Interference competition and invasion: spatial structure, novel weapons and resistance zones
aa r X i v : . [ q - b i o . P E ] S e p Interference competition and invasion: spatialstructure, novel weapons and resistance zones
Andrew Allstadt a,1 , Thomas Caraco a, ∗ , F. Moln´ar Jr. b , G. Korniss b a Department of Biological Sciences, University at Albany, Albany, NY 12222, USA b Department of Physics, Applied Physics and Astronomy, Rensselaer Polytechnic Institute,Troy, NY 12180, USA
Abstract
Certain invasive plants may rely on interference mechanisms (allelopathy, e.g .)to gain competitive superiority over native species. But expending resources oninterference presumably exacts a cost in another life-history trait, so that thesignificance of interference competition for invasion ecology remains uncertain.We model ecological invasion when combined effects of preemptive and inter-ference competition govern interactions at the neighborhood scale. We considerthree cases. Under “novel weapons,” only the initially rare invader exercisesinterference. For “resistance zones” only the resident species interferes, and fi-nally we take both species as interference competitors. Interference increasesthe other species’ mortality, opening space for colonization. However, a speciesexercising greater interference has reduced propagation, which can hinder itscolonization of open sites. Interference never enhances a rare invader’s growthin the homogeneously mixing approximation to our model. But interferencecan significantly increase an invader’s competitiveness, and its growth whenrare, if interactions are structured spatially. That is, interference can increasean invader’s success when colonization of open sites depends on local, ratherthan global, species densities. In contrast, interference enhances the common,resident species’ resistance to invasion independently of spatial structure, un-less the propagation-cost is too great. Increases in background mortality ( i.e .,mortality not due to interference) always reduce the effectiveness of interferencecompetition.
Keywords: biotic resistance, interference competition, invasion, pairapproximation, spatial ecology ∗ Corresponding author. Tel.: +1 518 442 4343
Email addresses: [email protected] (Andrew Allstadt), [email protected] (Thomas Caraco), [email protected] (F. Moln´ar Jr.), [email protected] (G. Korniss) Present address: Blandy Experimental Farm, University of Virginia, Boyce, VA 22620,USA
Preprint submitted to Journal of Theoretical Biology October 17, 2018 . Introduction
Both lateral and vertical interactions can affect the likelihood that an inva-sive species advances when rare (Levine et al., 2004; Going et al., 2009). But formany plants, competitive asymmetry between invader and resident species gov-erns both the outcome and the timescale of ecological invasion (Lavergne and Molofsky,2004; Vil`a and Weiner, 2004; O’Malley et al., 2006a; MacDougall et al., 2009).Given invader-resident competition, ecological superiority may depend on morethan one mechanism (Case and Gilpin, 1974; Ridenour and Callaway, 2001).Our analysis addresses the combined impact of preemptive and interference com-petition on invasion dynamics when biotic interactions are structured spatially.We assume that interference has a cost (Adams et al., 1979; Bazzaz and Grace,1997; Amarasekare, 2002); an increasing level of interspecific interference re-quires a reduction in propagation rate, diminishing that species’ capacity tocolonize unoccupied sites.In many plant communities the primary mode of competition is site preemp-tion (Schoener, 1983; Bergelson, 1990; Crawley et al., 1999; Yurkonis and Meiners,2004); i.e ., species interact through colonization of empty sites. Superior pre-emptive competitors have higher propagation rates or lower mortality rates(Korniss and Caraco, 2005; O’Malley et al., 2006b; Allstadt et al., 2007). Theformer increases colonization of open sites, and the latter decreases a com-petitor’s opportunities for colonization. Preemptive competitors have the sameniche in a spatially homogeneous environment (Amarasekare, 2003). Ordinarilythis precludes coexistence (Shurin et al., 2004; Allstadt et al., 2009), since self-regulation does not exceed interspecific competition. Case and Gilpin (1974)suggest that this niche similarity might favor evolution of interference mecha-nisms.An interference competitor inhibits another species’ access to a critical re-source, often by harming individuals of the other species. Examples include in-terspecific territoriality in animals and chemical competition in plants (Case et al.,1994; Callaway and Ascheghoug, 2000). Exotic invaders may suppress nativespecies’ densities through interference competition (D’Antonio and Vitousek,1992; Callaway and Ridenour, 2004; Cappuccino and Arnason, 2006). The “novelweapons” hypothesis proposes that some invasive plants release chemicals thatinhibit growth of native species (Callaway and Ascheghoug, 2000). Interest-ingly, allelopathic interference may act directly on individuals of the residentcompetitor, or may act indirectly through toxic effects on native species’ mi-crobial mutualists, particularly mycorrhizal fungi (Wolfe and Klironomos, 2005;Callaway et al., 2008). So, under the novel weapons hypothesis, invaders attaincompetitive superiority because their phytochemicals present novel challenges tonative species. Reasonably, in other communities, exotic species likely encounterinterference competition from natives; see comments in Von Holle et al. (2003).Our models address the three general cases where interference can affect theoutcome of resident-invader competition. We associate novel weapons with in-terference by the invader only. We refer to “resistance zones” when the residentspecies, but not the invader, exerts interference competition. And, of course,2nvaders and residents may each compete via interference (Case and Gilpin,1974). If neither species exhibits interference, our model leaves preemption asthe sole competitive mechanism. In this case the species that, when alone,maintains the greater equilibrium density will always displace its competitor(Amarasekare, 2003; O’Malley et al., 2006a; Allstadt et al., 2007).Discrete, stochastic spatial models and their deterministic analogues havebeen employed, commonly and successfully, to gain insight into collective behav-ior of multi-“species” interactions (Marro and Dickman, 1999; Murray, 2003) inphysics (Korniss et al., 1995, 1997), chemistry (Antal et al., 1996; Toroczkai et al.,1997; Ziff et al., 1986), and in the study of population dynamics (Ellner et al.,1998; McKane and Newman, 2004; O’Malley et al., 2006b, 2009). We use bothmethods to investigate the combined effects of preemptive and interference com-petition on invasion. We organize our paper as follows. First, we present adiscrete (individual-based), stochastic model where an invader and a residentspecies compete preemptively, and one or both species also employs interference.We let a species’ propagation rate depend functionally on its level of interference;we consider convex, linear and concave trade-offs. We explore the model by an-alyzing invasibility criteria of both a mean-field approximation (homogeneousmixing) and a pair approximation. Then we apply results of the approximationsto interpret simulations of the full spatial model. The Discussion compares ourresults to other spatial models incorporating allelopathy. Appendices collectmuch of the analytical detail.Our results find that interference by the invader increases its likelihood ofsuccessful invasion only when interactions are spatially structured. That is, thenovel weapons advantage, in our model, appears only when we account for localclustering of invader individuals. Interference by the resident, the initially com-mon species, can inhibit invasion with and without spatial structure. Increasesin background mortality rate (mortality before adjustment due to interference)diminish the competitiveness of the species relying more on interference. In-creased mortality reduces both local and global densities; that is, the frequencyof empty sites increases. Consequently, the value of interference, relative topropagation, declines. Our model assumes that interference increases mortalityof a preemptive competitor, but interference does not generate an alternativeniche. Consequently, we should not anticipate coexistence absent continuousintroduction or strong effects of spatial clustering (Allstadt et al., 2009).
2. A discrete, stochastic spatial model
When plant species compete, interactions regulating population growth gen-erally occur at the neighborhood scale (Goldberg, 1987; Uriarte et al., 2004). In-terference competition, including allelopathy, ordinarily has a local spatial struc-ture. And, when propagule dispersal distance (inter-ramet distance in clonalspecies) is limited, plants compete preemptively for space at the neighborhoodscale. Consequently, our model - which integrates preemptive and interferencecompetition - assumes that competitive interactions among nearest neighborsdrive invasion and the dynamics of species’ abundances (O’Malley et al., 2006a).3 .1. Model construction
Two clonal plant species compete on an L x × L y lattice with periodic bound-aries. Each lattice site represents the resources required to sustain a single in-dividual (a ramet) of either species. The local occupation number at site x is n i ( x ) = 0 , i = 1 ,
2, referring to the resident and invader species, re-spectively. During a single simulated time unit, one Monte Carlo step per site[MCSS], L x L y sites are chosen randomly for updating.An empty site may be occupied by species i through introduction from out-side the environment, or through local propagation. Introduction of species i at an open site occurs as a Poisson process with rate β . Each species has thesame introduction rate, to avert any effect of propagule pressure on the out-come of competition. Local propagation into an open site has rate α i η i ( x ),where α i is the individual-level propagation rate for species i , and η i ( x ) =(1 /δ )Σ x ′ ǫ nn( x ) n i ( x ′ ) is the density of species i in the neighborhood around opensite x . nn( x ) is the set of nearest neighbors of site x , and δ is the number ofsites in that neighborhood ( δ = | nn ( x ) | ). Since we equate colonization withpropagation of new ramets, we let δ = 4.Colonization can occur only at open sites. An occupied site opens throughmortality. Individuals of each species suffer density-independent mortality atrate µ (Cain et al., 1995). An individual occupying site x experiences an in-creased mortality rate due to interference if nn ( x ) includes any heterospecifics.That is, an individual of species i at site x has total mortality rate µ + θ j η j ( x ) , i = j . θ j ≥ j , and η j ( x ) is the density of species j on the neighborhood around site x .Summarizing transition rules for an arbitrary site x , we have0 β + α η ( x ) −→ , β + α η ( x ) −→ , µ + θ η ( x ) −→ , µ + θ η ( x ) −→ , (1)where 0, 1, 2 indicates whether a site is open, resident-occupied, or invader-occupied, respectively. Table 1 defines the symbols and notation we use. Weassume that interspecific competition drives the dynamics; i.e ., each species per-sists absent competition. Therefore, we restrict attention to the β ≪ α c ( µ ) < α i ( i = 1 ,
