1. 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

2. B. Peckham 2005 SIAM D.S.2 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

3. 3 3 models LV LCC LGR where

4. B. Peckham 2005 SIAM D.S.4 Nullcline comparison: LV vs. LGR LV: LG: Sketch (strong interaction and r 10 =0=r 20 ): x y x y

5. B. Peckham 2005 SIAM D.S.5 Weak mutualism: LV vs. LGR

6. B. Peckham 2005 SIAM D.S.6 Strong mutualism: LV vs.LGR 1->2 LV: LGR: