Competition Please do not use the images in these PowerPoint slides without permission. Photo of hyenas and lioness at a carcass from https://www.flickr.com/photos/davidbygott/4046054583

Intra-specific vs. Inter-specific Competition Non-trophic 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 Please do not use the images in these PowerPoint slides without permission. Note that resources contrast with other environmental factors that can limit organisms, but are not consumed nor depleted, e.g., pH. Space can be a resource. Oxygen can be a resource in aquatic ecosystems. 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 Please do not use the images in these PowerPoint slides without permission. From your textbook: “Competing species are more likely to coexist when they use resources in different ways.” Intraspecific competition causes logistic growth. Interspecific competition causes competitive exclusion of P. caudatum when with P. aurelia. Hardin, Garrett. 1960. The competitive exclusion principle. Science 131:1292-1297. Gause, G. F. 1934. Experimental analysis of Vito Volterra’s mathematical theory of the struggle for existence. Science 79:16-17. Bowman, Hacker & Cain (2017), Fig. 14.9, 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 Please do not use the images in these PowerPoint slides without permission. From your textbook: “Competing species are more likely to coexist when they use resources in different ways.” Intraspecific competition causes logistic growth. Interspecific competition causes competitive exclusion of P. caudatum when with P. aurelia. Gause, G. F. 1934. Experimental analysis of Vito Volterra’s mathematical theory of the struggle for existence. Science 79:16-17. Bowman, Hacker & Cain (2017), Fig. 14.9, after Gause (1934)

Lotka – Volterra Phenomenological Competition Models Alfred Lotka & Vito Volterra (1880-1949) (1860-1940) Please do not use the images in these PowerPoint slides without permission. Wikipedia “Vito Volterra” page; accessed 02-X-2014 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] Please do not use the images in these PowerPoint slides without permission. dN/dt = rN(1-(N/K)) Is the same as… dN/dt = rN((K-N)/K) See pg. 326 of textbook. Note that in the logistic population growth model, growth rate is reduced by something related to population density (predation, parasitism, competition) – that can be assumed to be “intraspecific competition” for the sake of the phenomenological model. A common interpretation is that K takes on a particular finite value as a result of intraspecific competition (i.e., we model intraspecific competition with K). Then, when terms for the other species are added to the population models, imagine what would happen at K for a given species (the addition of the competitor would make the pop. growth rate go negative).

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 Please do not use the images in these PowerPoint slides without permission. Note: sometimes the first equation’s alpha is labeled α12 for the per capita effect on Species 1 of Species 2. 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: N1 = 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) Please do not use the images in these PowerPoint slides without permission. dN1/dt = r1N1[(K1-N1-αN2)/K1] 0 = r1N1[(K1-N1-αN2)/K1] 0 / r1N1 = [r1N1[(K1-N1-αN2)/K1]] / r1N1 0 = (K1-N1-αN2)/K1 0 * K1 = [(K1-N1-αN2)/K1] * K1 0 = K1-N1-αN2 N1 = K1-αN2

Lotka – Volterra Phenomenological Competition Models The equations for equilibrium solutions become: Species 1: N1 = [K1 - αK2] / [1 - αβ] Species 2: N2 = [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) Please do not use the images in these PowerPoint slides without permission. Solve previous slide’s equations for N (use algebra). 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 Please do not use the images in these PowerPoint slides without permission. Gotelli, N. 2001. A Primer of Ecology. Sinauer, Sunderland, MA. 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 Please do not use the images in these PowerPoint slides without permission. 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 When N1 = 0, N2 = K1/α Isocline for Species 1 dN1/dt = 0 N2 Please do not use the images in these PowerPoint slides without permission. Zero net growth isocline = zero population growth isocline (textbook’s term) = isocline When N2 = 0, N1 is at its carrying capacity. If α>1 (i.e., intersp. > intrasp. comp.), then (K1/α) < K1 If 0<α<1 (i.e., intrasp. > intersp. comp.), then (K1/α) > K1 N1 = K1 - αN2 0 = K1 - αN2 αN2 = K1 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 When N2 = 0, N1 = K2/β Isocline for Species 2 dN2/dt = 0 N2 Please do not use the images in these PowerPoint slides without permission. K2 / β N1

Competitive exclusion of Lotka-Volterra Model Plot the isoclines for 2 species together to examine population trajectories K1/α > K2 K1 > K2/β For species 1: K1 > K2α (intrasp. > intersp.) For species 2: K1β > K2 (intersp. > intrasp.) Competitive exclusion of Species 2 by Species 1 K1 / α N2 K2 Please do not use the images in these PowerPoint slides without permission. Solve the first 2 equations for K1 and K2, respectively, to give the inequality for Sp. 1 and Sp.2, respectively. Remember that K represents intraspecific competition. K1 / α > K2 For Sp. 1: (K1 / α) · α > K2 · α K1 > K2 · α So, it makes sense that intraspecific competition is greater than interspecific competition for Sp. 1, since the carrying capacity number of individuals for Sp. 1 is larger than the number of individuals of Sp. 2 at its carrying capacity when scaled to equivalent Sp. 1 individuals (i.e., scaled by the competition coefficient, α). K1 > K2 / β For Sp. 2: K1 · β > (K2 / β) · β K1 · β > K2 So, it makes sense that interspecific competition is greater than intraspecific competition for Sp. 2, since the number of individuals for Sp. 1 at its carrying capacity when scaled to equivalent Sp. 2 individuals (i.e., scaled by the competition coefficient, β) is larger than the carrying capacity number of individuals for Sp. 2. = stable equilibrium K2 / β K1 N1