2) regime, where α c ( µ ) is the critical propagation rate below whicheither species, in the other’s absence, grows too slowly to avoid extinction(Oborny et al., 2005; O’Malley et al., 2006a). The essential lesson of life-history theory is that increased allocation of re-sources to one trait advancing survival or reproduction comes at the expenseof another trait (Bell and Koufopanou, 1986). Applying this concept, we as-sume that any increase in a species’ level of interference reduces that species’clonal propagation rate. The functional dependence (trade-off) has the form α Ri + θ Ri = C R , where C is a constant, equal to the maximal feasible rate ofpropagation, and R defines the shape of the trade-off (Giraldeau and Caraco,2000). If 0 < R <
1, the cost of increasing α (i.e., the decrease in θ ) decreases4 able 1: Definitions of model variables and parameters Symbols Definitions L x , L y (= L ) Lattice size x Location of lattice site n ( x ) Occupation number for residents at site x ; n ( x ) = 0 , n ( x ) Occupation number for invaders at site x ; n ( x ) = 0 , x ) Set of nearest neighbors around site x δ size of neighborhood around site x ( δ = | nn ( x ) | ) β Common introduction rate at empty sites η i ( x ) Density of species i on nn( x ) α i Individual rate of propagule production, species iµ Background mortality rate, both species θ j Mortality of species i due to interference by species jα c ( µ ) Minimal propagation rate for persistence C Maximal propagation rate; C = 0.8 R Curvature of α, θ trade-off ρ i ∗ Equilibrium single-species densityas α increases. If R = 1, the cost of increasing α is constant, and if R > α increases at greater α . More importantly, any increasein R increases competitiveness of a species that employs both preemption andinterference; the constraint moves farther from the origin as R increases. If aspecies does not exercise interference, we let its propagation rate vary on ( α c ( µ ),0.8]. Therefore, we let C = 0 . Under the novel weapons hypothesis, the invader exercises interference com-petition while the resident does not. That is, θ = 0, and0 < θ ≤ R q C R − α c ( µ ) R Invasive plants may interfere with native competitors through several mech-anisms. Some impact natives directly; others are mediated through a thirdspecies. Invaders can act as a disease reservoir (Eppinga et al., 2006; Borer et al.,2007), focus herbivory on native species (Dangremond et al., 2010), or produceallelopathic chemicals (Prati and Bossdorf, 2004; Cipollini et al., 2008). Ouranalysis emphasizes the invader’s trade-off between propagation and interferenceacross a range of native species lacking interference. Under the novel weaponshypothesis, the resident species does not compete through interference, and welet its reproductive rate vary to model variation in the strength of preemptionthe invader encounters.
To study resistance zones, we let the resident exercise interference competi-tion while the invader does not. That is, 0 < θ ≤ R q C R − α c ( µ ) R , and θ = 0.5 resident species exercising interference competition may strongly resist inva-sion since its initial density (by definition) will be relatively high. Given bioticresistance combining preemption and interference, we vary the invader’s propa-gation rate to consider a range of matches between the invader and the abioticenvironment invaded. Under the assumption of mutual interference competition, we have 0 <θ , θ ≤ R q C R − α c ( µ ) R . For this case we assume that each species’ resourceallocation between propagation and interference follows the same trade-off.
3. Mean-field approximation
The spatially-homogeneous mean-field approximation to our model ignoresany effects of locally clustered growth. The mean-field model (MF) so offers com-parison to results including interactions at the neighborhood scale; see Wilson(1998), Pascual and Levin (1999) or Cuddington and Yodzis (2000) for perspec-tive. ρ and ρ represent the global densities of species 1 and species 2, respec-tively. Ignoring continuous introduction (letting β = 0) we have a mean-fielddynamics: ˙ ρ = α ρ (1 − ρ − ρ ) − ρ ( µ + θ ρ ) (2)˙ ρ = α ρ (1 − ρ − ρ ) − ρ ( µ + θ ρ ) (3) θ i represents the increased mortality of species j induced by species i ; we scale θ i per unit density of species i . Species’ persistence absent competition in theMF approximation requires only that each α i > α c ( µ ) = µ . Letting θ i ≥
0, for i = 1 ,
2, we conduct a general stability analysis of the MF model’s equilibria.Thereafter, we consider competitive superiority when the specific propagation-interference constraint introduced above applies.
