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Predicting Naturalization vs. Invasion in Plant Communities using Stochastic CA Models Margaret J. Eppstein1 & Jane Molofsky2 1Depts. of Computer Science and Biology 2Dept. of Botany 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. Lots of recent evidence that local intra- and inter-specific positive and negative feedbacks in plant communities can 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 Standard Lotka-Volterra competition models ignore frequency dependent feedback effects on population growth rates Nj dNi ri Ni 1 ij 1 d j dt Ki j 1..s di Ni Frequency independent population growth rate Classic theoretical ecology: •Mean field assumptions (space ignored) •Equilibrium conditions emphasized We develop a model incorporating the influences of: •propagule pressure, •frequency independent components of growth, •frequency dependent feedback relationships, •resource competition, and •spatial scale of interactions. This model can be used to explore complex influences of spatially localized frequency dependence and competitive interactions on population dynamics. We extend standard Lotka-Volterra competition equations Nj dNi ri Ni 1 ij 1 d j dt Ki j 1..s di Ni to include frequency dependent growth rates. Nj i ij ij Ki Ki dN i j 1.. s dt Nj Nk ij N j i ij ij N i j jk jk Ki Kj j 1.. s j i k 1.. s Nj N i 1 ij 1 d j Ki j 1..s di N i 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 density Frequency independent component t H i where i Habitat quality dNi dt H it Dit N i t t t H D F j j i j 1.. s t F ij j 1.. s represents frequencydependent habitat quality (nonlinear functions could be Frequency substituted here…) dependence j Alternate model implementations: deterministic H, D computed over Field for the Mean neighborhood (4th order Runge-Kutta) each cell t t H D i i Fi t 1 t t H D j j j 1.. s stochastic Mean Field Local Neighborhoods (global neighborhood) (overlapping 33 cells) Spatially-Explicit Models (Stochastic Cellular Automata) 100100 cells each Probability of occupancy of a cell at next time step Stochastic Cellular Automata Model (shown for 2 species) Stochastic probability that cell at Pi t 1 x x is occupied by species i at time t+1 H it x H1t x 1H D1t x Dit x iH 1D iD H 2t x Species specific Interaction neighborhoods i 2H D2t x 2D H Species specific Dispersal neighborhoods D i x Neighborhoods can vary in size, shape, distribution e.g., uniform square neighborhood of size 3 3 For the results shown here, we assume uniform square neighborhoods of various sizes, that are species-symmetric and same for dispersal and frequency dependent interactions. Habitat quality Hi If maximum habitat quality is identical between two species… Frequency Fj …then invasiveness is a function of relative net frequency dependence of species and neighborhood size (smallest absolute frequency dependence wins, but rate of invasion also controlled by neighborhood size) Summary of Invasiveness predictions by frequency dependence 12 quadrants +- Resident positive, Exotic negative: Medium Invasiveness Smallest scale highest invasion success Smallest scale slowest invasion to extinction 0.5 M 0 L -0.5 H coexist VH -1 Smaller neighborhoods -1 reduce region of co-existence low -0.5 -- Resident negative, Exotic negative: Exotic becomes established and coexists. 0 22 +0.5 +1 invasiveness +1 quadrant map H M 11 Reddish shaded regions show where|1|>|2|, so Species 2 has a chance to invade. ++ Resident positive, Exotic positive: Least invasive Smallest scale highest invasion success Smallest scale slowest invasion to extinction medium high very high -+ Resident negative, Exotic positive: Most invasive region Intermediate scale highest invasion success Smallest scale fastest invasion to extinction Example: Single propagule of exotic in +- quadrant (invader negative) 0.8, 0.1 11 Tight clusters of invaders expand 22 +1 * 0.5 11 0 -0.5 33 cell -1 -1 -0.5 0 22 Out of 100 trials +0.5 +1 Average takeover time for invader is longest at shortest scale Invader wins Resident wins Example: Single propagule of exotic in -+ quadrant (e.g. after enemy release; residents negative, exotic positive) Loose clusters of invaders expand +1 0.5 11 0 -0.5 1111 cell -1 -1 * Very invasive: even a slight frequency dependent advantage promotes invasion 11 0.5, 22 0.4 Note long takeover times! Non-0.5 +1 equilibrium important. 0 +0.5 dynamics 22 Out of 100 trials Average takeover time for invader is longer at larger scale Invader wins Resident wins 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: t r Pop growth rate i Fi t 1 Fi t Outcomes 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) 1 r1 F 0.99 r2 F 0.99 1 1 2 r2 F 0.01 r1 F 0.01 1 1 growth rate differences at frequency extremes 11 a) 1122 :: b) 1122 :: 22 11 11 22 Conditional Invasion Extirpation of S2 c) 1122 :: d) 1122 :: 22 11 11 22 Naturalization 22 Invasion 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)! Invasiveness outcome quadrant Net feedbacks 12:+- 12:++ 12:-+ 12:-12:+- 12:++ 12:-+ 12:-- Where net feedbacks are: 1 11 12 , 2 22 21 Even if the resident community has net negative feedback (1<0) Specifically, the invasiveness outcomes are determined While the introduced exotic has net positive feedback (by 2>0) both frequency dependent andenemy frequency independent (e.g., following release), components all interacting species: are possible. all 4ofinvasiveness outcomes sign( i ) sign( j jj i ij ) Invasiveness outcomes change with the relative average fitness of the resident and exotic. Unstable equilibrium pt H i is the habitat (conditional invasion) suitability averaged H1 H 2 Invasiveness is very sensitive to perceived propagule pressure H1 H 2 H1 H 2 over all frequencies H1 H 2 H1 H 2 a) b) c) d) e) f) g) h) i) j) Exotic is less fit S1still wins but can establish (extirpation of S2) Although in naturalization Stableexotic equilibrium pt quadrant, is still a threat (naturalization) S2 wins (invasion) Histogram of perceived propagule pressure in cells with at least one propagule in its neighborhood a)a) Conditional Invasion quadrant 9 propagules introduced b) b) Meanfield (M): Can’t Invade c)c) Scattered (S): Stochastic invasion d) d) Clumped (C): Likely to invade 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.) Measure growth rates Experimental in existing patches of different System: densities of Phalaris, in both native and grass introduced ranges. Reed Canary Phalaris This may be a practical way arundinacea to assess invasive native to Europe, potential of newly introduced exotic plants, and/or to invasive in N. American wetlands. estimate range limits of invasive species. Should predict invasion quadrant Should predict naturalization quadrant Conclusions •Both 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 intraspecific 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. 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. For more details: Eppstein, M.J. and Molofsky, J. "Invasiveness in plant communities with feedbacks". Ecology Letters, 10:253-263, 2007. Eppstein, M.J., Bever, J.D., and Molofsky, J., "Spatio-temporal community dynamics induced by frequency dependent interactions", Ecological Modelling, 197:133-147, 2006.