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?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

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

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

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

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)

Jonathan• Data Collection• Analysis

– Area Method– Threshold Method

Kunj• Theoretical Model

– Details later

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

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

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

Threshold Method, ResultsFallopia Artemisia

020406080

100

1 2 3 4 5 6 7

Transect

Cov

er (

scal

ed %

)

Maximum Overlap

Difference at 50%

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

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.

( )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]

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

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

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

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

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.

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.

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.

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

Threshold Concept• A convenient way to relate above and below

ground interactions.

A dramatic example of the right side case observed at Morristown Park.

A Dramatic Observation

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.

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.

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

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.

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.

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.

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.

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).

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).

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.

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.

Future Work

• Mixed Strategy Optimization.

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