The dynamics has three boundary equilibria and one positive equilibrium.Mutual extinction, designated equilibrium E1, cannot be stable since each α i >µ . At Equilibrium E
2, species 1 competitively excludes species 2: E (cid:18) ρ ∗ = 1 − µα , ρ ∗ = 0 (cid:19) (4)Species 2 excludes species 1 at equilibrium E E (cid:18) ρ ∗ = 0 , ρ ∗ = 1 − µα (cid:19) (5)6inally, a positive (internal) equilibrium, E
4, can be expressed as: ρ ∗ = θ ( α − µ ) + µ ( α − α ) α θ + α θ + θ θ (6) ρ ∗ = θ ( α − µ ) + µ ( α − α ) α θ + α θ + θ θ (7)As noted above, we know that when competition is strictly preemptive ( θ = θ = 0), the species cannot coexist under homogeneous mixing.Suppose that the resident excludes the invader. From Appendix A, localstability of invader extinction requires: µ ( α α −
1) + θ ( µα − < µ < α , θ ( µα − ≤
0. Therefore, any interference by the residenttends to stabilize invader extinction by increasing invader mortality. That is,interference by the common species can help prevent advance of the rare species,even if the rare species has the greater propagation rate. Clearly, if α > α and θ ≥
0, the resident species repels the invader. Note that θ , the invader’slevel of interference competition, does not appear in Inequality (8). Hence theinvader cannot increase its growth rate when rare through interference; onlyincreased propagation promotes mean-field invasion.To elaborate, the invader advances from rarity only if Inequality (8) is re-versed. From Appendix A, assuming θ >
0, the rare species invades iff : α − α > θ ( α µ − > E Bistability requires that local-stability conditions for both single-species equi-librium nodes ( E E
3) hold simultaneously. Given bistability, initial condi-tions determine the outcome; at some point, the more abundant species “wins.”From Appendix A, the common species (1 or 2) repels its rare competitor if θ (1 − α µ ) < α − α and θ ( α µ − > α − α (10)If α > α , the first expression must hold, since θ (1 − α µ ) ≤
0. The secondinequality can hold simultaneously if θ is relatively large, and the differencebetween propagation rates is not too large. If α > α , the second expressionmust hold, since θ ( α µ − >
0, and symmetric conditions promoting bistability7re clear. Bistability can arise even if the species with the greater propagationrate does not exert interference. For example, if α > α , the system can bebistable when θ = 0 (provided, of course, θ > ρ , ρ ) phasespace. One example plots competitive asymmetry. Species 2 advances whenrare, and when common, excludes species 1. The other plots show bistability,one for a symmetric (identical species), and one for an asymmetric, domain ofattraction. Coexistence (local stability of a positive equilibrium) requires that eachspecies invade the other. Together, the conditions for mutual invasion indicatethat coexistence requires (Appendix A): θ (1 − α µ ) > α − α and θ (1 − α µ ) > α − α (11)Since each α i > µ , each (1 − α i µ ) <
0; the LHS of each of these inequalitiesmust then be non-positive. But one ( α i − α j ) must be positive, so that the twoinequalities cannot be true simultaneously. Hence the MF does not permit mu-tual invasion, and so does not admit competitive coexistence, when the speciesinteract through both preemption and interference. The preceding, general stability analyses did not treat interference as a func-tion of propagation rate. Invoking the life-history constraint generates a specialcase of the MF analysis. The condition for the invader’s advance when rare,Inequality (9), becomes: α > α + (cid:16) α µ − (cid:17) R p C R − α R (12)From the general MF analysis, any allocation to interference by the invader di-minishes its propagation rate ( α ), and so decreases the likelihood of successfulinvasion. Any increase in R , relaxing the propagation-interference tradeoff, in-creases the range of feasible parameter combinations where the resident repelsthe invader (see fig. 2). That is, the opportunity for successful invasion de-clines as R increases in the MF model, since only the resident can benefit frominterference under homogeneous mixing.Increasing background mortality ( µ ) increases the likelihood of successfulinvasion. Greater mortality, with θ and the α i fixed, decreases the resident’sdensity. Consequently, sites available for colonization increase in density, andthe impact of interference on the invader’s dynamics declines. Therefore, greaterbackground mortality, which affects both species, decreases the range of feasibleparameter combinations where the resident repels the invader (see fig. 2).If the competitors mix homogeneously, interference can help the residentrepel the invader, but cannot promote the rare species’ invasion. Interference,8s a mechanism of biotic resistance, has a greater effect when a small levelof interference does not cost too much in propagation, and when backgroundmortality is relatively low (so that resident density is relatively high).
