**Developing Models of Species Interactions and Population Dynamics**

In Concept 43.2, we pointed out that the per capita growth rate of a species can be affected not only by interactions between individuals of the same species (intraspecific interactions), but also by interactions with individuals of another species (interspecific interactions). We began with a word equation that described interspecific competition:

per capita growth rate (r) of species A = |
{maximum possible r for species A in uncrowded conditions– an amount that is a function of A's own population density} – {an amount that is a function of the population density of competing species B} |

Notice that the expression on the right contains two parts. The first part—in the first set of brackets—describes growth of the species A population when species B is absent and species A is by itself. The second part—in the second set of brackets—describes the effect of species B. This is a general feature of all models that describe interspecific interactions: there is a term that describes how each species grows when by itself, and another term that describes the effect of the interacting species.

We will now show how to express this word equation in a more formal mathematical form. We start with a model for population growth of species A when it is by itself. Let us assume that species A grows in a density-dependent fashion. In Animated Tutorial 42.2, we saw that the simplest form of density-dependent population growth—in which intraspecific competition causes a linear decline in per capita growth rate as population density increases—can be expressed as:

per capita growth rate = 1/*N* (d*N*/d*t*) = *r* – (*r*/*K*) *N*

This is the first term of our mathematical model of growth of species A. To make it clear that we are talking about species A, we will add subscripts to the equation to indicate that we are referring to the population density, per capita growth rate, and carrying capacity of species A:

1/*N*_{A} (d*N*_{A}/d*t*) = *r*_{A} – (*r*_{A}/*K*_{A}) *N*_{A}

To complete our model of growth of the population of species A, we need to include the second term that describes the additional effect of species B's population density on per capita growth rate of species A. What form should this term take? Let us assume that each individual of the competing species B uses some resources that individuals of species A could otherwise use, and so reduces the per capita growth rate of species A in the same way that each individual of species A does. However, let us also add a multiplier α to describe the effect of each individual of species B relative to the effect of an additional individual of species A. If α = 1, then an individual of species B reduces A's per capita growth rate exactly as much as an additional individual of species A; if α > 1, an individual of species A has a larger effect than an additional individual of species A, and if α < 1, an individual of species A has a smaller effect. (Note also that if α = 0, species B does not compete with species A and has no effect on its per capita growth rate).

To capture the total effect of species B on species A, the term we add must include the number of individuals of species B as well as the multiplier α that describes the effect of each individual on species A:

1/*N*_{A} (d*N*_{A}/d*t*) = {*r*_{A} – (*r*_{A}/*K*_{A}) *N*_{A}} – (*r*_{A}/*K*_{A}) α*N*_{B}

This, then, is a formal mathematical equivalent to the word equation we began with.

Now, to describe the interspecific interaction completely, we need to write an equation for the competing species B. Let us assume that species B grows in the same way as species A does when by itself, and that species A affects species B in the same way that species B affects species A. In this case, we weigh the effect of each individual of species A on species B by a multiplier β:

1/*N*_{B} (d*N*_{B}/d*t*) = {*r*_{B} – (*r*_{B}/*K*_{B}) *N*_{B}} – (*r*_{B}/*K*_{B}) β *N*_{A}

We can use a computer program to simulate the growth of two competing populations by selecting values for the following parameters of the equations:
starting values for population sizes of species A and B (*N*_{0A}, *N*_{0B}), values of *r*_{A} and *r*_{B}, values of *K*_{A} and *K*_{B}, and
values for α and β. We can use these equations to calculate the
change in population size in a tiny increment of time d*N*/d*t* and add that increment to the starting population. Now we repeat the calculation with the
new starting population and again add d*N*/d*t*. In this way, the computer calculates the trajectory of population growth.

To see how competition influences population growth, click on the **Activity** tab.

If you are interested in seeing how ecologists can use equations to make predictions about the outcome of competition, click on
the **When Will Competitors Coexist?** tab. (This is an advanced topic.)

**Activity: Exploring Growth of Competing Populations**

Go to the **Graph** tab to see the following properties of the growth of competing populations.

**Property 1:** When alone, each species grows in a density-dependent way.

- Click on the "Species A alone" checkbox and observe the growth curve. What is the shape of the curve? At what population size does the population stop growing?

What are the values of the parameters*r*_{A}and*K*_{A}?

- Click on the "Species B alone" checkbox and observe the growth curve. What is the shape of the curve? At what population size does the population stop growing?

What are the values of the parameters*r*_{B}and*K*_{B}?

**Property 2:** Each species grows more slowly when the competitor is present than when the competitor is absent, and each species reaches a lower population density when it stops growing.

- Click on the "Species A and B coexist" checkbox. Compare the slopes and final densities of each species in this graph with the curves and final densities of each species alone. Are the slopes higher when each species is alone? Does each species stop growing at a larger population size when alone?

**Property 3:** Competition can result in coexistence, or in the competitive exclusion of one species by the other.

- Note the values of the parameters
*r*_{A},*r*_{B},*K*_{A},*K*_{B}, α, and β for the "Species A and B coexist" scenario.

- Now click on the "Species A outcompetes species B" checkbox. What is the final population size of Species B? Of species A? How does the final population size of the "winner" compare to its carrying capacity?

Note the values of the parameters*r*_{A},*r*_{B},*K*_{A},*K*_{B}, α , and β for the "Species A outcompetes species B" scenario.

- Now click on the "Species B outcompetes species A" checkbox. What is the final population size of species A? What is the final population size of species B?

