Plant growth: The amount of growth of plants in an ungrazed pasture is a function of the amount of plant biomass already present and the amount of rainfall.18 For a pasture in the arid zone of Australia, the formula
gives an approximation of the growth. Here R is the amount of rainfall (in millimeters) over a 3-month period, N is the plant biomass (in kilograms per hectare) at the beginning of that period, and Y is the growth (in kilograms per hectare) of the biomass over that period. (For comparison, 100 millimeters is about 3.9 inches, and 100 kilograms per hectare is about 89 pounds per acre.)
a. Solve Equation (2.6) for R.
b. Ecologists are interested in the relationship between the amount of rainfall and the initial plant biomass if there is to be no plant growth over the period. Put Y = 0 in the equation you found in part a to get a formula for R in terms of N that describes this relationship.
c. Use the formula you found in part b to make a graph of R versus N (again with Y = 0). Include values of N from 0 to 800 kilograms per hectare. This graph is called the isocline for zero growth. It shows the amount of rainfall needed over the 3-month period just to maintain a given initial plant biomass.
d. With regard to the isocline for zero growth that you found in part c, what happens to R as N increases? Explain your answer in practical terms.
e. How much rainfall is needed just to maintain the initial plant biomass if that biomass is 400 kilograms per hectare?
f. A point below the zero isocline graph corresponds to having less rainfall than is needed to sustain the given initial plant biomass, and in this situation the plants will die back. A point above the zero isocline graph corresponds to having more rainfall than is needed to sustain the given initial plant biomass, and in this situation the plants will grow. If the initial plant biomass is 500 kilograms per hectare and there are 40 millimeters of rain, what will happen to the plant biomass over the period?