The classic Lotka–Volterra (LV) equations represent a simple model of two species competing for the same resources over ecological time. (Not to be confused with the LV predator-prey model.) It is assumed that each species grows logistically in the absence of the other species. Factors such as predation and seasonal effects are not incorporated into the model. The classic LV equations have been rescaled for this Demonstration by the carrying capacity of each population. You can vary the competition coefficients, intrinsic growth rates and the carrying capacities of each species to observe the effect on the phase plane.
Snapshot 1: The case of stable coexistence. Isoclines are superimposed on the phase portrait.
Snapshot 2: The phase portrait where coexistence is impossible, one species survives and one goes extinct regardless of initial population densities.
Snapshot 3: The principle of competitive exclusion. The phase portrait includes color-coded basins of attractions.
This Demonstration simulates the interaction between two competing species based on the classic LV equations. The snapshots illustrate three qualitative changes to the phase portrait that occur as parameters are varied.