As we saw on p. 251 of the textbook, competing species can coexist in only one of the four outcomes of competition in the Lotka–Volterra competition model. We can use the graphical analyses of the Lotka–Volterra competition model shown in textbook Figure 11.13 to derive the conditions under which coexistence occurs.
The single case in which competitors coexist is shown in textbook Figure 11.13D. In that graph, the N1 and N2 isoclines intersect. Note that for lower values of N1 (those to the left of the point where the two isoclines intersect), the N1 isocline is above the N2 isocline, but for higher values of N1, the reverse is true. Examining the y intercepts of the two isoclines, we see that the N1 isocline will be above the N2 isocline for lower values of N1 whenever the following inequality holds:
Similarly, examining the x intercepts of the two isoclines, we see that the N2 isocline will be above the N1 isocline for higher values of N1 (those to the right of the point where the two isoclines intersect) whenever the following inequality holds:
If we rearrange the terms in these inequalities, the first inequality indicates that
while the second inequality indicates that
Finally, if we combine these two inequalities into a single statement, we find that:
These are the conditions for coexistence in the Lotka–Volterra competition model given in Equation 11.2 (textbook p. 252).
