Interspecific Competition Involves Two or More Species.

A relationship that affects the populations of two or more species adversely (– –) is interspecific competition. In interspecific competition, as in intraspecific competition, individuals seek a common resource in short supply (see Chapter 11). But in interspecific competition, the individuals are of two or more species. Both kinds of competition may take place simultaneously. In the deciduous forest of eastern North America, for example, gray squirrels compete among themselves for acorns during a year when oak trees produce fewer acorns. At the same time, white-footed mice, white-tailed deer, wild turkey, and blue jays vie for the same resource. Because of competition, one or more of these species may broaden the base of their foraging efforts. Populations of these species may be forced to turn away from acorns to food that is less in demand.

Like intraspecific competition, interspecific competition takes two forms: exploitation and interference (see Section 11.3). As an alternative to this simple dichotomous classification of competitive interactions, Thomas Schoener of the University of California–Davis proposed that six types of interactions are sufficient to account for most instances of interspecific competition: (1) consumption, (2) preemption, (3) overgrowth, (4) chemical interaction, (5) territorial, and (6) encounter.

Consumption competition occurs when individuals of one species inhibit individuals of another by consuming a shared resource, such as the competition among various animal species for acorns. Preemptive competition occurs primarily among sessile organisms, such as barnacles, in which the occupation by one individual precludes establishment (occupation) by others. Overgrowth competition occurs when one organism literally grows over another (with or without physical contact), inhibiting access to some essential resource. An example of this interaction is when a taller plant shades those individuals below, reducing available light (as discussed in Chapter  4, Section  4.2). In chemical interactions, chemical growth inhibitors or toxins released by an individual inhibit or kill other species. Allelopathy in plants, in which chemicals produced by some plants inhibit germination and establishment of other species, is an example of this type of species interaction. Territorial competition results from the behavioral exclusion of others from a specific space that is defended as a territory (see Section  11.10). Encounter competition results when nonterritorial meetings between individuals negatively affect one or both of the participant species. Various species of scavengers fighting over the carcass of a dead animal provide an example of this type of interaction.

13.2 The Combined Dynamics of Two Competing Populations Can Be Examined Using the Lotka–Volterra Model

In the early 20th century, two mathematicians—the American Alfred Lotka and the Italian Vittora Volterra—independently arrived at mathematical expressions to describe the relationship between two species using the same resource (consumption competition). Both men began with the logistic equation for population growth that we developed previously in Chapter 11 :

Species1:dN1/dt=r1N1(1−N1/K1)Species2:dN2/dt=r2N2(1−N2/K2)Species1:dN1/dt=r1N1(1−N1/K1)Species2:dN2/dt=r2N2(1−N2/K2)

Next, they both modified the logistic equation for each species by adding to it a term to account for the competitive effect of one species on the population growth of the other. For species 1, this term is αN2, where N2 is the population size of species 2, and α is the competition coefficient that quantifies the per capita effect of species 2 on species 1. Similarly, for species 2, the term is βN1, where β is the per capita competition coefficient that quantifies the per capita effect of species 1 on species 2. The competition coefficients can be thought of as factors for converting an individual of one species into the equivalent number of individuals of the competing species, based on their shared use of the resources that define the carrying capacities (see Chapter 12, Section 12.2 and Figure  12.3, and Quantifying Ecology 12.1). In resource use, an individual of species 1 is equal to β individuals of species 2. Likewise, an individual of species 2 is equivalent to α individuals of species 1. These terms (α and β), in effect, convert the population size of the one species into the equivalent number of individuals of the other. For example, assume species 1 and species 2 are both grazing herbivores that feed on the exact same food resources (grasses and other herbaceous plants). If individuals of species 2 have, on average, twice the body mass as individuals of species 1 and consume food resources at twice the rate, with respect to the food resources, an individual of species 2 is equivalent to two individuals of species 1 (that is, α = 2.0). Likewise, consuming food resources at only half the rate as species 2, an individual of species 1 is equivalent to one-half an individual of species 2 (that is, β = 0.5).

Now we have a pair of equations that consider both intraspecific and interspecific competition.

Species1:dN1/dt=r1N1(1−(N1+αN2)/K1)Species2:dN2/dt=r2N2(1−(N2+βN1)/K2)   (1)   (2)Species1:dN1/dt=r1N1(1−(N1+αN2)/K1)Species2:dN2/dt=r2N2(1−(N2+βN1)/K2)   (1)   (2)

As you can see, in the absence of interspecific competition—either α or N2 = 0 in Equation (1) and β or N1 = 0 in Equation (2)— the population of each species grows logistically to equilibrium at K, the respective carrying capacity. In the presence of competition, however, the picture changes.

