1. 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

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 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

4. 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

5. 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.

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

8. Spatially-Explicit Models
Alternate model implementations:
deterministic
Mean Field
(4th order Runge-Kutta)
Local
Neighborhoods
(overlapping 33 cells)
stochastic
Mean Field
(global neighborhood)
H, D computed over the neighborhood for each cell
Spatially-Explicit Models
(Stochastic Cellular Automata)
100100 cells each
Probability 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+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.

10. 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)

11. Summary of Invasiveness predictions by frequency dependence 12 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

12. Example: Single propagule of exotic in +- quadrant (invader negative)
Tight clusters of invaders expand
33 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.

13. Example: Single propagule of exotic in -+ quadrant
(e.g. after enemy release; residents negative, exotic positive)
Loose clusters of
invaders expand
1111 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.

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 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

16. all 4 invasiveness outcomes are possible.
Given almost any of the four possible combinations of signs of net frequency dependence (the 12 quadrants), it possible to end up in almost any of the 4 possible invasiveness classes (the 12 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 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

18. 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

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 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

21. 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.

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: