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Competition Photo of hyenas and lioness at a carcass from - PowerPoint Presentation

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Competition Photo of hyenas and lioness at a carcass from - PPT Presentation

Competition Photo of hyenas and lioness at a carcass from https wwwflickrcomphotosdavidbygott4046054583 A B Competition Influence of species A Influence of Species B positive 0 neutralnull ID: 763624

competition species lotka volterra species competition volterra lotka equilibrium models population amp intrasp intersp tilman model trajectories phenomenological effects

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Competition Photo of hyenas and lioness at a carcass from https:// www.flickr.com/photos/davidbygott/4046054583

A B Competition - - Influence of species A Influence of Species B + (positive) 0 (neutral/null) - (negative) A B Amensalism 0 - A B Antagonism (Predation/Parasitism) + - A B Commensalism + 0 A B Neutralism (No interaction) 0 0 A B Commensalism 0 + A B Mutualism + + A B Amensalism - 0 A B Antagonism (Predation/Parasitism) - + - 0 + Redrawn from Abrahamson (1989); Morin (1999, pg. 21) Pairwise Species Interactions

Intra-specific vs. Inter-specific Competition Photo of hyenas and lioness at a carcass from https:// www.flickr.com/photos/davidbygott/4046054583 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

Intra-specific vs. Inter-specific Competition Cain, Bowman & Hacker (2014), Fig. 12.11, after Gause (1934); photomicrographs from Wikimedia Commons “ Complete competitors cannot coexist .” (Hardin 1960) Paramecium aurelia Paramecium caudatum

Intra-specific vs. Inter-specific Competition Resource partitioning – differences in use of limiting resources – can allow species to coexist Cain, Bowman & Hacker (2014), Fig. 12.11, after Gause (1934) P. aurelia & P. caudatum ate mostly floating bacteria; P. bursaria ate mostly yeast cells on the bottoms of the tubes

Alfred Lotka & Vito Volterra Lotka – Volterra Phenomenological Competition Models Photo of Lotka from http://blog.globe-expert.info; photo of Volterra from Wikimedia Commons (1860-1940) (1880-1949)

Logistic population growth model – growth rate is reduced by intraspecific competition: Species 1: dN1 /dt = r1N1[(K 1-N1)/K1] Species 2: dN2/dt = r2N2[(K2- N 2)/K2]Functions added to further reduce growth rate owing to interspecific competition : Species 1: dN1/ dt = r1N1 [(K1-N 1-f( N2))/K1] Species 2: dN2/dt = r2N2 [(K2-N2-f(N1))/K2] Lotka-Volterra Competition Equations: Lotka – Volterra Phenomenological Competition Models

Lotka-Volterra Competition Equations: The function ( f ) could take on many forms, e.g.: Species 1: dN1/dt = r1N1[(K 1 -N1-αN2)/K1] Species 2: dN2/dt = r2N2[(K2-N2-βN1 )/K 2] 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

This makes intuitive sense: The equilibrium for N 1 is the carrying capacity for Species 1 (K1) reduced by some amount owing to the presence of Species 2 (αN2) Find equilibrium solutions to the equations, i.e ., set dN/dt = 0: Species 1: N1 = K1 - αN2 Species 2: N2 = K2 - βN1 ^ ^ ^ ^ However, each species’ equilibrium depends on the equilibrium of the other species! So, by substitution… Species 1: N 1 = K 1 - α(K2 - βN1) Species 2: N2 = K2 - β(K1 - αN2) ^ ^ Lotka – Volterra Phenomenological Competition Models

These provide some insights into the conditions required for coexistence under the assumptions of the model The equations for equilibrium solutions become: Species 1: N 1 = [K1 - αK2] / [1 - αβ] Species 2: N2 = [ K2 - βK1] / [1 - αβ] ^ ^ 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

State-space graphs help to track population trajectories (and assess stability) predicted by models 4 time steps From Gotelli (2001) Lotka – Volterra Phenomenological Competition Models

State-space graphs help to track population trajectories (and assess stability) predicted by models Mapping state-space trajectories onto single population trajectories 4 time steps 4 time steps From Gotelli (2001) Lotka – Volterra Phenomenological Competition Models

Remember that equilibrium solutions require dN /dt = 0Species 1: N1 = K1 - αN 2 Isocline for Species 1 dN 1/dt = 0^ N 1 K 1 K 1 / α Therefore: When N 2 = 0, N 1 = K 1 When N 1 = 0, N2 = K1/α Lotka-Volterra Model N 2

Remember that equilibrium solutions require dN /dt = 0Species 2: N2 = K2 - βN 1 ^ N 1 N 2 K 2 / β K 2 Isocline for Species 2 dN 2 / dt = 0 Therefore: When N 1 = 0, N 2 = K 2 When N 2 = 0, N1 = K 2/β Lotka-Volterra Model

N 2 K 2 / β K 2 Plot the isoclines for 2 species together to examine population trajectories K 1 / α > K 2K1 > K2/ βFor species 1: K1 > K2α (intrasp. > intersp.)For species 2: K1β > K 2 (intersp. > intrasp.) K 1 K 1 / α = stable equilibrium Competitive exclusion of Species 2 by Species 1 N 1 Lotka-Volterra Model

N 2 K 2 / β K 1/ α Plot the isoclines for 2 species together to examine population trajectories K 2 > K 1 / αK2/β > K1For species 1: K2α > K1 (intersp. > intrasp.)For species 2: K2 > K1 β (intrasp. > intersp.) K 1 K 2 Competitive exclusion of Species 1 by Species 2 = stable equilibrium N 1 Lotka-Volterra Model

N 2 K 2 / β K 1 / α Plot the isoclines for 2 species together to examine population trajectories K 2 > K 1 /α K1 > K2/βFor species 1: K2α > K1 (intersp. > intrasp.)For species 2: K1 β > K2 (intersp. > intrasp.) K 1 K 2 Competitive exclusion with an unstable equilibrium = stable equilibrium = unstable equilibrium N 1 Lotka-Volterra Model

N 2 K 2 / β K 2 Plot the isoclines for 2 species together to examine population trajectories K 1 / α > K 2K2/β > K1For species 1: K1 > K2α (intrasp. > intersp.) For species 2: K2 > K1β (intrasp. > intersp.) K 1 K 1 / α = stable equilibrium Coexistence at a stable equilibrium Lotka-Volterra Model N 1

Mechanisms of Competition Dissecting exploitation competition reveals its indirect nature H - P Solid arrows = direct effects ; dotted arrows = indirect effects - + + - H Redrawn from Menge (1995) Exploitation competition Interference competition (direct aggression , allelopathy , etc .) P - P H - H

Mechanisms of Competition David Tilman 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/ Synedra Asterionella

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

Robert MacArthur (1930-1972) Classic Pattern Interpreted as Evidence for Competitively-Structured Assemblages Painting of “MacArthur’s warblers” by D. Kaspari for M. Kaspari (2008); anniversary reflection on MacArthur (1958)

Character Displacement Cain, Bowman & Hacker (2014), Fig. 12.19, after Lack (1947) The “ Ghost of Competition Past ” ( sensu Connell 1980) is hypothesized to be the cause of the beak size difference on Pinta Marchena