## O

Symmetric competition

Asymmetric competition

 Species 1 Intraspecific ™ Strong effects Species 2 -Interspecific <— Weak effects

Figure 1 Schematic illustrating different types of competitive interactions. (a) Illustrates intraspecific (dashed line) and interspecific (solid line) competition between two species, which are represented by black and white circles. (b) Illustrates symmetric (above) and asymmetric (below) competition. Arrow size indicates interaction strength. Redrawn from Keddy PA (1995) Competition. London: Chapman and Hall.

product of its intrinsic rate of increase (r1) and current population size (A!):

The 'intrinsic rate of increase' (r) is the per capita population growth rate that could potentially be realized by a species. In other words, it represents the potential for a species to increase in abundance. For example, mice can rapidly fill a landscape, and have greater intrinsic rates of increase than elephants, which proliferate more slowly. Equation  represents exponential growth, meaning growth accelerates through time without bound and is therefore unrealistic. The subscript 1 assigns each variable to a specific species. Different subscripts will be introduced when the effect of interspecific competition is added to the model.

As populations grow, intraspecific competition intensifies, which places limits on population growth. To incorporate the effects of crowding, a third term called the carrying capacity (K) is incorporated into the model. The carrying capacity is specifically defined as the

 1 1 1 1

O Paramecium aurelia £ Paramecium caudatum

Figure 2 Schematic illustrating carrying capacities and competition coefficients. The large box refers to the carrying capacity of two species, illustrated as different numbered tiles. Populations of both species are below carrying capacity because more tiles can fit within the larger box. Tiles representing species 2 are four times larger than species 1. Therefore, individuals of species 2 have four times the per capita resource consumption rate as species 1, indicating the competition coefficient of species 2 on species 1 (a12) is equal to 4. Redrawn from Krebs CJ (2001) Ecology: The Experimental Analysis of Distribution and Abundance. San Francisco: Benjamin Cummings.

maximum population size a given locale can support. Imagine a species' carrying capacity as a square box, which contains tiles representing individuals. When the box is filled completely by tiles, it represents a population at carrying capacity. When there are too many tiles to fit in the box, it represents an unstable population, which is larger than its carrying capacity. In this situation, the population will decline, or by analogy, tiles are taken away. If there is empty space within the box, more tiles can be added, representing population growth. Figure 2 illustrates the carrying capacity of two species, which are represented by different-sized boxes. Although the size of box (or the total amount of resources) is the same for both species, species 1 has a larger carrying capacity, because more of its tiles can fit inside the box.

Carrying capacity is incorporated in the model by including a new term [(K- N) / K] into eqn :

This term places a limit on exponential growth, and the resulting equation represents 'logistic growth', which is illustrated by the two Paramecium species shown in Figure 3a. When their population sizes are low (K>> N), the new term in eqn  approximates 1 , so the population behaves basically as in eqn [ 1 ] (zone 1 , Figure 3a). However, as the population grows, this new term becomes smaller and smaller, placing stronger and

50 0 Zone 2

Zone 2

Both species grown in seperate cultures

The same species when co-occuring in the same culture

The same species when co-occuring in the same culture 10 15

Days

Figure 3 Growth of two Paramecium species when growing alone (a) or together (b). Both species show similar logistic growth patterns when grown separately, but P. aurelia outcompetes P. caudatum when grown together. In (a), zone 1 shows changes in population size similar to exponential growth, while zone 2 illustrates the effects of intraspecific competition on population dynamics. Redrawn from Gause GF (1934) The Struggle for Existence. Baltimore: Williams & Wilkins.

10 15

Days

Figure 3 Growth of two Paramecium species when growing alone (a) or together (b). Both species show similar logistic growth patterns when grown separately, but P. aurelia outcompetes P. caudatum when grown together. In (a), zone 1 shows changes in population size similar to exponential growth, while zone 2 illustrates the effects of intraspecific competition on population dynamics. Redrawn from Gause GF (1934) The Struggle for Existence. Baltimore: Williams & Wilkins.

stronger limits on population growth (zone 2, Figure 3a). If the population overshoots its carrying capacity (K < N), it becomes negative and the population declines. When the population size equals its carrying capacity (K = N) the equation equals zero, representing a stable or equilibrium population size.

So how might competition between species influence population dynamics? To incorporate interspecific competition into the logistic equation, a fourth and final variable, called the 'competition coefficient' (a), is included. It represents the per capita, competitive effect of one species on another and converts individuals of one species into an equivalent number of individuals of another species. Figure 2, which describes the concept of a carrying capacity, also illustrates competition coefficients. Tiles representing species 2 are four times larger than species indicating that individuals of species 2 require four times more resources. Therefore, the competition coefficient representing the effect of species 2 on species 1 (a 12) is 4.00, while that of species 1 on species 2 (a2 1 ) is 0.25. The coefficients representing intraspecific competition (i.e., an and a22) are assumed to equal 1. To incorporate interspecific competition into the logistic equation, the competition

coefficients are multiplied by population sizes, and then subtracted from the carrying capacity. When the abundance of species 1 is multiplied by the competition coefficient corresponding to its per capita effects on species 2 (i.e., N1 a21), it represents the total competitive effect of species 1 on species 2. Similarly, the competitive effect of species 2 on species 1 is N2 a12. Taken altogether dNi „fKi -anNi -auN2\