4. Pair approximation
Pair approximation (PA) incorporates correlations of the occupation statusof nearest neighboring sites into a dynamics (Dickman, 1986; Matsuda et al.,1987; Bauch and Rand, 2000; van Baalen, 2000). By incorporating a minimalspatial structure, PA predicts equilibrium densities of individual-based modelsmore accurately than do MF models (Ellner et al., 1998; Caraco et al., 2001).For our purposes, PA addresses consequences of local clustering, generated bydispersal limitation, for invasion. Appendix B presents details.The PA tracks global densities ρ i ( t ), and conditional densities q j | i ( t ) rep-resenting the likelihood that a site is in state j , given that a neighboring sitein state i (Sato et al., 2000; O’Malley et al., 2006a). The three global densities( ρ ( t ) , ρ ( t ) , ρ ( t )) imply 9 local densities ( q j | i ). However, the PA’s dimensionis limited by simple constraints: ρ + ρ + ρ = 1 q | i + q | i + q | i = 1 (13) q i | j ρ j = q j | i ρ i , where i, j = 0 , , i = j ; we suppress time dependence for simplicity. Theseconstraints leave only 5 of the 12 total variables are independent. Since species2 is the invader, we track ρ , ρ , q | , q | , q | . Note that the dynamics of q | ,the conditional density of a resident given an invader at a neighboring site, candepend on a novel-weapons effect when θ > ρ = βρ + ρ α q | − ρ µ − θ q | ρ (14)Compared to the mean-field model, both local propagation and mortality dueto interference depend on local, rather than global, densities. Hence the PAmodels both site preemption and interference as effects of spatially clusteredgrowth.To model the dynamics of a local density, we first write the dynamics of adoublet ρ ii , a global density, where ρ ii = ρ i q i | i . The PA’s doublet dynamics( dρ ii /dt ; see Appendix B) introduces the triplets q | and q | . OrdinaryPA assumes that neighbors of neighboring sites are weakly correlated, and lets q | = q | and q | = q | . The approximation closes the system of equations,permitting us to write an invasion criterion (Iwasa et al., 1998).For the invasion analysis, assume that an introduction of species 2 has oc-curred. Invasion either succeeds or fails before the next introduction event9ccurs (since β ≪ µ < α i ). Successful invasion requires that the invader havea positive growth rate when rare; ˙ ρ > ρ →
0. From Appendix B, thiscondition yields the invasion criterion:1 − q | − q | > µ + θ q | α . (15)The left side represents the frequency of open sites neighboring an invader, q | . The right side represents the ratio of death rate to birth rate for theinvader. Successful invasion, then, requires that the invader colonize a neigh-boring, empty site before it dies. The death rate sums background mortalityand averaged mortality from the resistance zone about a resident neighboringthe invader. The seeming simplicity of the invasion criterion masks an impor-tant biological difference between the MF criterion and the neighborhood-scalecondition for invasion. In the PA condition for successful invasion, the localdensity q | , hence the local density of open sites q | , depends on θ , invaderinterference (see Appendix B). That is, the neighborhood-scale invasion criterionreveals a role for novel weapons, contrasting to the MF result. Consequently, aninterfering invader might invade a resident species despite the resident havingthe greater propagation rate. The right side of the criterion, of course, includethe resistance-zone effect of interference by the resident.To compare the PA invasion criterion to both the MF and simulation of thefull spatial model (see below), we evaluated Inequality (15) numerically. Weconstructed pairwise invasion plots for the three cases defined above: only theinvader interferes (novel weapons), only the resident interferes (resistance zones)and both species exercise interference competition. Whenever a species was aninterference competitor, we invoked the propagation-interference constraint.Figure 3 shows pairwise invasion results when the invader, and not the resi-dent, is the interference competitor. That is, the invader obeys the propagation-interference constraint, and θ = 0 independently of the resident’s propagationrate; we so isolate the novel weapons effect. When interactions are local, in-terference competition can promote an invader’s growth when rare. When thepropagation cost of interference is smaller ( R is larger) and µ is small, invaderswith a propagation rate much less than the resident’s rate succeed. Invader in-terference proves advantageous since it opens sites neighboring invaders. Henceinterference makes space available where the invader’s local density may matchor exceed the resident’s local density - despite the resident’s greater global den-sity. Hence, novel weapons, in our model, enhances invasion when individualscompete at the neighborhood scale, but has no effect under homogeneous mix-ing.Figure 4 shows pairwise invasion results when the resident, and not the in-vader, is the interference competitor. That is, the resident obeys the propagation-interference constraint, and θ = 0 independently of the invader’s propagationrate; we isolate the resistance zone effect. A resident can repel an invader with amuch higher propagation rate than its own through local interference. Increas-ing R now reduces the range of parameter combinations permitting invasion, andincreasing µ promotes invader success - results symmetrically opposite to those10or novel weapons. When only the resident can interfere, a greater value for R reduces the pleiotropic cost of interference. Increasing µ opens more space forthe invader; greater background mortality renders a given level of interferenceless effective as a competitive mechanism.Figure 5 shows pairwise invasion results when both species exercise interfer-ence, and so both are constrained by the propagation-interference tradeoff. Notsurprisingly, the resident species, because of its initial high density, can employinterference to repel the invader over a much greater range of parameters thanwhen both species lack interference (where greater propagation always wins).The utility of interference, from the resident species’ perspective, declines atsmall R , and at greater µ ; the invasion plot for ( R = 0 . , µ = 0 .
2) matches thecase where competition is strictly preemptive. In most of the plots substantialregions of bistability appear. A species with intermediate levels of both prop-agation and interference can repel a broad range of propagation-interferencecombinations. And, when rare, the intermediate combination is repelled byspecies within that broad range, as long as R ≥ R and small enough µ , a resident that invests solelyin propagation, and not in interference, can be invaded by a species that mixespropagation and interference (rightmost columns of invasion plots). The resi-dent, with an initially high density and no interference capacity, is vulnerableto a clustered invader that can locally open sites via interference competition.
