Appendix Spatial models
Consider a spatially structured environment consisting of a number of patches of suitable habitat embedded in an inhospitable matrix. There is permanent spatial heterogeneity in habitat quality as would occur if there were differences in soil, nutrient availability, or moisture content that would make some host plant patches or ponds more productive than others. These spatial differences are assumed to occur within a spatial scale that can be traversed by the organisms occupying these habitats.
Within each habitat patch we have a multitrophic interaction characterized by in-traguild predation or keystone predation. In both cases coexistence can occur within a habitat patch if there is an interspecific trade-off between competitive ability and susceptibility to predation. The expression of this trade-off, however, depends on on variability in resource and/or predator traits. At very low or very high resource productivity/predator mortality one species gains an overall advantage and excludes the other (Leibold 1996; Holt and Polis 1997; Noonberg and Abrams 2005). Coexistence in variable environments thus requires additional mechanisms besides the competition-predation trade-off.
I consider the simplest mathematical representation of a metacommunity with in-traguild predation or keystone predation: a three patch model with each patch exhibiting a level of resource productivity that leads to a qualitatively different outcome: (i) resource productivity is too low for the predator-resistant inferior competitor to invade when rare, (ii) resource productivity is too high for the predator-susceptible superior competitor to invade when rare, and (iii) resource productivity is within the range that allows both species to invade and coexist via a competition-predation trade-off (Holt and Polis 1997; Noonberg and Abrams 2005; Amarasekare 2006, 2007).
I consider a situation in which the resource species is sedentary. The two consumer species do disperse, as does the predator in the case of keystone predation.
2.6.1 Intraguild predation
The spatial dynamics of a community with IGP are given by:
^ = e±a±RjC±j — d\C\j — aCyC^j — /?iCy + J^Cy (2.2)
APPENDIX: SPATIAL MODELS 27
^ = e2a2RjC2j - d2C2j + faC^Cy - /32C2j + ^ ]T C2j (2.3)
j=i where Rj is the resource abundance in patch j, and Cij is the abundance of Consumer species i in patch j (i = 1,2, j = 1,2, 3; Amarasekare 2006). The parameter rj is the per capita rate of resource production in patch j and K is the resource carrying capacity. Resource productivity varies spatially while the resource carrying capacity remains invariant across patches. The parameter ai is the Consumer i's attack rate, ei is the number of its offspring resulting from resource consumption, and di is its background mortality rate. The parameter a is the Consumer 2's attack rate on Consumer 1, and f is the number of Consumer 2 offspring resulting from intraguild predation. Consumer 1 therefore is the IGPrey, and Consumer 2 is the IGPredator. The parameter f3i is the per capita emigration rate of Consumer i.
I nondimensionalize Equations (2.1)-(2.3) using scaled quantities. Nondimensional analysis not only reduces the number of parameters but also highlights the biologically significant scaling relations between parameters (Nisbet and Gurney 1982; Murray 1993).
I use the substitutions ri Rj ^ Cij „ rj „ eiOiK „ e2aK f ßi f ef
Rj = -77, Cij = —Tj = -f, a,i = ——, a = ——, ßi = —, / = -,
K eiK d1 di d2 di e2
S=^,r = d1t (di ¿0, i = 1,2, j = 1,2,3) di to transform the original variables into nondimensional quantities. The dimensionless time metric t expresses time in terms of the IGPrey's death rate. This time scaling allows for comparing systems that differ in their natural time scales. Resource abundance is expressed as a fraction of the resource carrying capacity, and varies from
0 to 1. The consumers' abundances are scaled by their respective conversion efficiencies and the resource carrying capacity. The scaled attack rates (ai) depend on the resource carrying capacity and the consumer death rate (di); the scaled interference parameter a shows that the per capita inhibitory effect of the IGPredator on the IGPrey depends on the IGPredator's conversion efficiency and mortality rate as well as the resource carrying capacity. The parameter S is the ratio of the consumers' mortality rates, and f3i is the per capita emigration rate of Consumer i relative to its within-patch mortality rate. The other important parameter is the efficiency metric f. On their own, the efficiency parameters ei and f have little meaning; as a composite they reveal important scaling relationships between conversion efficiencies for resource consumption and IGP. For instance, large values of f imply that for any value of ei, f >> e2, i.e., the IGPredator obtains a greater benefit from the IGPrey than from the basal resource.
