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Competition Photo of hyenas and lioness at a carcass from https://www.flickr.com/photos/davidbygott/4046054583 Pairwise Species Interactions Influence of species A Influence of Species B A - - (negative) 0 (neutral/null) - 0 - A - B Competition Amensalism - 0 A B 0 A B A B Antagonism (Predation/Parasitism) + A B 0 0 Amensalism Neutralism (No interaction) Commensalism - 0 + 0 + B + (positive) + A B + Antagonism (Predation/Parasitism) A B + Commensalism Redrawn from Abrahamson (1989); Morin (1999, pg. 21) A + Mutualism B Intra-specific vs. Inter-specific Competition Interaction between individuals in which each is harmed by their shared use of a limiting resource (which can be consumed or depleted) for growth, survival, or reproduction Photo of hyenas and lioness at a carcass from https://www.flickr.com/photos/davidbygott/4046054583 Intra-specific vs. Inter-specific Competition “Complete competitors cannot coexist.” (Hardin 1960) Paramecium aurelia Paramecium caudatum Cain, Bowman & Hacker (2014), Fig. 12.11, after Gause (1934); photomicrographs from Wikimedia Commons Intra-specific vs. Inter-specific Competition Resource partitioning – differences in use of limiting resources – can allow species to coexist P. aurelia & P. caudatum ate mostly floating bacteria; P. bursaria ate mostly yeast cells on the bottoms of the tubes Cain, Bowman & Hacker (2014), Fig. 12.11, after Gause (1934) Lotka – Volterra Phenomenological Competition Models Alfred Lotka & Vito Volterra (1880-1949) (1860-1940) Photo of Lotka from http://blog.globe-expert.info; photo of Volterra from Wikimedia Commons Lotka – Volterra Phenomenological Competition Models Lotka-Volterra Competition Equations: Logistic population growth model – growth rate is reduced by intraspecific competition: Species 1: dN1/dt = r1N1[(K1-N1)/K1] Species 2: dN2/dt = r2N2[(K2-N2)/K2] Functions added to further reduce growth rate owing to interspecific competition: Species 1: dN1/dt = r1N1[(K1-N1-f(N2))/K1] Species 2: dN2/dt = r2N2[(K2-N2-f(N1))/K2] Lotka – Volterra Phenomenological Competition Models Lotka-Volterra Competition Equations: The function (f) could take on many forms, e.g.: Species 1: dN1/dt = r1N1[(K1-N1-αN2)/K1] Species 2: dN2/dt = r2N2[(K2-N2-βN1)/K2] The competition coefficients α & β measure the per capita effect of one species on the population growth of the other, measured relative to the effect of intraspecific competition If α = 1, then per capita intraspecific effects = interspecific effects If α < 1, then intraspecific effects are more deleterious to Species 1 than interspecific effects If α > 1, then interspecific effects are more deleterious Lotka – Volterra Phenomenological Competition Models Find equilibrium solutions to the equations, i.e., set dN/dt = 0: ^ Species 1: N 1 = K1 - αN2 ^ Species 2: N2 = K2 - βN1 This makes intuitive sense: The equilibrium for N1 is the carrying capacity for Species 1 (K1) reduced by some amount owing to the presence of Species 2 (αN2) However, each species’ equilibrium depends on the equilibrium of the other species! So, by substitution… ^ ^ ^ ^ Species 1: N1 = K1 - α(K2 - βN1) Species 2: N2 = K2 - β(K1 - αN2) Lotka – Volterra Phenomenological Competition Models The equations for equilibrium solutions become: ^ Species 1: N 1 = [K1 - αK2] / [1 - αβ] ^ Species 2: N 2 = [K2 - βK1] / [1 - αβ] These provide some insights into the conditions required for coexistence under the assumptions of the model E.g., the product αβ must be < 1 for N to be > 0 for both species (a necessary condition for coexistence) But they do not provide much insight into the dynamics of competitive interactions, e.g., are the equilibrium points stable? Lotka – Volterra Phenomenological Competition Models 4 time steps State-space graphs help to track population trajectories (and assess stability) predicted by models From Gotelli (2001) Lotka – Volterra Phenomenological Competition Models 4 time steps State-space graphs help to track population trajectories (and assess stability) predicted by models 4 time steps Mapping state-space trajectories onto single population trajectories From Gotelli (2001) Lotka-Volterra Model Remember that equilibrium solutions require dN/dt = 0 ^ Species 1: N1 = K1 - αN2 Therefore: When N2 = 0, N1 = K1 K1 / α Isocline for Species 1 dN1/dt = 0 N2 When N1 = 0, N2 = K1/α K1 N1 Lotka-Volterra Model Remember that equilibrium solutions require dN/dt = 0 ^ Species 2: N2 = K2 - βN1 Therefore: When N1 = 0, N2 = K2 K2 Isocline for Species 2 dN2/dt = 0 N2 When N2 = 0, N1 = K2/β K2 / β N1 Lotka-Volterra Model Plot the isoclines for 2 species together to examine population trajectories Competitive exclusion of Species 2 by Species 1 K1/α > K2 K1 > K2/β K1 / α N2 For species 1: K1 > K2α (intrasp. > intersp.) K2 For species 2: K1β > K2 (intersp. > intrasp.) = stable equilibrium K2 / β N1 K1 Lotka-Volterra Model Plot the isoclines for 2 species together to examine population trajectories Competitive exclusion of Species 1 by Species 2 K2 > K1/α K2/β > K1 K2 N2 For species 1: K2α > K1 (intersp. > intrasp.) For species 2: K2 > K1β (intrasp. > intersp.) K1/ α = stable equilibrium K1 N1 K2 / β Lotka-Volterra Model Plot the isoclines for 2 species together to examine population trajectories Competitive exclusion with an unstable equilibrium K2 > K1/α K1 > K2/β K1/ α N2 For species 1: K2α > K1 (intersp. > intrasp.) K2 For species 2: K1β > K2 (intersp. > intrasp.) = stable equilibrium = unstable equilibrium K2 / β K1 N1 Lotka-Volterra Model Plot the isoclines for 2 species together to examine population trajectories K1/α > K2 K2/β > K1 Coexistence at a stable equilibrium K1 / α N2 For species 1: K1 > K2α (intrasp. > intersp.) For species 2: K2 > K1β (intrasp. > intersp.) K2 = stable equilibrium K2 / β K1 N1 Mechanisms of Competition Exploitation competition Dissecting exploitation competition reveals its indirect nature H - H - - + + P Interference competition (direct aggression, allelopathy, etc.) H - H P - P Solid arrows = direct effects; dotted arrows = indirect effects Redrawn from Menge (1995) Mechanisms of Competition David Tilman Synedra Asterionella Cain, Bowman & Hacker (2014), Fig. 12.4, after Tilman et al. (1981); photos of diatoms from Wikimedia Commons; photo of Tilman from http://www.princeton.edu/morefoodlesscarbon/speakers/david-tilman/ Mechanisms of Competition David Tilman Cain, Bowman & Hacker (2014), Fig. 12.4, after Tilman et al. (1981); photo of Tilman from http://www.princeton.edu/morefoodlesscarbon/speakers/david-tilman/ Asymmetric vs. Symmetric Competition Cain, Bowman & Hacker (2014), Fig. 12.7 Classic Pattern Interpreted as Evidence for Competitively-Structured Assemblages Robert MacArthur (1930-1972) Painting of “MacArthur’s warblers” by D. Kaspari for M. Kaspari (2008); anniversary reflection on MacArthur (1958) Character Displacement The “Ghost of Competition Past” (sensu Connell 1980) is hypothesized to be the cause of the beak size difference on Pinta Marchena Cain, Bowman & Hacker (2014), Fig. 12.19, after Lack (1947)