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 ]

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

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 )

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

Not ecological models No mechanisms of competition in the model –Phenomenological Environment not explicitly included Mechanistic models of Resource competition

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

Resources for 0 growth dN / N dt = 0 R*R* dN / N dt > 0dN / N dt < 0 R 0

Kinds of resources Consider 2 potentially limiting resources Illustrate zero growth isocline graphically Defines 8 types 3 types important –substitutable –essential –switching

Substitutable resources: Interchangeable R2R2 R1R1 Zero growth isocline dN / N dt < 0 dN / N dt > 0 Prey for most animals

Switching resources: One at a time R2R2 R1R1 Zero growth isocline dN / N dt < 0 dN / N dt > 0 Nutritionally substitutable Constraints on consumption

Essential resources: both required R2R2 R1R1 Zero growth isocline dN / N dt < 0 dN / N dt > 0 Soil nutrients for plants

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

Feeding rate R F F max K 1/2 Holling type 2 Functional response Michaelis- Menten enzyme kinetics Monod microbial growth

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

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 ]

Limitation by 1 resource R dN / N dt R*R* -m 0

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

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 ]

Competition for 1 resource sp. 1 R dN / N dt R*1R*1 -m 1 0 R*2R*2 -m 2 sp. 2

Dynamics of competition for 1 resource t N R*1R*1 R*2R*2 R R sp. 1 SP.2

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

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

Experiments Laboratory tests confirm this prediction Primarily done with phytoplankton Summarized by Tilman (1982) Grover (1997) Morin, pp Chase & Leibold, pp

Consumption of 2 resources consumption vector: resultant of consumption of each resource R1R1 R2R2 C i1 C i2 CiCi consumes more R 1

Essential resources consumption vectors are parallel (essential) R1R1 R2R2 C i1 C i2 C1C1

Substitutable resources consumption vectors are not parallel (substitutable) R1R1 R2R2 C i1 C i2 CiCi

Switching resources consumption vectors are perpendicular to isocline (switching) R1R1 R2R2 C1C1

Renewal for 2 resources supply vector: points at supply point S 1,S 2 R1R1 R2R2 S 1,S 2 U

Equilibrium: 1 sp. 2 resources consumption vector equal & opposite supply vector R1R1 R2R2 CiCi CiCi CiCi U S 1,S 2 U U

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

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

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

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