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Deja Vu Communities and Spatial Dynamics of Competing Plant
Species
Kunj Patel and Jonathan Lansey New Jersey Institute of Technology
Newark, NJ
Advisors: Claus Holzapfel, Amitabha Bose
Mathematical Biology Seminar - NJIT - Spring 2006
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What are Invasive Species?A living thing growing in a foreign environment
They come by . . .• Mussel clinging to boat• Intentionally to control another pest• Plant used as packing material
Why are they a problem?They . . .• Grow out of control• Reduce biodiversity• $72 billion damage to US crops
US annual loss of $138 billion overall-National Council for Science and the Environment
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Invasive Species Theories• Invasive Species Traits:
– rapid reproduction– high dispersal ability– highly competitive or aggressive behavior
• Enemy release hypothesis• Island ecosystems are Naïve• No one theory can explain all ramifications of invasive
species
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Invasive Plant Species
• Allopatric Pair: Plants from different regions
– Native/ Invasive
– Invasive/ Invasive
• Sympatric Pair: Plants from the same region
– Native/ Native
– Invasive/ Invasive (a Deja Vu community)
Our Hypothesis: Root Interactions• Sympatric plant’s roots will not grow into another's.
– They evolved methods to communicate– Sharp borders result
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Invasive Plant Species
• Allopatric Pair: Plants from different regions
• Sympatric Pair: Plants from the same region
Our Hypothesis: Root Interactions• Sympatric plant’s roots will not grow into another's.
– They evolved methods to communicate– Sharp borders result
• Allopatric plant’s roots will grow into another’s– Distinct gene pools– May have evolved incompatible signal mechanisms– Highly overlapped borders result
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Root Interactions
Root systems of neighboring guayule plants (Parthenium argentatum)root territories” Schenk et all.(1999)
Root exudates (exuded chemicals) Rice, 1973 Possibly many more undiscovered mechanisms Holzapfel, C. &
Alpert, P. (2003), Schenk et al.(1999)
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Jonathan• Data Collection• Analysis
– Area Method– Threshold Method
Kunj• Theoretical Model
– Details later
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How do we measure overlap?To quantify “overlap” we first quantify the borders.
• We assume that above ground borders correlate with root borders below.
– 5 sites in northern NJ– 9 Sympatric Borders– 6 Allopatric Borders– 2 Transects per Border
Data Taken:• Height• % Cover• # Stems (approx)
Plant A Plant B
Quadrat50 cm
3 m
3.5 m
Subplot
50 cm
12 3 4 5
67
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What is Overlap?
Area Computation
Fallopia Microstegium
020406080
100
1 2 3 4 5 6 7
Transect
Cov
er (
scal
ed %
)
Sympatric
Border
Fallopia Artemisia
020406080
100
1 2 3 4 5 6 7
Transect
Cov
er (
scal
ed %
)
Allopatric
Border
Plant A Plant B
Quadrat
A
B
Area B> Area A
Quadrat
Quadrat
12 3 4 5 6 7
Subplot
50 cm
50 cm
3 m
3.5 m
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Results of Area Computation
Fallopia Microstegium
020406080
100
1 2 3 4 5 6 7
Transect
Cov
er (
scal
ed %
)
Sympatric
Border
Fallopia Artemisia
020406080
100
1 2 3 4 5 6 7
Transect
Cov
er (
scal
ed %
)
Allopatric
Border
0
5
10
15
20
25
Pairing
Ove
rlap
(%
of
tran
sect
)
Allopatric pairs have significantlylarger %Cover overlap compared to
sympatric pairs
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Threshold Method, ResultsFallopia Artemisia
020406080
100
1 2 3 4 5 6 7
Transect
Cov
er (
scal
ed %
)
Maximum Overlap
Difference at 50%
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Room for Improvement
• No quantification of error- ocular estimation.
• The “Overlap” and “Area” methods are not the only possibilities.
Others will be possible with . . .• More accurate data.
• Larger data set
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Mathematical Assumptions and Formulation
• The basic nature of plant ecology simplifies the mathematical approach. – Growth and propagation are the only things under consideration– All other underlying plant physiology is captured by a small
number of parameters.
• So, let u(t) represent the density of plants of species u in a particular area. u(t) could also be the density of roots.
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( )du
ru c udt
The logistic equation• The logistic equation is
the simplest model of plant growth.
• The unstable fixed point at the origin is appropriate for plants. (Analogous to a shoot from a clonal plant, or a seed from a parent plant).
r Is the growth rate [“births”/ plant/season]
c is the carrying capacity [maximum number of plants that can occupy a single quadrat]
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The simplest competition model
i i
i i
c bv u
a a
1b1 1 1 1 1
2 2 2 2 2
( )
( )
dur u c a u b v k uv
dtdv
r v c a u b v k uvdt
The nullclines are 4 lines in the upper right quadrant.
0, 0u v
i=1,2
1k
Is the competition term
Is the inhibition term
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Nondimensionalization
1
2
(1 )
(1 )
dUU U V UV
dTdV
gV U V UVdT
Where
1 2
1 1 1 2
1 1 2 2 11 1 1 2, 2 1 1 2
1 1 1 2 2
1, ,
, , ,
c c
r c a bk b b a a
g c r r r c kr c c c c
Notice that and can be combined.1 2
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Conclusions of the ODE system
The Bendixson-Dulac negative criterion, which says that if there exists a function f(u,v) such that the divergence is negative for all values of u and v, then there can not exist a limit cycle.
1( , )f u v
uv
2
1
2 21 1
( *, *) (0,0)
( *, *) (0,1)
( *, *) (1,0)
11
( *, *) ,
( ) 1 ( ) 1
U V
U V
U V
gU V
g g
Fixed points:
1
2
(1 )
(1 )
dUU U V UV
dTdV
gV U V UVdT
System of Equations:
Is a such function.
1 1 If then the cooperative state for U* is negative. In this case (0,1) is stable.
2 1 1 2 0b c b c If then U* increases relative to V* as is increased.
1r
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The simplest Competition Model
Global behavior for interspecific competition between species u and v. The relative carrying capacities and strengths of competition dominate the behavior. Figure adapted from Neuhauser (2001)
1 1 12
2 2 21
( )
( )
dur u K u v
dtdv
r v K u vdt
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Distinguishing Properties of Sympatric and Allopatric Pairs
1 1 1 2
1 1 1 2
2 2 2 1
2 2 2 1
b k c a
a r a cb k c a
a r a c
With strong sympatric species interactions, both of these inequalities are satisfied. With allopatric species interactions, at least one of the inequalities are broken.
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Including Spatial Components
• The growth/competition terms adequately capture the dynamics at any particular location. Thus a single term is needed.
• A diffusive term provides a good approximation, since species propagate down their concentration gradients.
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Resulting PDE Model2
1 12
2
2 22
(1 )
(1 )
U UD U U V UV
T x
V VD gV U V UV
T x
0(0, )
( , ) 0
u t u
ul t
x
0
(0, ) 0
( , )
vt
xv l t v
( ,0) ( )
( ,0) ( )
u x f x
v x g x
With boundary conditions
And initial conditions
It has a clear analogy with reaction diffusion equations chemistry.
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Steady State Graphs for the cooperative state
In blue are the initial state (light blue) and final state (dark) of u and in red are the initial and final states of v.
Threshold
Overlap region
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Threshold Concept• A convenient way to relate above and below
ground interactions.
A dramatic example of the right side case observed at Morristown Park.
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A Dramatic Observation
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Numerical Experiments
0 0.2 0.4 0.6 0.8 10.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1Allopatric case k1=0 k2=0
0 0.2 0.4 0.6 0.8 10.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1Sympatric case k1=1 k2=1
In both cases, all parameters are equal, including threshold levels. In right, the inhibition is increased from 0 to 1 in both plants. The resulting overlap is shown in purple.
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Numerical Experiments
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1Worst case scenario k1=1 k2=0
Fallopia J aponica vs. Mustard Plant
0
20
40
60
80
100
120
140
160
1 2 3 4 5 6 7 8 9 10 11 12
Quadrat
Mustard Plant
Fallopia J aponica
A simulation of an allopatric border (left) where the red plant inhibits the left, and the blue plant doesn’t retaliate. Notice that there is nearly complete overlap. (right) A field observation which closely matches the simulation of the worst case scenario allopatric border.
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Parameter Effects: Inhibition
Inhibition of the red species to the blue species is increased from 0 to 1. The arrows show how the curve changes for any increase in k2.
0 0.2 0.4 0.6 0.8 1-0.2
0
0.2
0.4
0.6
0.8
1
1.2Worst case scenario k1=0 k2=5
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Parameter Effects: Growth
Growth of the red species to the blue species is increased from 0.5 to 2. The arrows show how the curve changes for any increase in r1.
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Parameter Effects: Diffusion
Diffusion of the red species to the blue species is increased from 0.01 to 0.04. The arrows show how the curve changes for any increase in the Diffusion constant.
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Cumulative Parameter Effects
0
10
20
30
40
50
60
70
Quadrat
Artemesia
J apanese Stiltgrass
(left) All parameters are equal. (top right) All parameters are doubled, so each parameter likely resides in a physiologically feasible range. (bottom right) Field observation.
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Conclusions of the PDE Model
• Inhibition constant > Diffusion constant > Growth.
• Sympatric species can likely reduce overlap of borders through increasing the existing inhibition and by making use of the need for a plant threshold.
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Future Work
• Implement the same model, though replacing the diffusive term with a term that is nonconservative, like an integral.
• Produce a model which accounts for differing types of competition on the basis of time scales (slow and fast processes).
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Future Work
Direct inhibition (large k1 and k2); fast process—small
Competition with no inhibition (k1=k2=0 or very small); slow process—large
Figure from Holzapfel et al (2001).
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Future Work, continued2
1 1 1 12
2
2 2 2 22
u uD r uc uv
T x
v vD r vc uv
T x
11 1 1 1 1 1 1
22 2 2 2 2 2 2
cc c a u b v M c
tc
c c a u b v M ct
The direct inhibition acts on a faster time scale than does the limitation offered by the limited carrying capacity.
1 is the dimensionless time scale.
1M is a bounding parameter
of c1.
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Future Work
• The inclusion of seasonal variation into the models.
1 sin(2 ) sin(2 )1 1 1 1
2 sin(2 )2 2 2 2
[ ( ( ) ( )) ) ]2
[ ( ( ) ) ]2
g mt f wt
s vt
dur u n e h e a u b v k uv
dtdv
r v c e a u b v k uvdt
In red is the superposition of two differing time scales.
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Future Work
• Mixed Strategy Optimization.
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A simulation of the Mixed Strategy Game for The special case when plant B (green) only propagates locally, whereas plant A propagates near and far. Notice that plant A begins to “encase” plant B. This is one example of many criteria outlining how one plant can out-compete another spatially.
A simulation