

Part 3.2: Competing Species II
One biological scenario for the systems considered in Part 3a is that of two species with unlimited access to food, but who tend to fight each other when they meet. The model indicates that this is an unstable situation. As soon as the system is not exactly at the equilibrium point, one species tends to be killed off. In fact, this happens in nature too. For this reason the scenario is unlikely to be observed.In fact, the assumption of unlimited resources is unreasonable. We should add terms to the differential equations describing the growth rates of the species that reflect the negative effect of crowding. Recall that, in the predatorprey models, we modeled the negative effect on the prey growth rate of encounters with the predators by a term that was proportional to the product of the two populations. Now we model crowding of the same species by considering it as a negative effect proportional to the probability of chance meetings between random members of the same population. So for the x growth rate we include a term proportional to x^{2} and for the y growth rate a term proportional to y^{2}.
This leads to a system of equations of the following form:
dx/dt = ax  bxy  qx^{2}
dy/dt = cy  pxy  ry^{2}
dx/dt = ax  bxy  qx
dy/dt = cy  pxy  ry
In the process of finding the equilibrium points in Step 1, you probably considered the set of points where dx/dt = 0. This set is called the xnullcline. For this system, the xnullcline consists of the xaxis together with the line
(b) For each subregion determine whether x is increasing or decreasing and whether y is increasing or decreasing.
(c) Use your determination in (b) to sketch on paper what trajectories for this system should look like. Include several starting points within each subregion.
(b) How do these trajectories compare with the ones you drew in Step 3?


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