Presentation on theme: "Predicting Naturalization vs"— Presentation transcript:
1 Predicting Naturalization vs Predicting Naturalization vs. Invasion in Plant Communities using Stochastic CA ModelsMargaret J. Eppstein1 & Jane Molofsky21Depts. of Computer Science and Biology2Dept. of Botany
2 What makes some plant species invasive in some communities? Lots of theories, e.g.:Enemy Release Hypothesis(Keane & Crawley, 2002)Evolution of Increased Competitive Ability(Blossey & Notzold, 1995)Biotic Resistance Hypothesis(Elton, 1958)Propagule pressure (number and frequency)(Von Holle & Simberloff, 2005; Lockwood et al, 2005)Despite the many important advances in understanding potential causes of invasiveness, it remains unclear how the various ecological influences interact, or how to predict invasiveness.
3 Lots of recent evidence that local intra- and inter-specific positive and negative feedbacks in plant communitiescan drive population dynamics and affect biodiversity(e.g, Wolfe & Klironomos, 2005; Reinhart & Callaway, 2006)Pollinators (+)Predators (-)Symbionts (+)Pathogens (-)Soil chemistry (+ or -)Emphasis has been on changes in feedbacks between native and invasive ranges of a species
4 Frequency independent population growth rate Standard Lotka-Volterra competition models ignore frequency dependent feedback effects on population growth ratesFrequency independent population growth rateClassic theoretical ecology:Mean field assumptions (space ignored)Equilibrium conditions emphasized
5 We develop a model incorporating the influences of: propagule pressure,frequency independent components of growth,frequency dependent feedback relationships,resource competition, andspatial scale of interactions.This model can be used to explore complex influences of spatially localized frequency dependence and competitive interactions on population dynamics.
6 We extend standard Lotka-Volterra competition equations to include frequency dependent growth rates.
7 represents frequency-dependent habitat quality In an example community of annual plants (di =1) where competition is for space (Ki=Kj=Nk,k) and all species require the same amount of space per individual (ij=1), this reduces to:Assume dispersal is proportional to species densityFrequency independent componentwhererepresents frequency-dependent habitat quality(nonlinear functions could be substituted here…)Habitat qualityFrequency dependence
8 Spatially-Explicit Models Alternate model implementations:deterministicMean Field(4th order Runge-Kutta)LocalNeighborhoods(overlapping 33 cells)stochasticMean Field(global neighborhood)H, D computed over the neighborhood for each cellSpatially-Explicit Models(Stochastic Cellular Automata)100100 cells eachProbability of occupancy of a cell at next time step
9 Stochastic Cellular Automata Model (shown for 2 species) Stochastic probability that cell at is occupied by species i at time t+1Species specific Interaction neighborhoodsSpecies specific Dispersal neighborhoodsNeighborhoods can vary insize, shape, distributionFor the results shown here, we assume uniform square neighborhoods of various sizes, that are species-symmetric and same for dispersal and frequency dependent interactions.
10 If maximum habitat quality is identical between two species… Habitat quality HiFrequency FjFigure 1. The linear frequency dependent interaction factor (Eqn. 1.2) as a function of frequency of species i, shown for five representative values of frequency dependence ii.…then invasiveness is a function ofrelative net frequency dependence of speciesand neighborhood size(smallest absolute frequency dependence wins, but rate of invasion also controlled by neighborhood size)
11 Summary of Invasiveness predictions by frequency dependence 12 quadrants +- Resident positive, Exotic negative:Medium InvasivenessSmallest scale highest invasion successSmallest scale slowest invasion to extinction++ Resident positive, Exotic positive:Least invasiveSmallest scale highest invasion successSmallest scale slowest invasion to extinctionquadrant mapReddish shaded regions show where|1|>|2|, so Species 2 has a chance to invade.+1lowHMML0.5medium11invasivenesshighH-0.5coexistVHvery high-1Smaller neighborhoods reduce region of co-existence-+ Resident negative, Exotic positive:Most invasive regionIntermediate scale highest invasion successSmallest scale fastest invasion to extinction-- Resident negative, Exotic negative:Exotic becomes established and coexists.-1-0.522+0.5+1
12 Example: Single propagule of exotic in +- quadrant (invader negative) Tight clusters of invaders expand33 cell -10.5+1-0.5*-1-0.5+0.5+1Average takeover time for invader is longest at shortest scaleOut of 100 trialsInvader winsResident winsFigure 8. Invasiveness by species 2 (black) in the loser positive region of the +- quadrant. Smaller neighborhoods promote invasiveness.
