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?
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Deja Vu Communities and Spatial Dynamics of Competing Plant Species
Kunj Patel and Jonathan Lansey
New Jersey Institute of Technology
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 . . .
Why are they a problem?
They . . .
Our Hypothesis: Root Interactions
Our Hypothesis: Root Interactions
Root systems of neighboring guayule plants (Parthenium argentatum)root territories” Schenk et all.(1999)
To quantify “overlap” we first quantify the borders.
Area B> Area A
Allopatric pairs have significantlylarger %Cover overlap compared to sympatric pairs
Maximum Overlap Difference at 50%
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).
Is the growth rate [“births”/ plant/season]
cis the carrying capacity [maximum number of plants that can occupy a single quadrat]
Is the competition term
Is the inhibition term
The nullclines are 4 lines in the upper right quadrant.
can be combined.
System of Equations:
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.
then the cooperative state for U* is negative. In this case (0,1) is stable.
then U* increases relative to V* as is increased.
Is a such function.
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)
With strong sympatric species interactions, both of these inequalities are satisfied. With allopatric species interactions, at least one of the inequalities are broken.
And initial conditions
With boundary conditions
It has a clear analogy with reaction diffusion equations chemistry.
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.
A dramatic example of the right side case observed at Morristown Park.
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.
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.
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.
(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.
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).
The direct inhibition acts on a faster time scale than does the limitation offered by the limited carrying capacity.
is the dimensionless time scale.
is a bounding parameter of c1.
In red is the superposition of two differing time scales.
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.