5. Simulation of individual-based model
We simulated the individual-based spatial model with L x = L y ≡ L = 256,and β = 0.001. We initiated simulations with the resident occupying each site.The resident’s density was allowed to decline to its single species equilibriumdensity, ρ ∗ , without any introduction. Then, at time t = 0, introduction ofboth invader and resident individuals began. We tracked the global densities ofeach species ρ i ( t ), and recorded a successful invasion when the invader reducedthe global density of the resident species to ρ ∗ / θ = 0 inde-pendently of the resident’s propagation rate, isolating the novel weapons effect.Invader interference promotes successful invasion more frequently in simulationthan under pair approximation. That is, we note a novel-weapons effect (absentin the MF). Invader clustering increases the advantage of local interference com-petition against a resident lacking interference. Even when the propagation-costof interference is steep ( R <
1) an invader may advance and exclude a residentwith a greater propagation rate. When interference is less costly (
R >
1) mostinvaders succeed, except those with a very low propagation rate. PA, then givesa reasonable, but understated, prediction of the impact of novel weapons in thedetailed spatial model.Figure 7 shows pairwise invasion results when the resident, and not theinvader, interferes competitively in the individual-based model. The residentobeys the propagation-interference constraint, and θ = 0 independently of theinvader’s propagation rate, isolating the resistance zone effect. Interferenceallows a resident to repel invasion by rare species with much greater rates oflocal propagation. For the resistance-zone effect, PA predicts the results of theindividual-based model very accurately across all parameter combinations wesimulated.The simulation model’s pairwise invasion results for the case where bothspecies exercise interference competition appear in Figure 8. When the cost ofinterference is high ( R = 0 .
5) the results are very close to those for preemptivecompetition only. That is, the species with the greater rate of propagation winsin most cases. Compared to PA, invader clustering in the simulation model pro-duces more cases of successful invasion at low background mortality. At lessercosts of interference ( R = 1 ,
2) residents mixing intermediate levels of propaga-tion and interference repel most invading species, particularly when backgroundmortality is not too large. Residents that invest either too much in interference(low α ) or too little are susceptible to invasion by species with a more balancedmix of propagation and interference.In a separate exercise, we released both species from the propagation-interferenceconstraint and searched the parameter space for competitive coexistence. We set µ = 0 . β = 0 . L = 64 to save computation time. The simulationsran for 100,000 MCSS, and the threshold for coexistence was ρ i > . i = 1 ,
6. Discussion
In the novel weapons scenario an invader gains competitive superiority overa resident through interference. Applications focus on allelopathic interference(Callaway and Ascheghoug, 2000). If species mix homogeneously, we find that arare invader gains no advantage through interference competition (Case and Gilpin,1974). But if competitive interactions occur at the local scale, an invader can12mploy interference to advance against a resident species with a greater rate oflocal propagation (figs. 3 and 6). Invaders are rare globally but locally clusteredin our spatial models. Invaders can use interference to reduce resident densityat the perimeter of these clusters, where local invader density is sufficiently highto compete effectively for open sites (Korniss and Caraco, 2005).In the resistance zone scenario, the resident holds competitive superiorityover an invader through interference. A resident’s competitive interference hasa strong effect under homogeneous mixing; few invaders advance successfullyunless interference is costly and background mortality is high (fig. 2). Underpair approximation, interference by the resident restricts invasion in two cases:when the resident, but not the invader, interferes, and when both species exerciseinterference competition. The resistance zone effect also appears in simulationof the individual-based model. But when both species can interfere, successfulinvasion is far more common in the simulation. The difference between the fullspatial model