1.  For each of the following graphs, fill in the intercepts that are not filled in and state what  the outcome of competition between the two species is predicted to be, or, if it cannot be predicted, state what the possible outcomes are.  On each graph, draw one or more trajectories, starting from very small population sizes of both species, that indicate the predicted outcome(s).  Then draw a graph of population size versus time for each possible outcome of competition for each case; include both species on each graph and label which line represents which species.

2.  Suppose you find that in many different areas, striped skunks competitively exclude spotted skunks, while spotted skunks never competitively exclude striped skunks, and only occur in areas without striped skunks.  Draw the isocline graph for the Lotka-Volterra competition model that best models competition between these skunk species; label the axes and intercepts. Plot the expected change in population size over time if a small number of each species is introduced to a new area where skunks did not previously exist (assume the competitive interaction between the species is the same in this area as in other areas)

3.  For each of the following cases, draw competition isoclines and give numerical values for all intercepts.  For which of the following cases can you determine the outcome of competition (either say which species will competitively exclude the other or state that the two species will co-exist) from the isocline graphs given?  For the case(s) in which you can, state the outcome, and plot population growth versus time for each species.  For the case(s) in which you cannot, what other parameters from the Lotka-Volterra models would affect the outcome? How would they affect it (i.e. based on them which species would be more likely to competitively exclude the other)?  Explain in words why they affect the outcome of competition, and plot population growth versus time for each possible outcome.

(a) K₁=50, K₂=60, a₁₂=.7, a₂₁=1.1   (b) K₁=50, K₂=60, a₁₂=1.0, a₂₁=1.3
(c) K₁=50, K₂=60, a₁₂=1, a₂₁=1.1

4.  Two species of pocket mouse compete for food, which they collect in pouches in their cheeks.  Perognathus biggus is larger and can stuff more seeds into its cheek pouches; Perognathus agillus is smaller and can pick up and use smaller seeds.  As a result, the two species use resources differently and can co-exist.  In areas where only one species occurs, P. biggus reaches a carrying capacity of 20 mice per hectare and P. agillus reaches a carrying capacity of 30 mice per hectare.  In a one hectare plot where the two mice coexist, they have stationary (unchanging) population sizes of 10 P. biggus and 20 P. agillus.  Use the appropriate form of the Lotka-Volterra competition models to graphically determine the unknown intercepts and from the intercepts estimate alpha values.  Explain in words what these alpha values mean.

5.  The following three graphs show the pattern of population growth for three species of Daphnia (water fleas) grown in liter jars in constant laboratory conditions when each species is grown alone:

When Daphnia species 1 and species 2 are grown together (both species in the same jar) their populations grow as follow:

 When Daphnia species 1 and species 3 are grown together their populations grow as follows:

 

Estimate values from these graphs as appropriate to draw competition isoclines for species 1 and species 2, and for species 1 and species 3; be as numerically accurate as possible.  For the interaction between species 1 and species 3, estimate alpha values from the graph.

6.  In a large lab experiment, many combinations of different densities of two species of Paramecium were set up; population numbers were counted a week later.  The arrows on the graph below represent the changes in numbers of the two species for each density.
 

 
(a)  Based on these arrows, draw competition isoclines for the two species on the graph.

(b)  Graphically estimate values for K₁, K₂, a₁₂, and a₂₁.

(c)  Suppose populations of these two species were started together with 2 individuals of each species. On graph paper, plot the predicted change in numbers for the two species over time.