Predicting Predicting Naturalization vs. Naturalization vs. Invasion in Plant Invasion in Plant

Communities using Communities using Stochastic CA ModelsStochastic CA Models

Margaret J. Eppstein1 & Jane Molofsky2

1Depts. of Computer Science and Biology2Dept. 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 (-)

Soil chemistry (+ or -)

Symbionts (+)Pathogens (-)

Emphasis has been on changes in feedbacks between native and invasive ranges of a species

1..

1 1 jii i ij j i i

j s i

NdNr N d d N

dt K

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

•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 develop a model incorporating the influences of:

We extend standard Lotka-Volterra competition equations

1..

1 1 jii i ij j i i

j s i

NdNr N d d N

dt K

1..

1.. 1..

1..

1 1

ji ij ij i

j s ii

j ki ij ij i j jk jk ij j

j s j i k si j

ji ij j i i

j s i

NK

KdN

dt N NN N

K K

NN d d N

K

to include frequency dependent growth rates.

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:

1..

t ti i i i

t t tj j i

j s

dN H D N

dt H D F

1..i j

t ti ij

j s

H F

where represents frequency-dependent habitat quality(nonlinear functions could be substituted here…)

Habitat quality

Frequency independent component

Frequency dependence

Assume dispersal is proportional to species

density

Alternate model implementations:

deterministic Mean Field

(4th order Runge-Kutta)

stochastic Mean Field

(global neighborhood)

Spatially-Explicit Models(Stochastic Cellular Automata)

100100 cells each

1

1..

t tt i i

i t tj j

j s

H DF

H D

Probability of occupancy of a cell at next time step

H, D computed over the neighborhood for

each cell

Local Neighborhoods

(overlapping 33 cells)

x. ., uniform square neighborhood

of size 3 3

e g

Species specific Interaction neighborhoods Hi

Species specific Dispersal neighborhoods Di

Stochastic Cellular Automata Model (shown for 2 species)

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.

Neighborhoods can vary in size, shape, distribution

Stochastic probability that cell at is occupied by species i at time t+1x

1 1 2 2

1

1 1 2 2

H Di i

H D H D

t ti i

ti t t t t

H DP

H D H D

x xx

x x x x

If maximum habitat quality is identical between two species…

…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)

Hab

itat

qual

ity H

i

Frequency Fj

++ Resident positive, Exotic positive:Least invasiveSmallest scale highest invasion success Smallest scale slowest invasion to extinction

+- Resident positive, Exotic negative:Medium InvasivenessSmallest scale highest invasion success Smallest scale slowest invasion to extinction

-+ Resident negative, Exotic positive:Most invasive regionIntermediate scale highest invasion success Smallest scale fastest invasion to extinction

-- Resident negative, Exotic negative:Exotic becomes established and coexists.

Summary of Invasiveness predictions by frequency dependence 12 quadrants

-1

0.5

+1

-1 -0.5 0 +0.5 +1

0

-0.5

22

quadrant map

coexist

11

low

very high

medium

high

inva

sive

ness

L

M

M

H

H

VH

Reddish shaded regions show where|1|>|2|, so Species 2 has a chance to invade.

Smaller neighborhoods reduce region of co-existence

22-1 -0.5 0 +0.5 +1

-1

0.5

+1

0

-0.5

11

*

11 220.8, 0.1

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

Out of 100 trials

Invader wins

Resident wins

Tight clusters of invaders expand

33 cell

Average takeover time for invader is longest at shortest scale

22-1 -0.5 0 +0.5 +1-1

0.5

+1

0

-0.5

11

*

11 220.5, 0.4

Example: Single propagule of exotic in -+ quadrant (e.g. after enemy release; residents negative, exotic positive)

Out of 100 trials Invader wins

Resident wins

Loose clusters of invaders expand

1111 cell

Average takeover time for invader is longer at larger scale

Very invasive: even a slight frequency dependent advantage promotes invasion

Note long takeover times! Non-equilibrium dynamics

important.

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.

1

i

i

i

tt

t

Fr

F

Pop growth rate

1 1

1 1

2

2 0.01 0

1

2

1 0.99 0.99

1 .01

F F

F F

r

r

r

r

growth rate differences at frequency extremes

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)

Consider species’ population growth rates r:

1

1

1

11

2

1 2 : 1 2 :

1 2 : 1 2 :

b)

c) d)

a)

Extirpation of S2 Conditional Invasion

InvasionNaturalization

2

2

2

2

1

1

1

11

2

1 2 : 1 2 :

1 2 : 1 2 :

b)

c) d)

a)

Extirpation of S2 Conditional Invasion

InvasionNaturalization

2

2

2

2

1

1

1

11

2

1 2 : 1 2 :

1 2 : 1 2 :

b)

c) d)

a)

Extirpation of S2 Conditional Invasion

InvasionNaturalization

2

2

2

2

1

1

1

11

2

1 2 : 1 2 :

1 2 : 1 2 :

b)

c) d)

a)

Extirpation of S2 Conditional Invasion

InvasionNaturalization

2

2

2

2

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)!

( ) ( )i j jj i ijsign sign

Specifically, the invasiveness outcomes are determined by both frequency dependent and frequency independent components of all interacting species:

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.

1 11 12 2 22 21, Where net feedbacks are:

12:--

12:-+

12:++

12:+-

12:--12:-+12:++12:+-

12:--

12:-+

12:++

12:+-

12:--12:-+12:++12:+-Ne

t fee

dbac

ksInvasiveness outcome quadrant

1 2H H 1 2H H1 2H H1 2H H1 2H H

U n s t a b l e e q u i l i b r i u m p t( c o n d i t i o n a l i n v a s i o n )

S t a b l e e q u i l i b r i u m p t( n a t u r a l i z a t i o n )

S 1 w i n s( e x t i r p a t i o n o f S 2 )

S 2 w i n s( i n v a s i o n )

e )d )c )b )a )

j )i )h )g )f )

1 2H H 1 2H H1 2H H1 2H H1 2H H

U n s t a b l e e q u i l i b r i u m p t( c o n d i t i o n a l i n v a s i o n )

S t a b l e e q u i l i b r i u m p t( n a t u r a l i z a t i o n )

S 1 w i n s( e x t i r p a t i o n o f S 2 )

S 2 w i n s( i n v a s i o n )

e )d )c )b )a )

j )i )h )g )f )

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

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

is the habitat suitability averaged over all frequencies

iH

a)

b)

c)

d)

a)

b)

c)

d) Clumped (C): Likely to invade

Scattered (S): Stochastic invasion

Meanfield (M): Can’t Invade

Conditional Invasion quadrant

a)

b)

c)

d)

a)

b)

c)

d)

9 propagulesintroduced

His

togr

am

of p

erce

ived

pro

pag

ule

pre

ssu

re

in c

ells

with

at

leas

t on

e pr

opag

ule

in it

s ne

ighb

orho

od

(Black arrows indicate direction of increasing perceived propagule pressure.)

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)

Likelihood of early extirpation of exotic either increases or decreases with perceived propagule pressure, depending on the quadrant.

Experimental System:Reed Canary grass Phalaris arundinacea native to Europe, invasive in N. American wetlands.

Should predictinvasion quadrant

Should predict naturalization quadrant

Measure growth rates in existing patches of different densities of Phalaris, in both native and introduced ranges.

This may be a practical way to assess invasive potential of newly introduced exotic plants, and/or to

estimate range limits of invasive species.

•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.

Conclusions

•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.

Conclusions continued…

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.

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