In an island chain, species richness (S) increases with island area (A) according to the equation, $S = 4.3A^{0.55}$. Which one of the following graphs best represents this equation?
Show Hint
Power law relationships between area and species richness result in concave upward curves, indicating diminishing returns as the area increases. If the exponent is less than 1, the curve will flatten as the area grows.
The equation given is $S = 4.3A^{0.55}$, where $S$ represents species richness and $A$ represents the island area. This is a power law relationship, where the exponent 0.55 indicates how species richness increases with the area of an island.
Step 1: Analyzing the relationship
The equation shows that species richness $S$ increases with island area $A$, but the increase in $S$ is not linear. Instead, it follows a diminishing rate of increase. This means that as the area of the island increases, the species richness does increase, but at a slower rate. This type of relationship is characteristic of power laws with exponents less than 1, where the curve becomes flatter as the independent variable (in this case, area) increases.
Step 2: Visualizing the curve
To visualize the curve, let's break down the equation:
- When $A$ is small, the value of $A^{0.55}$ will be significantly lower than $A$. As $A$ increases, the impact of the exponent 0.55 becomes more apparent, and the rate at which species richness increases slows down.
- This results in a concave upward curve, which indicates rapid increases in species richness for small island areas, but as the area grows, the increase in species richness becomes less pronounced.
Step 3: Identifying the correct graph
Among the given graphs, graph (ii) best represents the behavior described by the equation $S = 4.3A^{0.55}$. It shows a steady increase in species richness with island area, but the rate of increase gradually slows down as the island area increases, which is consistent with a power law with an exponent less than 1.
- Graph (i) shows a linear relationship, which is not correct because the equation is not linear.
- Graph (iii) is an exponential curve, which would imply that species richness increases rapidly at a constant rate, which is not the case here.
- Graph (iv) shows a curve that starts high and then decreases, which does not align with the expected behavior of the equation.
Thus, the correct graph is (ii).
