deja vu communities and spatial dynamics of competing plant species
DESCRIPTION
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. What are Invasive Species? - PowerPoint PPT PresentationTRANSCRIPT
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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