Appendix A. Coexistence conditions and derivation of .
We seek conditions that determine the outcome of invasion of N_{2} in a food web containing the three native species, R, N_{1}, and P (Fig. 1). Conditions for the three possible outcomes (coexistence of N_{1} and N_{2}, extirpation of N_{1}, and inability of N_{2} to invade) are equivalent to coexistence conditions from general competition theory. For instance, inability of N_{2} to invade corresponds to competitive exclusion of N_{2}. For the model in Eq. 1, these coexistence conditions are mathematically identical to invasion conditions, i.e., N_{1} and N_{2} can coexist at equilibrium if each competitor can invade the system with the other species at equilibrium (see, e.g., Noonburg and Abrams 2005, for a more detailed analysis of this model). Hence, we must solve for two invasion scenarios: (1) invasion of N_{2} into the community consisting of R, N_{1}, and P; and (2) invasion of N_{1} into a community in which N_{2} is the resident consumer. Although the condition governing scenario 2 is calculated by invasion analysis, it is merely the mathematical criterion for persistence of N_{1} given successful invasion by N_{2}, and it does not imply that N_{1} is in fact invading the community.
Invasion of N_{2} influences persistence of N_{1} through direct and apparent competition; therefore, condition 2 specifies two requirements for persistence of N_{1}. First, N_{1} must be able to invade the food web composed of R, N_{2}, and P when these three species are at equilibrium. Second, because N_{2} potentially extirpates the predator (when N_{2} is a poor food resource), N_{1} must also be able to invade the food web composed of only R and N_{2}. Here again, this condition does not correspond to an additional invasion scenario—it is a condition for persistence of N_{1} after invasion of N_{2}.
In the following sections we derive expressions for conditions 1 and 2. We first rewrite the model with the two parameter combinations, =a_{2}/a_{1} and = b_{2}/b_{1}. Substituting and into Eq. 1 gives
(A.1) |
All of the following analyses are performed on Eq. A.1.
Condition 1: Invasion of N_{2}
The lower boundary (_{min}) in Fig. 2 is the minimum value of c for which the per capita growth rate of an infinitesimally small population of N_{2} is positive when the other three species are at equilibrium:
(A.2) |
In Eq. A.2 the superscripts for R^{*(1)} and P^{*(1)} denote the equilibria when competitor N_{1} is present but N_{2} is absent. We rearrange Eq. A.2 to obtain
(A.3) |
Condition 2: Invasion of N_{1}
The upper boundary (_{max}) is the maximum value of for which N_{1} can invade the system with N_{2} present at equilibrium. If the predator's attack rate on N_{2} is low (small ), the predator can not persist on N_{2}, (i.e., if N_{2} can invade and it extirpates N_{1}, the predator will also be extirpated). In this case N_{1} must have a positive per capita growth rate when R and N_{2} are at equilibrium and the other two species are absent: a_{1}_{1}R^{*(3)}-m_{1}>0, where R^{*(3)} is the resource equilibrium when N_{2} is the only other species present. This condition yields
(A.4) |
For larger , the effect of resource competition alone is sufficient for N_{2} to exclude N_{1}.
Larger values of allow the predator to persist on N_{2}. We find the point at which predator persistence on N_{2} is possible by solving P^{*(2)}>0 for which gives
(A.5) |
When this inequality is satisfied, the upper coexistence boundary depends on apparent competition as well as resource competition. In this case the invasion condition implies that N_{2} excludes N_{1} if the per capita growth rate of N_{1} is negative when the other three species are at equilibrium (R^{*(2)}, N_{2}^{*(2)}, P^{*(2)}):
(A.6) |
Substituting for the equilibria in Eq. A.6 and rearranging yields,
(A.7) |
The inequality in Eq. A.7 can be solved to obtain boundaries of region 2 (Fig. 2) in terms of or . For clarity of exposition in the text, we first solve Eq. A.2 for left and right values of (_{l} and _{r}; see Model and Results):
(A.8) |
Derivation of
The width of region 2 (Fig. 2) is equal to _{r}-_{l}, which is twice the second term on the right-hand side of Eq. A.8. (For sufficiently small , i.e., below the minimum of region 2, the solution to Eq. A.7 does not exist.) The upper value of that bounds region 1 is identical to _{min}. Hence, we rearrange Eq. A.2 to an expression for as a function of ,
(A.9) |
which is the width of region 1. The exact value of (width of region 2 relative to the width of region 1) is therefore
(A.10) |
for such that >0.
To obtain the approximation for in Eq. 4, we make the following assumption. In the absence of predators the per capita growth rate of either N_{1} or N_{2} when alone is high if the resource is at its carrying capacity, i.e., With these approximations,
(A.11) |
From Eq. A.5, we have in region 2, so the smaller root must be the result of addition in the numerator of Eq. A.11,
(A.12) |
and the larger root is
(A.13) |
If the parameter values are such that the relative size of the two roots is reversed, then Eq. A.6 can not be satisfied and Eq. A.4 is the only condition for persistence of N_{1}. In this case, increasing never causes extinction of N_{1}. The same approximation implies that _{max}≈_{r}.
The approximations in Eqs A.10 and A.11 are not accurate for small ; however, we can show that _{l} and _{r} bound the region defined by Eq. A.6 when <_{max}. The arithmetic is simpler if we rewrite the boundary of region 2 (Eq. A.8) in terms of instead of . Rearranging Eq. A.6 as a quadratic in and solving for the roots gives
(A.14) |
From Eq. A.14 we see that the approximation m_{i}≈0 results in boundaries that fall outside the exact boundaries of region 2 if <m_{2}/m_{1}, which must be true because the lower root in Eq. A.14 crosses _{max} at =m_{2}/m_{1}.
Noonburg, E. G., and P. A. Abrams. 2005. Transient dynamics limit the effectiveness of keystone predation in bringing about coexistence. American Naturalist 165:322–335.