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NOTES for educators.
Some of the following materials may be of use in the classroom..  See also the page entitled "miscellaneous" under the purple loosestrife page, for facts about that plant and its bio-control agents, which might be useful in modeling growth.   See also the large quantity of teaching materials at the Purple Loosestrife Project at Michigan State University website. 

Notes on Modeling the growth of the PL Invasion
Discussion Leader - Robert Norman (Dartmouth College Math Department) Discussion held at 3/27/00 meeting on mapping held at the Montshire Museum. 

The group discussed how to proceed, when you have almost no data about the plant's rate of spread. We have only two facts -- a rough map, and the knowledge that the Purple Loosestrife plant has arrived in living memory. They looked at our incomplete1999 map of PL in the Upper Valley, and speculated on how the plant spreads (wind, water and transportation corridors), and where the densest places are. 

Questions arose as to how many plants exist in the Upper Valley. The group speculated about rate, if one assumes that the plant arrived here 10-15 years ago. (For example, assuming 2 sq km of flowering plants here now, and a density of 100 plants per sq meter, this suggests that there are 108 plants here now. Further, assuming 15 years of growth, then the plant is spreading about 3.5 - fold per year, assuming that growth is exponential) 

How can one estimate the number of plants in a small area? Look at a small area, and multiply up to total area, independent of clumps? Is it OK to assume, say, 100 plants per square meter, or should we just count stems? Does it matter? For a model to be useful in this situation, is just order of magnitude sufficient? Perhaps measures of the relative rates of increase is more important. 

There was also some discussion as to how the growth rate and growth patterns might suggest how to target control measures (cutting, digging, or beetles) ---- start at periphery where density is low? Start at center where density is greatest? Catch isolated outbreaks first? Aggressive monitoring in unaffected areas? Some factors which might modify the rate of growth are -- predators, destruction of individual plants (mechanical or other means), limited habitat, and dispersal of seeds (by wind, animals, or humans -- which allow the plant to "jump" to a new site). 

Generic models for other invasive plants [see references] suggest radial growth, where population increases by a constant radius from the previous year's area (and thus would be modeled by a quadratic), and density growth where population increases in proportion to the current number of individuals (and thus represented as an exponential function).   And so, what characteristics of a model would lead to these two results -- number of seeds produced?  the distance of seed propagation?  the probability distribution of seed propagation. 

We can measure area, or count individuals. Or, we can do both and combine results -- all create some great math problems for the classroom, and allow some discussion about growth rates and explorations with inverse functions. 

How can one estimate area? rate of growth of radius of an area? population density? What measures in the field might help? How can we decide which growth model fits? 

There were several handouts at this meeting: 
one comparing exponential and quadratic models in general; and several from Hengeveld text (pages46,64,65) on neighborhood diffusion, illustrating linear display of the log if counting individuals, and linear display of the square root of the area affected by an invasion (in this case, muskrats in Europe). 

Also, a set of general discussion questions on this topic. 

References (available in Dartmouth's Dana Library; check call numbers QH 353 and SB 613 for related texts): 

  • Hengeveld, Rob. (1989) Dynamics of biological Invasions. Chapman and Hall, NY 
  • Luken, JO and Thieret, JW (eds). (1997) Assessment and Management of Plant Invasions. Springer-Verlag, New York