For example, the carrying capacity for species 1 is K1, and as N1 approaches K1, the population growth (dN1/dt) approaches zero. However, species 2 is also vying for the limited resource that determines K1, so we must consider the impact of species 2. Because α is the per capita effect of species 2 on species 1, the total effect of species 2 on species 1 is αN2, and as the combined population N1 + αN2 approaches K1, the growth rate of species 1 approaches zero as well. The greater the population size of the competing species (N2), the greater the reduction in the growth rate of species 1 is (see discussion in Section 12.2 and Figure 12.3).

The simplest way to examine the possible outcomes of competition using the Lotka–Volterra equations presented is a graphical approach in which we first define the zero-growth isocline for each of the two competing species. The zero-growth isocline represents the combined values of population size for species 1 (N1) and species 2 (N2) at which the population growth rate of the respective species is zero (dN/dt = 0). This occurs when the combined population sizes are equal to the carrying capacity of that species (see Figure 12.3). We can begin by defining the zero-growth isocline for species 1 (Figure  13.1a). The two axes in the graph shown in Figure  13.1a define the population size of species 1 (x-axis, N1) and species 2 (y-axis, N2). We must now solve for the combined values of N1 and N2 at which the growth rate of species 1 is equal to zero (dN1/dt = 0). This occurs when: (1 – (N1 + αN2)/K1) = 0 or K1 = N1 + αN2 (see Equation 1). In effect, we are determining the combined values of N1 and N2 that equal the carrying capacity of species 1 (K1). This task is made simple because K1 = N1 + αN2 represents a line and all that is necessary to draw the line is to solve for two points. The two simplest solutions are to solve for the two intercepts (where the line intersects the two axes). The x-intercept occurs when N2 = 0, giving us a value of N1 = K1. The y-intercept occurs when N1 = 0, giving us a value of αN2 = K1, or N2 = K1/a. Given these two points (values for N1, N2), we can draw the line defining the zero isocline for species 1 (Figure 13.1a). For any combined value of N1, N2 along this line, N1 + αN2 = K1 and dN1/dt = 0. For combinations of (N1, N2) that fall below the line (toward the origin: 0, 0), N1 + αN2 < K1 and the population of species 1 can continue to grow. An increase in the population of species 1 is represented by a green horizontal arrow pointing to the right. The arrow is horizontal because the x-axis represents the population of species 1. For combinations of N1 and N2 that fall above the line, N1 + αN2 > K1, the population growth rate is negative (as represented by the green horizontal line pointing to the left), and the population size declines until it reaches the line.

We can take this same approach and define the zero isocline for species 2 (Figure 13.1b). The x-intercept is N2 = 0 and N1 = K2/β, and the y-intercept is N2 = K2 and N1 = 0. As with the zero-growth isocline for species 1, for combinations of N1 and N2 that fall below the line, N2 + βN1 < K2 and the population of species 2 can continue to grow. The yellow vertical arrow pointing up represents an increase in the population of species 2. The arrow is vertical because the x-axis represents the population of species 2. For combinations of (N1, N2) that fall above the line, N2 + βN1 > K2, the population growth rate is negative (yellow vertical arrow pointing down), and the population size declines until it reaches the line (see Figure 13.1b). We can now combine the two zero-growth isoclines onto a single graph and examine the combined population dynamics of the two species for different values of N1 and N2.

13.3 There Are Four Possible Outcomes of Interspecific Competition

To interpret the combined dynamics of the two competing species, their isoclines must be drawn on the same x–y graph. Although there are an infinite number of isoclines that can be constructed by using different values of K1, K2, α, and β, there are only four qualitatively different ways in which to plot the isoclines. These four possible outcomes are shown in Figure  13.2. In the first case (Figure 13.2a), the isocline of species 1 is parallel to, and lies completely above, the isocline of species 2. In this case, the isoclines define three areas of the graph. In the lower left-hand area of the graph (point A), the combined values of N1 and N2 are below the zero-growth isoclines for both species, and the populations of both species can increase. The green horizontal arrow representing species 1 points right, indicating an increase in the population of species 1, whereas the orange vertical arrow representing species 2 points up, indicating an increase in the population of species 2. The next point representing the combined values of N1 and N2 must therefore lie somewhere between the two arrows and is represented by the black arrow pointing away from the origin. In the upper right-hand corner of the graph, the combined values of N1 and N2 are above the zero-growth isoclines for both species. In this case, the populations of both species decline (black arrow points toward the origin).

