So far all of our population models involved one species, but in the real world several species usually share a habitat. In this module we look at models in which two species share a habitat. Different species can interact in many different ways.

• Competitive Interaction -- they compete for some of the same resources.

• Aggressive Interaction -- they actively fight each other -- like the Montagues and Capulets.

• Cooperation or symbiosis -- they help each other in some way.

• Predator - Prey Interaction -- one species (the predator), eats the other (the prey).

• Weak - Strong Interaction -- one species is simply better suited to survive than the other.

The mathematical models we develop in this module will be able to describe all of these different kinds of interaction.

Because these models are so rich and can be used to study so many different kinds of interaction we build them carefully step-by-step. The simplest model for a single species is a model of the form

where R is a constant, called the relative growth rate. Models based on this differential equation predict that

and, if R is positive, the population skyrockets without bound. Thus, these models are not very realistic. We must replace the constant relative population growth rate by a function R(p) that reflects the effects crowding has on food, water, and shelter, and, thus, on the population growth. We work with the simplest function of this kind, a function of the form

-- for example, the function R(p) = 0.50 (1 - 0.01 p), whose graph is shown at the right. Notice that when the population, p, is zero then the relative population growth rate is a very profilic 0.50 but for every new individual who enters the habitat the relative population growth rate drops 0.005. By the time the population reaches 100 the population growth rate has dropped to zero. This is the nonzero equilibrium population for this species in this habitat.

If a second species is present in the same habitat then that species is likely to compete with the first species for some resources. For example, the two species might compete for water but not for food or shelter. We will use the letter q to denote the population of the second species. Now we need a function R(p, q), that reflects the effect of both p and q on the relative population growth rate for the species p. For example, we might have a function like

The figure at the right can help us understand this function. It shows two graphs. The blue graph shows the function R(0, q) and it gives us a picture of how the population q affects the relative population growth rate. When both p and q are zero the relative population growth rate is 0.50 and as the population, q, rises to 200 the relative population growth rate drops to zero. The red graph shows the function R(p, 0) and it gives us a picture of how the population p affects the relative growth rate. Once again, when both p and q are zero, the relative population growth rate is 0.50 but now as the population p rises to 200 the population growth rate falls much more steeply and drops all the way to -0.50.

The biological reason for this is that because two members of the species p compete with each other for absolutely everything -- they eat the same food, look for shelter in the same places, and are able to reach the same water supply -- they have a more negative effect on each other than two members of different species who generally compete with each other for some resources but not for others.

You can see this same effect numerically. Compute, for example, R(25, 25), and compare it to R(30, 25) and R(25, 30) notice that an additional five members of the species p has a greater effect than an additional five members of the species q.

In this example the population growth of the species p can be described by the differential equation

We need another differential equation to describe the population growth of the species q. We will use the equation

The crucial part of this differential equation is the factor

on the right hand side. This factor is the relative growth rate for the species q. Notice that this factor is more sensitive to q than to p because when another q enters the habitat he competes with the other q's for absolutely everything but when another p enters the habitat he competes with the q's for some but not all resources.

Putting this all together we have the pair of differential equations

The most important feature of these differential equations are the numbers or coefficients circled below.

Notice first that they are all negative because each new individual has a negative impact on individuals of both species. The pair of coefficients circled in red represent the intraspecies interactions -- the effects that each species has on its own growth rate. The pair of numbers circled in blue represent the interspecies interactions -- the effects that each species has on the growth rate for the other species. Because we are talking about two different species that compete in a relatively benign way, the intraspecies interactions are stronger than the interspecies interactions.

Look at the pairs of differential equations below and look at the list of different kinds of interactions at the beginning of this module. Decide for each pair of differential equations what kind of interaction it represents.

