Question

You are studying a density-dependent elephant population that has the following relationships for the birth rate (b) and the death rate (d), as functions of N.

b=0.10+0.03N-0.0005N^2

d=0.20+0.01N

Plot these functions on the same graph (on a separate page). Using Excel can help you with this: plug into the equation a range of numbers for N from 1 to 60 or so elephants. How many equilibrium points are shown by your graph? You do not have to calculate the values of the equilibria, but indicate where they occur on the graph with arrows that show the population size at the equilibrium point(s) (2 points)

Stability is a quality of an equilibrium point such that when the population is perturbed away from that point, it will return to that equilibrium density. To determine the stability, predict what will happen when N is near, but not at, each equilibrium point and determine the trajectory of the population by comparing the relative magnitudes of b and d at your chosen points. Which one(s) of the equilibrium points are stable? Explain your reasoning. (2 points)

How does this model differ from the logistic growth equation? (1 point)

Answer 1

There are two equilibrium points shown in the graph, where b and d graph meets.

To determine the stability, predict what will happen when N is near, but not at, each equilibrium point and determine the trajectory of the population by comparing the relative magnitudes of b and d at your chosen points.

At point 1 the population grows in size as b > d, whereas at point 2 b < d hence population declines.

Which one(s) of the equilibrium points are stable? Explain your reasoning.

Point 2 is stable as the population tends to decline but at point 1 it tends to increase.