1 substitute the nondimensional quantities into Equations (2.1)-(2.3) and drop the hats for convenience. This yields the nondimensional system:
dC 3
—^ = a\RjC\j - Cij - SaCljC2j - /3iCy + ^ Cy (2.5) T j=i
Unless otherwise noted, all variables and parameters from this point on are expressed as nondimensional quantities.
Because the goal is to understand the possible interplay between local coexistence mechanisms and those mediated by dispersal, I restrict attention to the situation where local coexistence via a competition-IGP trade-off is possible in at least one patch. The trade-off is such that the IGPrey is the superior resource competitor (i.e., it has a lower R*; Tilman 1982), but the IGPredator can prey on the IGPrey (a > 0). From Equations (2.4)-(2.6) R* in the absence of dispersal is and hence competitive superiority of the IGPrey translates into having a higher attack rate than the IGPredator. Throughtout, I use ai as the measure of competitive ability and a as a measure of the strength of IGP while keeping the mortality ratio (S) and conversion efficiency (f) fixed.
I introduce spatial variation by setting the resource productivity in each patch to a level that leads to one of the three outcomes observed in the absence of dispersal: (i) IGPrey only, (ii) Coexistence, (iii) IGPredator only. Adopting the convention that patches 1, 2, and 3 represent increasing levels of resource productivity we have n = (0,ro2), r2 = (ro2 ,tq1 ), and rs = (r^ ,rmax) where tq^ and ra2 are, respectively, the threshold resource productivities required for the IGPrey and IG-Predator to invade when rare, and rmax is the maximum resource productivity.
Because the focus is on the interplay between IGP and dispersal, I assume that the two consumer species differ in their attack rates (ai) and dispersal propensities (pi) but have similar background mortality rates (i.e., S = 1). Further details of model analyses are given in Amarasekare (2006).
2.6.2 Keystone prédation
The spatial dynamics of a community with keystone predation are given by:
= e2a2RjC2j - d2C2j - a2C2j-i>- - p2C2j + Ç ]T C2j (2.9)
j=i where Rj is the resource abundance, Cj is the abundance of consumer species i, and Pj is the predator abundance in patch j (i = 1, 2,j = 1,2, 3; Amarasekare 2008). The parameter rj is the per capita rate of resource production in patch j, and K is the spatially invariant resource carrying capacity; ai is consumer species i's attack rate on the resource, ei is the number of its offspring resulting from resource consumption, and di is its background mortality rate. The parameter ai is the predator's attack rate on consumer i, and fi is the number of resulting predator offspring, and dP is the predator's background mortality rate. The parameters f3i and f3P are, respectively, the per capita emigration rates of consumer i and the predator.
I nondimensionalize Equations (2.7)-(2.10) using the scaled quantities
Ri r
Cn p ejT
flf2*iK
(Note that I have separated the nondimensional parameter / j^. (/. k 1.2. i /
k) intoej = ^ and / = because it allows for a more biologically meaningful scaling relationship; Amarasekare 2008) The nondimensional time metric t expresses time in terms of the superior competitor's death rate. The nondimensionalized attack rate of the predator on consumer i (a) depends on the predator's conversion efficiencies (fi), consumer i's mortality rate, and the resource carrying capacity. The parameter dP is the predator's mortality rate relative to that of the superior competitor, and f3P is the predator's per capita emigration rate scaled by the predator's death rate. Other nondimensional quantities have the same meaning as in the IGP model (Equations (2.4)-(2.6)). Substituting these quantities into Equations (2.7)-(2.10) yields the following nondimensional system:
ij dt dC
e\aiRjCij — Cij — a^C^Pj — ß\C\j + ^^ Cy j=i 3
2j dt
j2j j=1
^ = dfcuCyPj + ^aiCijPj - dPPj - dpßpPj + <2'14)
Further details of model analyses are given in Amarasekare (2008).
eiOiK
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CHAPTER 3
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