The conditions under which both populations are stable are identified by setting both equations equal to zero and solving for the abundance of both species at equilibrium. Equilibrium conditions are then plotted in 'state space', which illustrates the population size of both species. The abundance of one species is plotted on the x-axis, while the other is shown on the j-axis (Figure 4). Let us consider species i first. After setting eqn  equal to zero and rearranging for species i when species 2 is absent, we get Ni = Ki. This point can be plotted in state space where x = Ki and j = 0. Similarly, eqn  at equilibrium can be rearranged to obtain the abundance of species 2 when species i is absent, which is N2 = Ki/ai2. This point can also be plotted in state space where x= 0 and j = Ki/ai2. If we assume that the relationship between these two points is linear, we can plot a straight line in state space that represents the equilibrium population growth rate of species i. This is called an 'isocline' and it represents combinations of abundances of both species where population growth of species i is stable (i.e., dN/dt = 0). The isocline for species i is plotted in Figure 4a. The same procedure can be followed for species 2, which is shown in Figure 4b.

K1/a12

K1/a12  KJ Q2i

Figure 4 Zero-growth isoclines of two competing species under the Lotka-Volterra model. (a) Shows the isocline for species 1 and (b) Shows the isocline for species 2. In (a) points above the isocline (zone 1) represent population sizes above equilibrium. These points will move horizontally toward the origin, until they reach the isocline representing population equilibrium, indicating population decline. Points in zone 2 move in the opposite direction, which illustrates population growth. Similar conditions occur in (b), which illustrates species 2, so trajectories are vertical.

Four hypothetical population sizes are shown as points in Figures 4a and 4b. Arrows illustrate predictions of how population size will change. In Figure 4a, any point in zone i will move horizontally toward the origin until it reaches the isocline, where it will stop and the population size ofspecies i will remain stable. Points in zone 2 will move horizontally away from the origin, or increase in size, until it reaches the isocline where it will remain stable. Similar circumstances occur with species 2 (Figure 4b); however, points move vertically because species 2 is plotted on the j-axis.

Notice that the state space graphs for species i and species 2 have the same axes. Therefore, we can plot both species' isoclines on the same graph, and compare their population dynamics simultaneously. Several hypothetical outcomes can be observed for species dynamics by plotting both isoclines together in state space. One outcome is that the isoclines will be more or less parallel, with one species having greater intercept values. This situation represents asymmetric interspecific competition or that one species is competitively dominant over the other. In Figure 5 a, the isocline for species i lies above the isocline for species 2. Under these circumstances (Ki/ai2 > K2 and Ki > K2/a2i), species i is competitively dominant. At small populations of both species, both will increase in size, which is represented by the point and arrow closest to the origin. When both species have large population sizes, both will decline. However, the relative abundances represented by points between the isoclines move diagonally, away from the origin towards the point on the x-axis representing equilibrium population size of species i (Ki), which is represented by a gray circle. The analogous situation is shown is Figure 5b. Here, species 2 is a stronger competitor than species i, and the isocline for species 2 lies above species i (K2> Ki/ai2, and K2/a2i> Ki), and species 2 excludes species i. The competitive dynamics of two Paramecium species provides a good example of this situation (Figure 3b). Although both have similar dynamics when grown separately, when grown together P. aurelia outcompetes P. caudatum, driving it to local extinction.

Another possible outcome is that the isoclines differ in slope and intersect somewhere in state space, as in Figure 5c. Here, the isocline for species i has a steeper slope, greater j-intercept, and lower x-intercept than species 2. Under these conditions (Ki/ai2> K2 and K2/a2i> Ki), intraspecific competition is stronger than interspecific competition. Both can stably coexist under these circumstances and their population sizes converge on a single point at the intersection of the two species' isoclines. Desert grainivores provide a good example of this situation in the wild. Many species of ants and rodents eat seeds as their primary food source in of North American deserts. While both types of competitors stably coexist under normal circumstances, when one type of seed-eater is removed from large field enclosures the other increases in abundance.    Figure 5 Four potential outcomes of the Lotka-Volterra model. Isoclines for two species are plotted against their respective population sizes. Gray points represent equilibrium population sizes. In (a) species 1 outcompetes species 2 and population sizes will follow the trajectory of arrows until it reaches K1 on the x-axis. (b) Illustrates competitive exclusion by species 2. (c) Illustrates coexistence, with populations reaching an equilibrium at the intersection of both species' isoclines. (d) Represents two possible outcomes, competitive exclusion by species 2 (gray point, y-axis) and exclusion by species 1 (gray point, x-axis).

A second outcome of intersecting isoclines is that the positions of each species are switched (Figure 5d). In this instance (K2 > K1/a12 and K1 > K2/a21) interspecific competition is greater than intraspecific competition. This situation is unstable and populations come to equilibrium on either the x- or j-axis, effectively with the local extinction of one species (Figure 5d). Small islands off the coast of New Guinea house a variety of bird species. Yet similar species (i.e., similar in size, diet, and habitat use) often fail to co-occur with each another. Islands house either one or the other putative competitors, which is what would be expected if interspecific competition was stronger than intraspecific competition. 