13 Example: Single propagule of exotic in -+ quadrant (e.g. after enemy release; residents negative, exotic positive)Loose clusters ofinvaders expand1111 cell Very invasive: even a slight frequency dependent advantage promotes invasion-10.5+1-0.5*Note long takeover times! Non-equilibrium dynamics important.-1-0.5+0.5+1Invader winsResident winsOut of 100 trialsAverage takeover time for invader is longer at larger scaleFigure 9. Invasiveness by species 2 (black) for in the loser negative region of the -+ quadrant. Intermediate sized-neighborhoods promote invasiveness.
14 S1 (resident community) and S2 (introduced exotic) HOWEVER, if we also consider differences in frequency independent components , the picture changes.Again, consider 2 idealized species:S1 (resident community) and S2 (introduced exotic)As with Lotka-Volterra competition equations,4 outcomes are possible.Consider species’ population growth rates r:Pop growth rateOutcomes are governed by the 4 possible combinations of signs of the pop growth rate differences , at the two frequency extremes (not the 4 possible quadrants)growth rate differences at frequency extremes
16 all 4 invasiveness outcomes are possible. Given almost any of the four possible combinations of signs of net frequency dependence (the 12 quadrants), it possible to end up in almost any of the 4 possible invasiveness classes (the 12 quadrants)!Where net feedbacks are:Even if the resident community has net negative feedback (1<0)While the introduced exotic has net positive feedback (2>0)(e.g., following enemy release),all 4 invasiveness outcomes are possible.Specifically, the invasiveness outcomes are determined by both frequency dependent and frequency independent components of all interacting species:
17 Invasiveness outcomes change with the relative average fitness of the resident and exotic. is the habitat suitability averaged over all frequenciesInvasiveness is very sensitive to perceived propagule pressureExotic is less fit but can still establishAlthough in naturalization quadrant, exotic is still a threat
18 9 propagules introduced Meanfield (M): Can’t Invade Scattered (S): Clumped (C):Likely to invadeScattered (S):Stochastic invasionMeanfield (M):Can’t InvadeConditional Invasion quadrant9 propagulesintroducedin cells with at least one propagule in its neighborhoodHistogram of perceived propagule pressure
19 Growth rate of exotic increases with its frequency Likelihood of early extirpation of exotic either increases or decreases with perceived propagule pressure, depending on the quadrant.Growth rate of exotic increases with its frequency(in conditional invasion quadrant)Growth rate of exotic decreases with its frequency(in naturalization and invasion quadrants)(Black arrows indicate direction of increasing perceived propagule pressure.)
20 Should predict naturalization quadrant Measure growth rates in existing patches of different densities of Phalaris, in both native and introduced ranges.Experimental System:Reed Canary grassPhalaris arundinaceanative to Europe,invasive in N. American wetlands.This may be a practical way to assess invasive potential of newly introduced exotic plants, and/or to estimate range limits of invasive species.Should predictinvasion quadrantShould predict naturalization quadrant
21 ConclusionsBoth frequency dependent and independent interactions have a big impact on invasiveness.Its not the change in interactions from native to introduced ranges that determines invasiveness, but the relative frequency dependent growth rates of exotic as compared to resident community.Spatial scale of interactions dramatically affects community structure and population dynamics.Understanding cluster formation and density and the relative inter and intra-specific dynamics in the interiors, exteriors, and boundaries of self-organizing clusters of con-specifics can provide insights into mechanism governing invasiveness.Importance of non-equilibrium dynamics in invasiveness; time scales of environmental change may exceed time to equilibrium.
22 Conclusions continued… Measuring relative growth rates in small patches with different frequencies of exotic species may help to predict invasiveness and/or range limits of invader.We have developed a stochastic cellular automata model that facilitates study of complex influences of spatially localized frequency dependent and competitive interactions.Eppstein, M.J. and Molofsky, J. "Invasiveness in plant communities with feedbacks". Ecology Letters, 10: , 2007.Eppstein, M.J., Bever, J.D., and Molofsky, J., "Spatio-temporal community dynamics induced by frequency dependent interactions", Ecological Modelling, 197: , 2006.For more details: