Systems of Differential Equations: Models of Species Interaction

Part 3.3: Competing Species III

We now take a different approach to the problem of modeling the interaction between two competing species. Recall the logistic model for the growth of a single population with a given "carrying capacity." If x represents the number of individuals in the population and M represents the carrying capacity of the environment, we modeled the rate of population growth by

dx/dt = rx(M - x)

For populations that are small with respect to the carrying capacity, the growth rate is essentially exponential

dx/dt = (rM)x

As the population approaches the carrying capacity, the rate of growth approaches 0, and the graph of the population levels off. The resulting logistic growth curve is displayed below.

Logistic Growth Curve

For more information on the logistic growth model, refer to the PostCALC module Population Growth Modules, Part 4.

The situation we want to model is that of two very similar populations (A and B) of individuals competing for the same niche in the environment -- e.g., two varieties of the same fish in a lake or two varieties of the same bird on an island. We make the following assumptions:

• The growth of each population in isolation can be described by a logistic growth model.

• The two populations may have different natural growth rates.

• There are different carrying capacities for the two populations. (For example, one species may be more efficient in obtaining energy from the available food.)

• To a member of Population A, the presence of a member of Population B represents the same crowding as the presence of another member Population A.

Suppose we let x represent the number of members of Population A and y represent the number of members of Population B. Our assumptions lead to the following system of equations

dx/dt = rx(M - x - y)

dy/dt = sy(N - x - y)

Here M and N represent the carrying capacities for populations A and B respectively, and rM and sN are the natural growth rate constants for the two populations.

1. Explain how each of the assumptions is reflected in this system.

2. Show that these systems are special cases of the family of systems investigated in Part 3.2. In what way are they special?

• (a) Using M=10, N=5, r=0.5, and s=1, construct the direction field in the phase plane.

(b) Look at trajectories for a variety of initial conditions.

3. Vary the coefficients M and N. How does the ratio M/N affect the trajectories?

4. Vary the coefficients r and s. How do changes in these coefficients affect the trajectories?

Let's look at the nullclines for the general system

dx/dt = rx(M - x - y)

dy/dt = sy(N - x - y)

The x-nullcline consists of the x-axis and the line M - x - y = 0. The y-nullcline consists of the y-axis and the line N - x - y = 0. The lines M - x - y = 0 and N - x - y = 0 are parallel and divide the first quadrant of the phase plane into three regions as indicated in the figure below. For the rest of this part, we assume that M is larger than N.

Regions Determined by the Nullclines

1. For each of the three regions determined by the nullclines, indicate whether x is increasing or decreasing and whether y is increasing or decreasing.

2. Explain why any trajectory that starts in Region I must eventually move into Region II. (Hint: What is the smallest value of dx/dt in Region I?)

• (a) Explain why it is impossible for a trajectory to leave Region II.

(b) What happens to trajectories once they are in Region II?

3. What can you say about trajectories starting in Region III? Must they stay in Region III? Must they leave Region III?

• (a) What do all the trajectories for this system have in common?

(b) What does this model imply about two similar species competing for the same ecological niche?