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Systems of Differential Equations: Models of Species Interaction

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 predator-prey 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 x2 and for the y growth rate a term proportional to y2.

This leads to a system of equations of the following form:

dx/dt = ax - bxy - qx2

dy/dt = cy - pxy - ry2

  1. You might imagine that it would be enough to include terms proportional to the first power of the population rather than the second, i.e., a system of the form

    dx/dt = ax - bxy - qx

    dy/dt = cy - pxy - ry

    Why would that modification not accomplish what we want?

Let's consider the following system:

dx/dt = x - xy - x2

dy/dt = 0.8y - 0.4xy - y2

  1. Find the four equilibrium points of this system. (Here we are allowing x or y or both to be 0. So, x = 0 and y = 0 identifies one equilibrium point; there are three others.)

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 x-nullcline. For this system, the x-nullcline consists of the x-axis together with the line

1 - x - y = 0

Similarly the y-nullcline is the set of points where dy/dt = 0. In this example, it consists of the y-axis together with the line

0.8 - 0.4x - y = 0.

The two nullclines divide the first quadrant into four subregions.

Division Into Subregions by the Nullclines

  1. For this step, use your computer algebra system to create a display with the two nullcline lines superimposed on the direction field.

      (a) What is the significance of the point where the two lines intersect?

      (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.

    • (a) Use your computer algebra system to display a number of trajectories on the direction field for this system. Include several trajectories that start in each of the four subregions.

      (b) How do these trajectories compare with the ones you drew in Step 3?

  2. How do the trajectories of solutions to the system change as you vary the ratio q/r? Does this seem reasonable in terms of the competing populations? Explain why or why not.

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