Much of the early theoretical work in ecology on the interaction between species started with the Lotka-Volterra model of competition (pg 431). That model was itself a modification of the logistic model of the growth of a single population (pg 430).

Understanding the Lotka-Volterra model's predictions of the behavior of interacting populations depends on understanding the conditions at equilibrium, i.e., when the change in population density is zero.

Continuing the discussion in the textbook and following the predictions in the lesson, there are two equations at equilibrium that are algebraically derived from the Lotka-Volterra model on page 430 by setting dN/dt to zero (the condition at equilibrium).

Ni = Ki - aijNj Nj = Kj - ajiNi

The density of a species is equal to its carrying capacity minus the impact on its resources exerted by a competitor.

These two equations lead to four sets of inequalities.

Ki > Kj/aji and Kj < Ki/aij	Population i wins
Ki < Kj/aji and Kj > Ki/aij	Population j wins
Ki < Kj/aji and Kj < Ki/aij	Both populations coexist
Ki > Kj/aji and Kj > Ki/aij	Unstable equiblibrium; one population wins by chance

1. Suppose that the carrying capacity of Population i is 7 and that of Population j is 5. One individual of Population j uses 0.4 as much of Population i's resources as a member of Population i does. One individual of Population i uses 1.0 times as much of Population j's resources as a member of Population j does.

a) Draw a graph like the one in the lesson under Theory of Competition - Equilibrium Conditions. You can draw your graph on a piece of paper or go to the Section of Animal Ecology at Leiden University in The Netherlands). Note: This site works with Netscape but will NOT work with Internet Explorer. Choose "The Lotka-Volterra competition model" from the drop-down list.

Put Population i on the x-axis and Population j on the y-axis (a phase plane graph). Do the equilibrium lines cross? What are the locations of the equilibrium lines relative to each other? What is the outcome of competition between these two species under these conditions?

a) What values did you choose for these parameters? How did you verify that coexistence occurs with these values? Note: Finding that the product of the two competition coefficients is less than zero does not always guarantee coexistence.

a) Leave the values of intrinsic rate of increase as you find them, and make sure the starting values for each of the two populations is 0.10. Run the simulation.

b) Change the values of intrinsic rate of increase to 0.5 for each population and rerun the simulation. Did changing the rates of population growth change the pattern in the time series graph? What difference did changing the rates of population growth make in the outcome of competition? Explain why the result you obtained is consistent with the Lotka-Volterra model of competition.

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