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**Predicting Naturalization vs**

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

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**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.

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**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

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**Frequency independent population growth rate**

Standard Lotka-Volterra competition models ignore frequency dependent feedback effects on population growth rates Frequency independent population growth rate Classic theoretical ecology: Mean field assumptions (space ignored) Equilibrium conditions emphasized

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**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.

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**We extend standard Lotka-Volterra competition equations**

to include frequency dependent growth rates.

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**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 density Frequency independent component where represents frequency-dependent habitat quality (nonlinear functions could be substituted here…) Habitat quality Frequency dependence

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**Spatially-Explicit Models**

Alternate model implementations: deterministic Mean Field (4th order Runge-Kutta) Local Neighborhoods (overlapping 33 cells) stochastic Mean Field (global neighborhood) H, D computed over the neighborhood for each cell Spatially-Explicit Models (Stochastic Cellular Automata) 100100 cells each Probability of occupancy of a cell at next time step

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**Stochastic Cellular Automata Model (shown for 2 species)**

Stochastic probability that cell at is occupied by species i at time t+1 Species specific Interaction neighborhoods Species specific Dispersal neighborhoods Neighborhoods can vary in size, shape, distribution 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.

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**If maximum habitat quality is identical between two species…**

Habitat quality Hi Frequency Fj Figure 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 of relative net frequency dependence of species and neighborhood size (smallest absolute frequency dependence wins, but rate of invasion also controlled by neighborhood size)

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**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 ++ Resident positive, Exotic positive: Least invasive Smallest scale highest invasion success Smallest scale slowest invasion to extinction quadrant map Reddish shaded regions show where|1|>|2|, so Species 2 has a chance to invade. +1 low H M M L 0.5 medium 11 invasiveness high H -0.5 coexist VH very high -1 Smaller neighborhoods reduce region of co-existence -+ Resident negative, Exotic positive: Most invasive region Intermediate scale highest invasion success Smallest scale fastest invasion to extinction -- Resident negative, Exotic negative: Exotic becomes established and coexists. -1 -0.5 22 +0.5 +1

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**Example: Single propagule of exotic in +- quadrant (invader negative)**

Tight clusters of invaders expand 33 cell -1 0.5 +1 -0.5 * -1 -0.5 +0.5 +1 Average takeover time for invader is longest at shortest scale Out of 100 trials Invader wins Resident wins Figure 8. Invasiveness by species 2 (black) in the loser positive region of the +- quadrant. Smaller neighborhoods promote invasiveness.

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**Example: Single propagule of exotic in -+ quadrant **

(e.g. after enemy release; residents negative, exotic positive) Loose clusters of invaders expand 1111 cell Very invasive: even a slight frequency dependent advantage promotes invasion -1 0.5 +1 -0.5 * Note long takeover times! Non-equilibrium dynamics important. -1 -0.5 +0.5 +1 Invader wins Resident wins Out of 100 trials Average takeover time for invader is longer at larger scale Figure 9. Invasiveness by species 2 (black) for in the loser negative region of the -+ quadrant. Intermediate sized-neighborhoods promote invasiveness.

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**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 rate 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) growth rate differences at frequency extremes

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**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:

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**Invasiveness outcomes change with the relative average fitness of the resident and exotic.**

is the habitat suitability averaged over all frequencies Invasiveness is very sensitive to perceived propagule pressure Exotic is less fit but can still establish Although in naturalization quadrant, exotic is still a threat

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**9 propagules introduced Meanfield (M): Can’t Invade Scattered (S):**

Clumped (C): Likely to invade Scattered (S): Stochastic invasion Meanfield (M): Can’t Invade Conditional Invasion quadrant 9 propagules introduced in cells with at least one propagule in its neighborhood Histogram of perceived propagule pressure

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**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.)

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**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 grass Phalaris arundinacea native 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 predict invasion quadrant Should predict naturalization quadrant

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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 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.

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**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:

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