A%20Bifurcation%20Analysis%20of%20a%20Differential%20Equations%20Model%20for%20Mutualism

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A Bifurcation Analysis of a Differential
Equations Model for Mutualism

• Bruce Peckham
• Department of Mathematics and Statistics
  University of Minnesota Duluth
• Collaborators
• Wendy Gruner Graves
• Rainy River Community College
• John Pastor
• Department of Biology, University of Minnesota
  Duluth

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Abbreviated History of 2-species Mutualism Models

• 1978 Vandermeer and Boucher (J. Theor. Biol.)
  Description of different dynamics for
  Lotka-Volterra (LV) model
• Weak mutualism populations survive at slightly
  higher populations than on their own
• Strong mutualism threshold for obligate models,
  but unbounded growth bend isoclines to
  prevent unbounded growth.
• 1979 B. S. Goh (Am. Nat.) Generalized to Taylor
  Series.
• 1983 A. Dean (American Naturalist) Limited
  Carrying Capacity (LCC) model
• Weak same as LV
• Strong singularity when either species is
  obligate
• 2004 GPP (under review, J. Math. Biol.) The
  limited growth rate (LGR) model
• Weak mutualism same as LV
• Strong mutualism threshold for obligate models,
  but no unbounded growth
• Allows for smooth transitions between facultative
  and obligate

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3 models
LV
LCC
where
LGR
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Nullcline comparison LV vs. LGR
LG
LV
y
y
Sketch (strong interaction and r100r20)
x
x
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Weak mutualism LV vs. LGR
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Strong mutualism LV vs.LGR
1-gt2
LV
LGR