and the population is constant. Now suppose the population experiences a brief disease epidemic, which causes 4% of individuals to perish. What will happen to the population if harvesting continues at the same rate? (4 marks) 5. Suppose a population of fish that follows this model is being harvested sustainably, as before. What will happen in the long run if, one day, a fisherman decides to throw one of the fish back? (4 marks) 6. This type of model is often said to have a “knife-edge” equilibrium. Explain what this means. (4 marks) 7. Governments often levy fishing quotas in areas where populations are at risk. Does this model shed any light on the effectiveness (or otherwise) of these policies? (Hint: what if someone cheats?) (4 marks)
A model linking these ideas was first proposed by a French mathematician Pierre Verhulst in 1838. The Verhulst model can be written as: ??(??) = ??0?? ???? 1+ ??0(?? ????-1) ?? (4) where ??0 is the initial population level. For simplicity, we’ll write the net reproduction rate simply as ??, rather than in terms of births and deaths. 1. What is the growth rate if the initial population is zero? (4 marks) 2. When the population is initially at its carrying capacity, i.e. ??0 = ??, what is the population at time ??? How does the population level change when the growth rate r increases? (4 marks) 3. Differentiate (4) to show that the rate of population growth at time t is ????(??) ???? = ????(??) (1 – ??(??) ?? ) (5) Remember to use the quotient rule to differentiate (4). You will find it easier if you carefully look for the terms that you expect to see in the final expression, and factor those out. (5 marks) 4. Does the population grow faster when it is above the environment’s carrying capacity (N>K), or below it (??