Ecological Archives E086-137-A1

Erik G. Noonburg and James E. Byers. 2005. More harm than good: when invader vulnerability to predators enhances impact on native species. Ecology 86:2555–2560.

Appendix A. Coexistence conditions and derivation of omega.

We seek conditions that determine the outcome of invasion of N2 in a food web containing the three native species, R, N1, and P (Fig. 1). Conditions for the three possible outcomes (coexistence of N1 and N2, extirpation of N1, and inability of N2 to invade) are equivalent to coexistence conditions from general competition theory. For instance, inability of N2 to invade corresponds to competitive exclusion of N2. For the model in Eq. 1, these coexistence conditions are mathematically identical to invasion conditions, i.e., N1 and N2 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 N2 into the community consisting of R, N1, and P; and (2) invasion of N1 into a community in which N2 is the resident consumer. Although the condition governing scenario 2 is calculated by invasion analysis, it is merely the mathematical criterion for persistence of N1 given successful invasion by N2, and it does not imply that N1 is in fact invading the community.

Invasion of N2 influences persistence of N1 through direct and apparent competition; therefore, condition 2 specifies two requirements for persistence of N1. First, N1 must be able to invade the food web composed of R, N2, and P when these three species are at equilibrium. Second, because N2 potentially extirpates the predator (when N2 is a poor food resource), N1 must also be able to invade the food web composed of only R and N2. Here again, this condition does not correspond to an additional invasion scenario—it is a condition for persistence of N1 after invasion of N2.

In the following sections we derive expressions for conditions 1 and 2. We first rewrite the model with the two parameter combinations, chi=a2/a1 and rho= b2/b1. Substituting chi and rho into Eq. 1 gives

(A.1)

 

All of the following analyses are performed on Eq. A.1.

Condition 1: Invasion of N2

The lower boundary (chimin) in Fig. 2 is the minimum value of c for which the per capita growth rate of an infinitesimally small population of N2 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 N1 is present but N2 is absent. We rearrange Eq. A.2 to obtain

 

(A.3)

 

Condition 2: Invasion of N1

            The upper boundary (chimax) is the maximum value of chi for which N1 can invade the system with N2 present at equilibrium. If the predator's attack rate on N2 is low (small rho), the predator can not persist on N2, (i.e., if N2 can invade and it extirpates N1, the predator will also be extirpated). In this case N1 must have a positive per capita growth rate when R and N2 are at equilibrium and the other two species are absent: a1epsilon1R*(3)-m1>0, where R*(3) is the resource equilibrium when N2 is the only other species present. This condition yields

 

(A.4)

 

For larger chi, the effect of resource competition alone is sufficient for N2 to exclude N1.

            Larger values of rho allow the predator to persist on N2. We find the point at which predator persistence on N2 is possible by solving P*(2)>0 for rho 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 N2 excludes N1 if the per capita growth rate of N1 is negative when the other three species are at equilibrium (R*(2), N2*(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 rho or chi. For clarity of exposition in the text, we first solve Eq. A.2 for left and right values of rho (rhol and rhor; see Model and Results):

 

(A.8)

 

Derivation of omega

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 N1 or N2 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 N1. In this case, increasing never causes extinction of N1. The same approximation implies that maxr.

            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 mi≈0 results in boundaries that fall outside the exact boundaries of region 2 if <m2/m1, which must be true because the lower root in Eq. A.14 crosses max at =m2/m1.

 

LITERATURE CITED

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.



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