Note the values of the parameters*r*_{A},*r*_{B},*K*_{A},*K*_{B}, α , and β for the "Species B outcompetes species A" scenario.

What different values of the parameters are associated with the different outcomes of competition?

If you are interested in seeing how these values are related to the criteria for coexistence of two competing species that we expressed in the text on p. 888 of Concept 43.2, go to the

We saw in the **Graph** tab that there are three possible outcomes of the competitive interaction between species A and species B:

- the two species can coexist,
- species A can drive species B to extinction, or
- species B can drive species A to extinction.

What determines the outcome of competition?

In Concept 43.2, we indicated that two competitors will coexist when individuals of each species have a negative effect on their own per capita growth rate that is greater than the negative effect they have on the per capita growth rate of the competitor. This condition allows each species to enjoy a growth advantage when it is rare and the competitor is at its carrying capacity. In other words, each species will be able to invade a system that formerly contained only the other species.

To see how ecologists have arrived at this conclusion about the conditions for coexistence, let us take another tour of mathematics land, and write out equations for the condition. We will begin by writing equations for per capita growth of species A and species B and setting these equations greater than zero to show that the population of each species is growing:

1/*N*_{A} (d*N*_{A}/d*t*) = *r*_{A} {1 – (1/*K*_{A}) (*N*_{A}) – (α/*K*_{A}) (*N*_{B})} > 0 for species A

1/*N*_{B} (d*N*_{B}/dt) = *r*_{B} {1 – (1/*K*_{B}) (*N*_{B}) – (β/*K*_{A}) (*N*_{A})} > 0 for species B

Notice that we've rearranged the equations in the **Introduction** tab, for reasons that will become apparent soon.

What must be true for the per capita growth rate to be positive? First, *r*_{A} and *r*_{B} must be
positive—but this will always be true if the environment is suitable for the two species. Second, the terms in parentheses
must be positive. Let us focus on conditions that make the terms in parentheses greater than zero when each species is rare and
the other is at its carrying capacity:

1 – (1/*K*_{A}) (*N*_{A}) – (α/*K*_{A}) (*N*_{B}) > 0 when *N*_{A} is near zero and *N*_{B} is at *K*_{B}

1 – (1/*K*_{B}) (*N*_{B}) – (β/*K*_{A}) (*N*_{A}) > 0 when *N*_{B} is near zero and *N*_{A} is at *K*_{A}

Let us solve the first of these equations using a common mathematical approach. We will set *N*_{A} to zero and
*N*_{B} equal to *K*_{B}, so that the condition for species A to invade becomes:

1 – (1/*K*_{A}) (0) – (α /*K*_{A}) (*K*_{B}) = 1 – (α /*K*_{A}) (*K*_{B}) > 0

Rearranging we have:

1 > (α /*K*_{A}) (*K*_{B})

Dividing through by *K*_{B} we have:

1/*K*_{B} > α/*K*_{A}

This must be true for species A to invade a system dominated by species B.

Taking the same steps with the equation for species B, we obtain:

1/*K*_{A} > β/*K*_{B}

This must be true for species B to invade a system dominated by species A.

If both of these inequalities are true, then A and B can coexist because neither species can drive the other extinct locally—each species can increase even when it is very rare.

Let us translate these inequalities back into words. Looking at the equations that we began with above, we can see that 1/*K*_{A}
is the effect of each individual of species A on the per capita growth rate of species A, 1/*K*_{B} is the effect of each individual of
species B on the per capita growth rate of species B, α/*K*_{A} is the effect of individuals of species B on the per capita growth rate
of species A, and β/*K*_{B} is the effect of individuals of species A on the per capita growth rate of species B. Therefore,
what the final two inequality equations say in words is that for the two competitors to coexist,

- individuals of species B suppress their own per capita growth rate more than they suppress the per capita growth rate of species A; and

- individuals of species A suppress their own per capita growth rate more than they suppress the per capita growth rate of species B.

These are exactly the criteria for two competing species to coexist, as given in the text on p. 888 of Concept 43.2!

Now look at the parameter values shown in the **Graph** tab. Are the criteria for coexistence met in the case where the two competitors coexist?
Are they met in the case where species B outcompetes species A? How about the case where species A outcompetes species B?

We have used the example of interspecific competition to illustrate the general strategy that ecologists use to understand how interspecific interactions influence the per capita growth rate and population density of the interacting species.

The basic approach used here for competition can also be used for consumer–resource interactions (such as predator–prey interactions) and for mutualistic interactions. In those cases, too, we would write out equations for the per capita growth rate of each of the interacting species when it is by itself, and then add terms to the equation that describe the effect of the other species it is interacting with.

We can graph the trajectory of growth of the species with different sets of parameter values to explore the properties of growth of the
interacting species. In the **Graph** tab, you saw trajectories for several sets of parameters. These illustrate three properties of growth:

**Property 1**: When alone, each species grows in a density–dependent way.

**Property 2**: Each species grows more slowly when the competitor is present than when the competitor is absent, and each species reaches a lower population density when it stops growing.

**Property 3**: Competition can result in coexistence, or in the competitive exclusion of one species by the other.

Finally, as the **When Will Competitors Coexist?** tab explains, we can also use the mathematical equations to understand the outcomes of the interactions.

By studying mathematical models of species interactions, we discover many interesting details about how the interactions that all species engage in affect their population dynamics and their local abundances.

**Textbook Reference:** Concept 43.2 Interactions within and among Species Affect Population Dynamics and Species Distributions