A spatial model of spartina invasion

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A spatial model of spartina invasion
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Spartina

• An exotic marsh grass
• First clone estimated to have been born in
  Willapa Bay in 1890
• grows from individual plants that form clones
  that grow in a circular fashion
• reproduces by seed, by plant fragments and
  lateral expansion of clones
• Forms continuous fields when clones merge
• Now covers large areas of the estuary

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Data available

• Ph.D. thesis of Blake Feist
• Has historic air photos in last 30 years that
  show both the covered and one can track the
  growth of individual clones.

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Key parameter

• Clones appear to grow at a uniform 0.8 meters in
  radius per year
• From this rate one can back calculate the number
  of new clones born each year

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The dynamics of an individual clone without
competition
Where Aa is the area covered by a clone of age a
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Key assumptions

• That the clone grows at 0.8 m/yr in radius at all
  ages (including age 1, 2 )
• That there is no overlap with other clones or
  unsuitable habitat

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The dynamics of a cohort
Where Ta is the total area covered by clones of
age a, and Na is the number of clones of age a
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If we hypothesize (or model) the number born in
each year

• We can easily calculate the area that would be
  covered, if there was no overlap between clones
• But clearly as the area gets more covered, the
  frequency of overlap will be higher

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Imagine the area available in a site is k and
each clone is thrown on a map randomly

• What is the probability that an individual point
  in the habitat will not be covered
• The probability that an individual point will be
  covered by a single clone is p, where p is the
  area of the clone divided by the total area (k).

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This is like survival

• The probability of surviving a large number of
  individual clones is

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Imagine 10 clones placed at random, each covering
10 of the area

• The probability that none of these clones would
  have landed on top of an individual site is
  exp(-1) or 37

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The area not covered is therefore
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Recruitment of new clones

• Assume that the number of seeds that will land in
  the area is proportional to the total area
  covered by Spartina
• Assume the landing site is random, and only those
  landing on uncovered (by spartina) ground will
  germinate

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Recruitment model
Fraction of area uncovered
Area now covered
The three terms are r, the seeds produced per
unit area of existing spartina, the current area
covered by spartina, and the proportion of the
area that is uncovered
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We now have a model that will predict the area
covered and the number of new clones!
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Model fitting

• Estimate 1 parameter, r, the seeds produced per
  area of spartina
• Use sum of squares, observed recruits per year
  compared to predicted

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Not great

• Blake noticed that the years with lots of new
  clones corresponded to warm periods

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Model with SST
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Diamond Point

• Started much later

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Diamond point area covered
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Anomalies/problems

• The above were done with a 4 year lag on SST
• Current thinking is that this corrects for
  slower that 0.8 m per year lateral growth when
  they are young
• Jensens beach wants more area to be covered
  early on,

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Uses of model

• Suggests specific experiments on growth and
  survival of clones
• Can test herbicide spraying as a control measure