• p' = 0.50 (1 - 0.005 p - 0.01 q) p

  q' = 0.50 (1 - 0.01 p - 0.005 q) q

• p' = 0.50 (1 - 0.01 p - 0.005 q) p

  q' = 0.50 (1 - 0.001 p - 0.002 q) q

• p' = 0.50 (1 - 0.005 p + 0.002 q) p

  q' = 0.50 (1 + 0.002 p - 0.005 q) q

For the rest of this module we analyze the model

Your job will be to apply the techniques we use to analyze this model to the three models above.

We will use a two-dimensional graph to represent the two populations p and q. A point on this graph tells us the two populations in the habitat at a particular time. The first coordinate represents the p population and the second coordinate represents the q population. For example, the point shown in the graph at the right represents a p population of 50 and a q population of 100. The two differential equations describe how the two populations are changing. For example, at the point (50, 100) shown in the graph above we have p'(50, 100) = 0 q'(50, 100) = -12.5

Since p'(50, 100) = 0, at this point the p population is unchanging. Since q'(50, 100) = -12.5, at this point the q population is decreasing. We indicate all this by the arrow shown in the graph at the right. This arrow is pointing downward because q is decreasing and, since q is drawn on the vertical axis, the direction of decreasing q is downward. The arrow is neither pointing to the left nor to the right because at this point p is unchanging and, since p is drawn on the horizontal axis, changing p would be to the right (if p were increasing) or left (if p were decreasing). This arrow is called the direction vector. You should think of this graph as the map of a lake and the direction vector at each point as indicating the direction of the current at that point.

For each of the points below compute the derivatives p' and q' and determine which way the direction vector should be drawn at that point. You can check your answer by clicking at the corresponding point on the graph at the right. The p-axis and the q-axis on this graph both run from zero to 200. The grid lines are spaced 20 units apart. p = 20, q = 20 p = 20, q = 60 p = 20, q = 100 p = 60, q = 20 p = 100, q = 20 p = 60, q = 60 p = 100, q = 100

Now we want to analyze the differential equation

that tells us how the population p is changing. Our first question is -- for what values of p and q is p temporarily still. That is, we want to solve the equation

or

The product on the left is zero when either of the two factors p or (1 - 0.01 p - 0.005 q) is zero. The first factor is zero anyplace along the q-axis and the second factor is zero any place along the line 1 - 0.01 p - 0.005 q = 0 These lines are marked in red in the graph at the right.

Now we want to analyze the differential equation

that tells us how the population q is changing. Our first question is -- for what values of p and q is q temporarily still. That is, we want to solve the equation

or

The product on the left is zero when either of the two factors q or (1 - 0.005 p - 0.01q) is zero. The first factor is zero anyplace along the p-axis and the second factor is zero anyplace along the line 1 - 0.005 p - 0.01 q = 0 These lines are marked in blue in the graph at the right.

Next we put these two graphs together to obtain the graph at the right. At any point along either of the red lines the population p is temporarily unchanging. At any point along either of the blue lines the population q is temporarily unchanging. There are four points where a red line and a blue line intersect. These points are the equilibrium points for this dynamical system. If the two populations are at the levels represented by these points they will stay there, unchanging. The first of these points -- the point (0, 0) -- is not particularly interesting. This point represents an empty habitat with no p's and no q's.

The next two equilibrium points -- (0, 100) and (100, 0) -- are a bit more interesting. These points represent situations in which just one of the two species is living in this habitat.

The most interesting point is the point where the two lines

and

intersect. A little algebra shows that this point is (66.6667, 66.6667). This point represents coexistence with the two species sharing the habitat.

Now we know there are four possible equilibrium points -- one corresponding to an empty habitat, two corresponding to a habitat with just one of the two species, and one corresponding to a habitat with the two species coexisting. But we don't know how these two populations are likely to evolve -- would we expect them both to die out? -- would we expect just one of the species to die out? -- or would we expect them to coexist peacefully.

In order to answer these questions we analyze the currents in our lake -- that is, we analyze the direction vectors.