and its pair approximation lies in the impact of invader clusterslarger than neighborhood size (hence, longer correlation distances) generatedby the individual-base model. Clustering increases an invader’s benefit frominterference against an interfering resident.When both species exhibit interference, the pairwise invasion plots indicateregions of bistability (if only one species interferes, one cannot meaningfullyreflect invasion plots about the diagonal). We found large regions of bistabilityunder homogeneous mixing. Bistability declined as spatial structure increased(Chao and Levin, 1981), and nearly disappeared in the simulation model (fig.8). Given a sufficiently long time, we would anticipate that the spatial processon a large, but finite, lattice would result in competitive exclusion. The timerequired, however, may exceed population-dynamic scales.In a three-species spatial model, where a non-interfering species interactedwith both a weak interferer and a strong interferer, Durrett and Levin (1997)found cyclic coexistence. Our models for two-species competition, by construc-tion, do not admit coexistence. In fact, when we assumed that both speciesexercise interference competition, and when increased interference reduces prop-agation, certain parameter sets suggested that a single combination of propaga-tion and interference might repel all other feasible combinations. Suppose westart the resident’s propagation rate, α , at minimal (maximal) levels. Thenlarger (smaller) propagation rates successively invade and exclude the residentuntil pairwise competition leaves a resident species that can exclude all others. Acknowledgments
The National Science Foundation supported this research through GrantsDEB-0918392 and DEB-0918413. We thank J. Newman, G. Robinson, I.-N.Wang and A.C. Gorski for comments. D. Yoakam kept us energized.13 ppendix A. Mean-field stability
The Jacobian for the spatially homogeneous MF dynamics is: J = (cid:20) α − µ − α ρ ∗ − ρ ∗ ( α + θ ) − ρ ∗ ( α + θ ) − ρ ∗ ( α + θ ) α − µ − α ρ ∗ − ρ ∗ ( α + θ ) (cid:21) (A.1)Evaluated at E
1, mutual extinction, the Jacobian becomes: J ( E
1) = (cid:20) α − µ α − µ (cid:21) (A.2)Since α i > µ for i = 1 ,
2, by assumption, mutual extinction is unstable.At E J ( E
2) = (cid:20) α − µ − α (1 − µα ) − (1 − µα )( α + θ )0 α − µ − (1 − µα )( α + θ ) (cid:21) (A.3)The two eigenvalues are: λ ( E
2) = µ ( α α −
1) + θ ( µα −
1) (A.4) λ ( E
2) = µ − α < λ ( E < λ ( E
2) must be positive (implying thatextinction of species 2 is unstable). Note that if α > α , then ( α α − >
0, acondition promoting invasion by species 2 when species 1 rests its single-speciesequilibrium E
2. When θ >
0, species 2 invades iff : α − α > θ ( α µ − > E
3, where species 2 competitively excludes species 1. Evalu-ating the Jacobian, we have: J ( E
3) = (cid:20) α − µ − (1 − µα )( α + θ ) 0 − (1 − µα )( α + θ ) α − µ − α (1 − µα ) (cid:21) (A.7)The symmetry between J ( E
2) and J ( E
3) extends to the eigenvalues. FromEq A.7, we obtain: λ ( E
3) = µ ( α α −
1) + θ ( µα −
1) (A.8) λ ( E
3) = µ − α < iff α − α > θ ( α µ − > λ ( E <
0. Hence, ifspecies 2 can invade species 1, E λ ( E <
0. When species 2 is common, it repels invasion by species 1 if λ ( E <
0. Both conditions hold, and the dynamics is bistable, iff : θ (1 − α µ ) < α − α and θ ( α µ − > α − α (A.11)From our analysis of the single-species equilibria, the two species can coexist iff λ ( E > λ ( E >
0. The first inequality implies that when species 1rests at its single-species equilibrium, species 2 can advance from rarity. Thesecond inequality reverses roles of common and rare. In the text we show thatthe mean-field dynamics does not admit competitive coexistence.
Appendix B. Pair-correlation dynamics
We develop our pair approximation (PA) by modifying methods described byIwasa et al. (1998). We write a dynamics for five state variables: ρ , ρ , q | , q | , q | .The first two are global densities, and the other three are local densities. In-voking constraints listed in the text, Expression (14), we express the remainingPA variables in terms of the five state variables. In particular: ρ = 1 − ρ − ρ q | = ρ − ρ − ρ (cid:0) − q | − q | ρ ρ (cid:1) q | = (cid:0) ρ − ρ − ρ (cid:1) (1 − q | − q | ) q | = 1 − q | − q | q | = 1 − q | − q | ρ ρ q | = q | ρ ρ (B.1)We omit q | , also easily calculated, since it does not appear in the analysis.To begin, we rewrite the dynamics of the resident’s global density, Eq. (14),in terms of our five state