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Alternative Lotka-Volterra competition Absolute competition coefficients dN i / N i dt = r i [1 – ii N i - ij N j ] equivalent to: dN i / N i dt = r i [K i - N i - j N j ] / K i = r i [K i /K i - N i /K i - j N j /K i ] = r i [1- (1/K i )N i – ( j /K i )N j ]
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Absolute Lotka-Volterra N1N1 0 1/ 21 1/ 22 dN 2 / N 2 dt = 0 1/ 11 dN 1 / N 1 dt = 0 1/ 12 Stable coexistence N2N2
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Competitive effect vs. response Effect: impact of density of a species –Self density (e.g., 11 ) –Other species density (e.g., 21 ) Response: how density affects a species –Self density (e.g., 11 ) –Other species’ density (e.g., 12 ) Theory: effects differ ( 11 > 21 ) Experiments: responses ( 11, 12 )
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Absolute Lotka-Volterra N1N1 0 1/ 21 1/ 22 dN 2 / N 2 dt = 0 1/ 11 dN 1 / N 1 dt = 0 1/ 12 Stable coexistence N2N2
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Not ecological models No mechanisms of competition in the model –Phenomenological Environment not explicitly included Mechanistic models of Resource competition
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Resources component of the environment availability increases population growth can be depleted or used up by organisms A resource is limiting if it determines the growth rate of the population –Liebig’s law: resource in shortest supply determines growth
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Resources for 0 growth dN / N dt = 0 R*R* dN / N dt > 0dN / N dt < 0 R 0
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Kinds of resources Consider 2 potentially limiting resources Illustrate zero growth isocline graphically Defines 8 types 3 types important –substitutable –essential –switching
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Substitutable resources: Interchangeable R2R2 R1R1 Zero growth isocline dN / N dt < 0 dN / N dt > 0 Prey for most animals
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Switching resources: One at a time R2R2 R1R1 Zero growth isocline dN / N dt < 0 dN / N dt > 0 Nutritionally substitutable Constraints on consumption
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Essential resources: both required R2R2 R1R1 Zero growth isocline dN / N dt < 0 dN / N dt > 0 Soil nutrients for plants
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Modeling resource-based population growth dN / N dt = p F - m –F = feeding rate on the resource –m = mortality rate (independent of R ) –p = constant relating feeding to population growth F = F max R / [K 1/2 + R ] –F max = maximal feeding rate –K 1/2 = resource level for 1/2 maximal feeding 1/2 saturation constant
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Feeding rate R F F max K 1/2 Holling type 2 Functional response Michaelis- Menten enzyme kinetics Monod microbial growth
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Modeling resource-based population growth dN / N dt = p F max R / [K 1/2 + R ] - m resource dynamics dR / dt = a ( S - R ) - (dN / dt + mN ) c –S = maximum resource supplied to the system –a = a rate constant –c = resource consumption / individual N = 0 if S = R then dR / dt = 0
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Equilibrium dN / N dt = 0 and dR / dt = 0 –resource consumption just balances resource renewal –growth due to resource consumption just balances mortality Equilibrium resource density: –R * = K 1/2 m / [ pF max - m ]
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Limitation by 1 resource R dN / N dt R*R* -m 0
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Conclusion 1 species feeding on 1 limiting resource reduces that resource to a characteristic equilibrium value R * R * determined by functional response and mortality –increases as K 1/2 increases –increases as m increases –decreases as p or F max increase
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Two consumers competing for one resource dN i / N i dt = p i F max i R / [K 1/2 i + R ] - m i dR / dt = a ( S - R ) - (dN i / dt + m i N i ) c i each species has its own R * [ R * 1 and R * 2 ]
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Competition for 1 resource sp. 1 R dN / N dt R*1R*1 -m 1 0 R*2R*2 -m 2 sp. 2
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Dynamics of competition for 1 resource t N R*1R*1 R*2R*2 R R sp. 1 SP.2
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Prediction for 2 species competing for 1 resource The species with the lower R * will eliminate the other in competition Independent of initial numbers Coexistence not possible –unless R * 1 = R * 2 R * rule
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Competitive exclusion principle Two species in continued, direct competition for 1 limiting resource cannot coexist Focus on mechanism Coexistence (implicitly) requires 2 independently renewed resources
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Experiments Laboratory tests confirm this prediction Primarily done with phytoplankton Summarized by Tilman (1982) Grover (1997) Morin, pp. 40-49 Chase & Leibold, pp. 62-63
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Consumption of 2 resources consumption vector: resultant of consumption of each resource R1R1 R2R2 C i1 C i2 CiCi consumes more R 1
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Essential resources consumption vectors are parallel (essential) R1R1 R2R2 C i1 C i2 C1C1
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Substitutable resources consumption vectors are not parallel (substitutable) R1R1 R2R2 C i1 C i2 CiCi
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Switching resources consumption vectors are perpendicular to isocline (switching) R1R1 R2R2 C1C1
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Renewal for 2 resources supply vector: points at supply point S 1,S 2 R1R1 R2R2 S 1,S 2 U
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Equilibrium: 1 sp. 2 resources consumption vector equal & opposite supply vector R1R1 R2R2 CiCi CiCi CiCi U S 1,S 2 U U
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Equilibrium Equilibrium (R 1,R 2 ) falls on isocline therefore, dN / N dt =0 U and C vectors equal in magnitude, opposite direction therefore dR 1 / dt = 0 and dR 2 / dt = 0
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Competition for 2 resources R1R1 R2R2 sp. 1 S 1,S 2 sp. 1 always excludes sp. 2 sp. 2 cannot survive neither spp. can survive
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Competition for 2 resources R1R1 R2R2 S 1,S 2 neither spp. can survive sp. 2 cannot survive sp. 1 always excludes sp. 2 S 1,S 2 coexistence sp. 1 sp. 2 sp. 1
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Equilibrium sp. 1 – needs less R 1 (limited by R 2 ) –consumes more R 2 sp. 2 –needs less R 2 (limited by R 1 ) –consumes more R 1 consumes more of the resource limiting to itself
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