In the interior region between the two isoclines, the dynamics of the two populations diverge. Here (at point C) the combined values of N1 and N2 are below the isocline for species 1, so its population increases in size, and the green horizontal arrow points to the right. However, this region is above the isocline for species 2, so its population is declining, and the yellow vertical arrow is pointing down. The black arrow now points down and toward the right, which takes the populations toward the carrying capacity of species 1 (K1). Note that this occurs regardless of where the initial point (N1, N2) lies within this region. If the isocline of species 1 lies above the isocline for species 2, species 1 is the more competitive species and species 2 is driven to extinction (N2 = 0).

In the second case (Figure 13.2b), the situation is reversed. The zero-growth isocline for species 2 lies above the isocline for species 1, and therefore species 2 “wins” leading to the extinction of species 1 (N1 = 0). Note that in the interior region (between the isoclines), the combined values of N1 and N2 are now below the isocline for species 2 allowing its population to grow (yellow vertical arrow pointing up), whereas it is above the isocline for species 1, causing its population to decline (green horizontal arrow pointing to the left). The result is a movement of the populations toward the upper left (see black arrow), the carrying capacity of species 2 (K2).

In the remaining two cases (Figures 13.2c and 13.2d), the isoclines of the two species cross, dividing the graph into four regions, but the outcomes of competition for the two cases are quite different. As with the previous two cases, we determine the outcomes by plotting the arrows, indicating changes in the two populations within each of the regions. However, the point where the two isoclines cross represents an equilibrium point, a combined value of N1 and N2 for which the growth of both species 1 and species 2 is zero. At this point, the combined population sizes of the two species are equal to the carrying capacities of both species (N1 + αN2 = K1 and N2 + βN1 = K2).

The third case is presented in Figure 13.2c. The region closest to the origin (point A) is below the isocline of both species, and therefore the growth of both populations is positive and the arrows point outward. The upper right-hand region (point B) is above the isoclines for both species, so both populations decline and the arrows point inward toward the axes and origin. In the bottom right-hand region of the graph (point C), we are above the isocline for species 1, but below the isocline for species 2. In this region, the population of species 1 declines (green horizontal arrow points to left), whereas the population of species 2 increases (yellow vertical arrow points up). As a result, the combined dynamics (black arrow) point toward the center of the graph where the two isoclines intersect. The upper left-hand region of the graph (point D) is above the isocline for species 2 but below the isocline for species 1. In this region, the population of species 2 declines, and the population of species 1 increases. Again, the combined dynamics (black arrow) point toward the center of the graph where the two isoclines intersect. The fact that the arrows in all four regions of the graph point to where the two isoclines intersect indicates that this point (combined values of N1 and N2) represents a “stable equilibrium.” The equilibrium is stable when no matter what the combined values of N1 and N2 are, both populations move toward the equilibrium value.

In the fourth case (Figure 13.2d), the isoclines cross, but in a different manner than in the previous case (Figure 13.2c). Again, both populations increase in the region of the graph closest to the origin (point A). Likewise, both populations decline in the upper right-hand region (point B). However, the dynamics differ in the remaining two regions of the graph. In the lower right-hand region, the combined values of N1 and N2 (point C) are below the isocline for species 1 but above the isocline for species 2. In this region, the population of species 1 decreases, whereas the population of species 2 continues to grow. The combined dynamics (black arrow) move away from the equilibrium point where the two isoclines intersect (point E) and toward the carrying capacity of species 1 (K1 on x-axis). In the upper left-hand region of the graph, the combined values of N1 and N2 (point D) are below the isocline for species 2 but above the isocline for species 1. In this region of the graph, the combined dynamics (black arrow) move away from the equilibrium point where the two isoclines intersect (point E) and toward the carrying capacity of species 2 (K2 on y-axis). This case represents an “unstable equilibrium.” If the combined values of N1 and N2 are displaced from the equilibrium (point E), the populations move into one of the two regions of the graph that will eventually lead to one species excluding the other (driving it to extinction: N = 0). Which of the two species will “win” is difficult to predict and depends on the initial population values (N1 and N2) and the growth rates of the populations (r1 and r2