variables. Doing the same for the invader’s globaldensity yields:˙ ρ = β (1 − ρ − ρ ) + ρ (cid:2) α (1 − q | − q | ρ ρ ) − µ − θ q | ρ ρ (cid:3) (B.2)˙ ρ = β (1 − ρ − ρ ) + ρ (cid:2) α (1 − q | − q | ) − µ − θ q | (cid:3) (B.3)To write the dynamics of the three conditional densities, we first require thedynamics of a corresponding doublet. By definition, q j | i = ρ ij ρ i , where ρ ij is theunordered doublet density. The doublet ρ ij has dynamics:˙ q j | i = 1 ρ i ˙ ρ ij − q j | i ρ i ˙ ρ i (B.4)15pplying the recipe to q | , we have:˙ ρ = 2 ρ β + 2 ρ α h δ + δ − δ q | i − ρ h µ + δ − δ θ q | i (B.5)The first two terms represent generation of new invader pairs through intro-duction and birth, and the third term represents the loss of invader pairs dueto background mortality and interference competition. This equation intro-duces triplets q | and q | into the dynamics. Ordinary PA takes neighbors ofneighboring sites as weakly correlated, and so we assume that q | = q | and q | = q | . The resulting closure of the equations allows the analysis withoutincluding the 27 types of triplets, or any higher order spatial correlations.Using the closure assumption and converting terms with equation (B.1),equation (B.5) becomes˙ ρ = 2 ρ (1 − q | − q | ) h β + α (cid:0) δ + ρ ( δ − δ (1 − ρ − ρ ) (1 − q | − q | ) (cid:1)i − ρ q | h µ + δ − δ θ q | i (B.6)Substituting Eqq. (14) and (B.6) into Eq. (B.4) yields:˙ q | = 2(1 − q | − q | ) h β + α δ (cid:0) ρ ( δ − − ρ − ρ (1 − q | − q | ) (cid:1)i − q | h βρ (1 − ρ − ρ ) + α (1 − q | − q | ) − µ − θ q | i − q | h µ + δ − δ θ q | i (B.7)Similarly,˙ q | = 2(1 − q | − q | ρ ρ ) h β + α δ (cid:0) ρ ( δ − − ρ − ρ (1 − q | − q | ρ ρ ) (cid:1)i − q | h βρ (1 − ρ − ρ ) + α (1 − q | − q | ρ ρ ) − µ − θ q | ρ ρ i − q | h µ + δ − δ θ q | ρ ρ i (B.8)˙ ρ is slightly more complicated. There are two ways to make a (1, 2) pair, andeach member of a (1, 2) pair interferes with the other. Proceeding:˙ ρ = ρ (1 − q | − q | ) h β + α ρ ( δ − δ (1 − ρ − ρ ) (1 − q | − q | ρ ρ ) i + ρ (1 − q | − q | ρ ρ ) h β + α ρ ( δ − δ (1 − ρ − ρ ) (1 − q | − q | ) i − ρ q | h µ + θ δ (cid:0) δ − q | (cid:1) + θ δ (cid:0) δ − q | ρ ρ (cid:1)i (B.9)16hen, following substitution:˙ q | = (1 − q | − q | ) h β + α ρ ( δ − δ (1 − ρ − ρ ) (1 − q | − q | ρ ρ ) i + ρ (1 − q | − q | ρ ρ ) h βρ + α ( δ − δ (1 − ρ − ρ ) (1 − q | − q | ) i − q | h µ + θ δ (cid:0) δ − q | (cid:1) + θ δ (cid:0) δ − q | ρ ρ (cid:1)i − q | h β (1 − ρ − ρ ) ρ + α (1 − q | − q | ) − q | θ i (B.10)Equations (B.2), (B.3), (B.7), (B.8), and (B.10) constitute the pair-approximationdynamics.To analyze invasion, we introduce species 2 at near-zero density. Invasionsucceeds or fails before the next introduction event occurs (since β ≪ µ i < α i ).Taking β = 0, the PA dynamics becomes:˙ ρ = ρ (cid:2) α (1 − q | − q | ρ ρ ) − µ − θ q | ρ ρ (cid:3) (B.11)˙ ρ = ρ (cid:2) α (1 − q | − q | ) − µ − θ q | (cid:3) (B.12)˙ q | = 2 α (1 − q | − q | ) h δ + ρ ( δ − δ (1 − ρ − ρ ) (1 − q | − q | ) i (B.13) − q | h α (1 − q | − q | ) + µ + θ q | (cid:0) δ − δ − (cid:1)i ˙ q | = 2 α (1 − q | − q | ρ ρ ) h δ + ρ ( δ − δ (1 − ρ − ρ ) (1 − q | − q | ρ ρ ) i (B.14) − q | h α (1 − q | − q | ρ ρ ) + µ + θ q | ρ ρ (cid:0) δ − δ − (cid:1)i ˙ q | = (1 − q | − q | )(1 − q | − q | ρ ρ ) (cid:0) ρ ( δ − δ (1 − ρ − ρ ) (cid:1) ( α + α ) (B.15) − q | h µ + α (1 − q | − q | ) i − q | h θ (cid:0) δ + δ − δ q | − q | (cid:1) + θ (cid:0) δ + δ − δ q | ρ ρ (cid:1)i The next step of the invasion analysis addresses the frequency of open sites whenthe invader is rare. Since the invader is rare, species 2 has no effect on either theresident’s equilibrium global density ( ρ ∗ ) or the equilibrium frequency of pairedresidents ( q ∗ | ). Setting ρ = 0 in Eqq. (B.11) and (B.14) yields: ρ ∗ = δ − − µα δδ − − µα (B.16)and q ∗ | = 1 − µα . (B.17)17ow we use these results to find the other conditional probabilities. Let x = q | , y = q | , and w = q | = 1 − x − y . Then:˙ x = 2 α wδ + xyθ h − δ − δ i − α wx − µx ˙ y = (cid:0) − δ − µα (cid:1)(cid:0) α + α (cid:1) w + y h α w − µ − θ δ − θ δ i (B.18)+ y θ h − δ − δ i Solving for the equilibria, we have: x ∗ = 2 α wα δw + µδ + y ∗ θ ( δ −
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25 in eachplot. Top: Species 2 excludes species 1 independently of initial condition; Expression (9)holds. α = 0 . θ = 0 .
2, and α = 0 .
75. Middle: Condition for bistability, Expression (10),holds. α = 0 . α = 0 . θ = θ = 0 .
1. Bottom: Bistable dynamics; α = 0 . α = 0 . θ = 0 . , θ = 0 .