# A21 And A12 Population

According to these results, the competition coefficients are determined by the ratio of the carrying capacities.

To begin the graphical analysis, we start with Equation 7.5. As we have seen from Equation 7.7c, when species two = 0, then N1 = K1. Similarly, when species one = 0, N2 = K1la12 (7.7d). If we graph N1 versus N2 (Fig. 7.2), we can use the two points (K1, 0) and (0, K1la12) to produce a line that represents saturation levels for species one. Combinations of populations 1 and 2 below (to the left of) the line are such that population 1 continues to increase. Combinations above the line (to the right) lead to a decrease in the size of population 1 (Fig. 7.2). The line itself is known as the zero isocline for population 1.

Population of species 1 Figure 7.2 Zero isocline for species 1.

Population of species 1 Figure 7.3 Zero isocline for species 2.

In a similar fashion, from Equation 7.8c we know that if population 1 = 0, then N2 = K2. From Equation 7.8d, we have that if population 2 = 0, then N1 = K2/a21. From the resultant points (0, K2 and K2/a21, 0), we can draw a second line or zero isocline, representing saturation levels for species two. Again, combinations of populations 1 and 2 to the left or below the line result in increases toward the carrying capacity for population 2; combinations above the line lead to a decrease in population 2 (Fig. 7.3).

We now place these zero isoclines or saturation levels on the same graph. Four combinations are possible when drawing these lines:

1 K,/a12 > K2 combined with K1 > K2/a21 (Fig. 7.4);

2 K2 > K,/a12 combined with K2/a21 > K1 (Fig. 7.5);

3 K1 > K2/a21 combined with K2 > K,/a12 (Fig. 7.6); and

4 K1/a12 > K2 combined with K2/a21 > K1 (Fig. 7.7).

In Fig. 7.4, both populations increase when their numbers are below both saturation lines. However, when combinations of species one and two are above the species-two zero isocline, but below the species-one zero isocline, the resultant vectors move the combined number of individuals toward only one equilibrium point, and that is when N1 = K1. In case one, then, we have competitive exclusion and species one is the winner.

Figure 7.5 illustrates the opposite situation. When combinations of species one and two are between the two saturation lines, the resultant vector moves the combined number of individuals toward an equilibrium at N2 = K2. Again we have competitive exclusion, but now species two is the winner.

Figure 7.6 represents a more complex, ambiguous situation. If the combination of N1 and N2 is along the line 0-S, the resultant vector moves along the line to a temporary, unstable equilibrium at S, where both species coexist. However, if anything in the environment moves the combinations of individuals to the right or the left of the line 0-S, then the resultant vectors move rapidly toward competitive exclusion. However, in this

Figure 7.4 Species 1 wins.
Figure 7.5 Species 2 wins.

case there are two stable equilibrium points. If the combinations of the two species produce points to the left of the line 0-S, the result is competitive exclusion with species two as the winner. If the points are to the right of the line 0-S, species one is the winner. Therefore this set of conditions will produce an unstable equilibrium or competitive exclusion with an indefinite winner (a stochastic result).

Finally, in Fig. 7.7 we have a situation in which each species slows its own growth more than that of its competitor. This allows a stable equilibrium with coexistence of the two species at point E.

Figure 7.6 Unstable equilibrium (S) or competitive exclusion with an indefinite winner.
Figure 7.7 Coexistence of both species at the stable equilibrium, E.

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