SUMMER 2004

Keith Manion and Lisa Song

My dissertation focused on a model of two competing species called the 'competition LPA model'. This model exhibited previously unseen dynamics, such as those seen in this graph. The 4 colors represent 4 distinct orbits; two of which (gray and pink) lead to coexistence of two species and the other two (brown and blue) lead to extinction of one species or the other. (The axes represent the total populations of two distinct species as predicted by the model.) The goal of Lisa and Keith's project was to isolate the same interesting dynamics in two simpler 'toy' competition models. Lisa worked on a 2-stage version of the Leslie/Gower model, and Keith worked on a 2-stage competition version of the Ricker model. With each model they were able to recreate the dynamics in question, via three specific 'non-Lotka/Volterra' properties:

• coexistence attractor(s) in the presence of extinction attractor(s)

• saddle-node bifurcation of a 2-cycle in the presence of extinction attractor(s)

• creating coexistence through an increase  in inter-specific competition

Lisa took the initiative in writing a program which generated 3-dimensional bifurcation diagrams, allowing us to see the dynamics for a full range of 2 different parameter values simultaneously. In the graph below, the vertical axis shows the long-term behavior of two distinct species (one in red, the other in green) for all combinations of two inter-specific competition parameters up to 1.0. The black region along the floor shows combinations which yield coexistence. So, we can see that by travelling parallel to either the CJY- or the CYJ-axis, we can find coexistence through an increase in competition, contradicting classical assumptions about competing species.