Competitive exclusion of Lotka-Volterra Model Plot the isoclines for 2 species together to examine population trajectories K2 > K1/α K2/β > K1 For species 1: K2α > K1 (intersp. > intrasp.) For species 2: K2 > K1β (intrasp. > intersp.) Competitive exclusion of Species 1 by Species 2 K2 N2 K1/ α Please do not use the images in these PowerPoint slides without permission. = stable equilibrium K1 K2 / β N1

Competitive exclusion with an Lotka-Volterra Model Plot the isoclines for 2 species together to examine population trajectories K2 > K1/α K1 > K2/β For species 1: K2α > K1 (intersp. > intrasp.) For species 2: K1β > K2 Competitive exclusion with an unstable equilibrium K2 K1/ α N2 Please do not use the images in these PowerPoint slides without permission. Starting conditions determine the winner. = stable equilibrium K2 / β K1 = unstable equilibrium N1

Coexistence at a stable equilibrium Lotka-Volterra Model Plot the isoclines for 2 species together to examine population trajectories K1/α > K2 K2/β > K1 For species 1: K1 > K2α (intrasp. > intersp.) For species 2: K2 > K1β Coexistence at a stable equilibrium K1 / α N2 K2 Please do not use the images in these PowerPoint slides without permission. Note that intraspecific competition must be greater than interspecific competition in both species for this to happen. Note also that the equilibrial densities are less than the carrying capacities of the two species, meaning that population regulation for each species operates around values lower than their respective carrying capacities (as in Gause’s experiments). = stable equilibrium K1 K2 / β N1

Mechanisms of Competition Exploitation competition Dissecting exploitation competition reveals its indirect nature H - H - - + + P Interference competition (direct aggression, allelopathy, etc.) Please do not use the images in these PowerPoint slides without permission. For this figure I am using P=plant & H=herbivore. From the phenomenological definition of competition, if 1 herbivore increases and the other decreases (and vice versa) we would consider this competition. So, under exploitation competition, indirect competition results in these reciprocal changes mediated by the resource (plant [or could be prey]) species. Menge, Bruce A. 1995. Indirect effects in marine rocky intertidal interaction webs: Patterns and importance. Ecological Monographs 65:21-74. H - H P - P Solid arrows = direct effects; dotted arrows = indirect effects Redrawn from Menge (1995)

Mechanisms of Competition David Tilman Synedra Asterionella Please do not use the images in these PowerPoint slides without permission. Tilman, David et al. 1981. Competition and nutrient kinetics along a temperature gradient: an experimental test of a mechanistic approach to niche theory. Limnology & Oceanography 26:1020-1033. Bowman, Hacker & Cain (2017), Fig. 14.6, 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 Please do not use the images in these PowerPoint slides without permission. Tilman, David et al. 1981. Competition and nutrient kinetics along a temperature gradient: an experimental test of a mechanistic approach to niche theory. Limnology & Oceanography 26:1020-1033. Bowman, Hacker & Cain (2017), Fig. 14.6, after Tilman et al. (1981); photo of Tilman from http://www.princeton.edu/morefoodlesscarbon/speakers/david-tilman/

Asymmetric vs. Symmetric Competition Please do not use the images in these PowerPoint slides without permission. Bowman, Hacker & Cain (2017), Fig. 14.7

Classic Pattern Interpreted as Evidence for Competitively-Structured Assemblages Robert MacArthur (1930-1972) Please do not use the images in these PowerPoint slides without permission. Wikipedia “Robert MacArthur” page; accessed 02-X-2014 Kaspari, Michael. 2008. Knowing your warblers: thoughts on the 50th anniversary of MacArthur (1958). Bulletin of the Ecological Society of America. October:448-458. MacArthur, Robert H. 1958. Population ecology of some warblers of northeastern coniferous forests. Ecology 39:599-619. 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 Please do not use the images in these PowerPoint slides without permission. G. fuliginosa was absent from Daphne Island & G. fortis was absent from Los Hermanos Island when these data were collected. Character displacement occurs when directional selection operates in each species, but in opposite directions in the two species. Connell, Joseph H. 1980. Diversity and the coevolution of competitors, or the ghosts of competition past. Oikos 35:131-138. Lack, David. 1947. Darwin’s Finches. Cambridge University Press, Cambridge, UK. Bowman, Hacker & Cain (2017), Fig. 14.12, after Lack (1947)