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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 =

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Presentation on theme: "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 ="— Presentation transcript:

1 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 ]

2 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

3 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 )

4 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

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

6 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

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

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

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

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

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

12 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

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

14 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

15 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 ]

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

17 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

18 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 ]

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

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

21 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

22 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

23 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

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

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

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

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

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

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

30 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

31 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

32 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

33 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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