We begin by looking at the species p and the differential equation p' = 0.50 (1 - 0.01 p - 0.005 q) p The graph at the right shows the points at which p' is zero. Notice these two red lines divide the "lake" into two regions -- a small enclosed triangular region and everything else. In each of these two regions the sign of p' must be the same at every point. This fact is an interesting one to discuss in a high school algebra class. We determine which sign p' has in each region by evaluating it at one point in the region.

For example, if we compute p' at the point (1, 1) in the small enclosed triangular region we see that its sign is positive in this region. Similarly if we compute p' at the point (200, 200) in the other region we see that its sign is negative. We indicate this on the graph at the right with red arrows pointing to the right (the direction of increasing p) inside the enclosed triangular region and with red arrows pointing to the left (the direction of decreasing p) in the other region.

Next we look at the species q and the differential equation q' = 0.50 (1 - 0.005 p - 0.01 q) q The graph at the right shows the points at which q' is zero. Notice these two blue lines divide the "lake" into two regions -- a small enclosed triangular region and everything else. In each of these two regions the sign of q' must be the same at every point. We determine which sign q' has in each region by evaluating it at one point in the region.

For example, if we compute q' at the point (1, 1) in the small enclosed triangular region we see that its sign is positive in this region. Similarly if we compute q' at the point (200, 200) in the other region we see that its sign is negative. We indicate this on the graph at the right with blue arrows pointing upward (the direction of increasing q) inside the enclosed triangular region and with blue arrows pointing downward (the direction of decreasing q) in the other region.

Now we want to combine the information from the two sets of graphs above into one graph that describes the currents in the lake. The graph at the right shows the points at which either p' is zero (the red lines) or q' is zero (the blue lines). Notice that these lines divide the lake into four numbered regions -- a quadrilateral with the origin in its lower left corner, two triangular regions, and a region containing everything else. We can determine what is happening in each of these four regions by looking at the two sets of graphs above.

In region number 1 the arrows are pointing upward and to the right. In region number 2 the arrows are pointing downward and to the right. In region number 3 the arrows are pointing upward and to the left. In region number 4 the arrows are pointing downward and to the left. This information is shown in the rough graph at the right.

You can gain a better understanding of the diagram above by working with the Java applet at the right. This applet covers the same lake as the diagram above. Both axes run from 0 to 200 with gridlines at 50, 100, 150, and 200. You can figuratively drop a cork into this lake by positioning the cursor at the point you want to place the cork and then clicking the mouse button. Notice that the cork is carried by the currents and that at each point it is moving generally in the direction indicated in diagram above. Try several different initial conditions -- that is, place the cork at several different points in the lake. If you want to clean up the applet and start with a clean lake click the grey button under the applet. Notice that with very few exceptions the cork is always carried to the equilibrium point that represents coexistence of the two species. If you place the cork very carefully on one of the two axes it will stay on that axis. This is what happens when only one species is present.

This model illustrates some of the advantages of biodiversity. Two species that are somewhat different will often coexist peacefully. Notice that the combined population 133.33333 of the two coexisting species is higher than the population of either species alone. In addition, the diversity in a habitat with several different species makes it more likely that some will survive in the event of a major change in the habitat.

Use the techniques we developed in this section -- the rough direction vector diagrams -- to analyze the three models below. Describe the long term behavior of each of these models and relate it to the real world interaction represented by the model. There are no CAS worksheets or notebooks for this module but the CAS buttons are active in case you want to launch your CAS system.

• p' = 0.50 (1 - 0.005 p - 0.01 q) p

  q' = 0.50 (1 - 0.01 p - 0.005 q) q

• p' = 0.50 (1 - 0.01 p - 0.005 q) p

  q' = 0.50 (1 - 0.001 p - 0.002 q) q

• p' = 0.50 (1 - 0.005 p + 0.002 q) p

  q' = 0.50 (1 + 0.002 p - 0.005 q) q