Curvilinear isoclines might be more realistic

Mutualism Equations (pp. 234-235, Chapter 11) dN1 /dt = r1 N1 ({X1 – N1 + a12 N2 }/X1) dN2 /dt = r2 N2 ({X2 – N2 + a21 N1 }/X2) (X1 – N1 + a12 N2 )/X1 = 0 when N1 = X1 + a12 N2 (X2 – N2 + a21 N1 )/X2 = 0 when N2 = X2 + a21 N1 If X1 and X2 are positive and a12 and a21 are chosen so that isoclines cross, there is a stable joint equilibrium Intraspecific self damping must be stronger than interspecific positive mutualistic effects.

N2* N1* Note: when together, both species reach higher equilbrium population density

Evidence of Competition in Nature. often circumstantial. 1 Evidence of Competition in Nature often circumstantial 1.Resource partitioning among closely-related sympatric congeneric species (food, place, and time niches) Complementarity of niche dimensions 2. Character displacement 3. Incomplete biotas: niche shifts 4. Taxonomic composition of communities

Major Foods (Percentages) of Eight Species of Cone Shells, Conus, on Subtidal Reefs in Hawaii __________________________________________________________ Gastro- Entero- Tere- Other Species pods pneusts Nereids Eunicea belids Polychaetes flavidus 4 64 32 lividus 61 12 14 13 pennaceus 100 abbreviatus 100 ebraeus 15 82 3 sponsalis 46 50 4 rattus 23 77 imperialis 27 73

MacArthur’s Warblers (Dendroica) Robert H. MacArthur

John Terborgh

John Terborgh

Time of Activity Seasonal changes in activity times

Complementarity of Niche Dimensions, page 276 Anolis Thomas W. Schoener

Prey size versus predator size

Peter R. Grant David Lack

Character Displacement in Hydrobia mud snails in Denmark Snail shell length, mm

Hutchinsonian ratios among short wing Accipiter hawks Thomas W. Schoener

Hutchinsonian ratios among Australian Varanus lizards