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Alternative Lotka-Volterra competition

Alternative Lotka-Volterra competition. Absolute competition coefficients dN i / N i dt = r i [1 – b ii N i - b ij N j ]. equivalent to: dN i / N i dt = r i [ K i - N i - a j N j ] / K i = r i [ K i / K i - N i / K i - a j N j / K i ]

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Alternative Lotka-Volterra competition

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  1. Alternative Lotka-Volterra competition • Absolute competition coefficients dNi/ Nidt= ri[1 – bii Ni - bijNj] equivalent to: dNi/ Nidt= ri[Ki - Ni - ajNj] / Ki = ri[Ki/Ki - Ni/Ki - ajNj/Ki] = ri[1-(1/Ki)Ni – (aj/Ki)Nj]

  2. 1/b12 Stable coexistence dN1 / N1dt = 0   1/b22 dN2 / N2dt = 0   0 1/b11 1/b21 N1 Absolute Lotka-Volterra N2

  3. Competitive effect vs. response • Effect: impact of density of a species • Self density (e.g., b11) • Other species density (e.g., b21) • Response: how density affects a species • Self density (e.g., b11) • Other species’ density (e.g., b12) • Theory: effects differ (b11 > b21) • Experiments: responses (b11, b12)

  4. 1/b12 Stable coexistence dN1 / N1dt = 0   1/b22 dN2 / N2dt = 0   0 N1 1/b11 1/b21 Absolute Lotka-Volterra N2

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

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

  9. Zero growth isocline R2 dN / N dt > 0 dN / N dt < 0 R1 Substitutable resources: Interchangeable Prey for most animals

  10. Zero growth isocline R2 dN / N dt > 0 dN / N dt < 0 R1 Switching resources: One at a time Nutritionally substitutable Constraints on consumption

  11. R2 Zero growth isocline dN / N dt > 0 dN / N dt < 0 R1 Essential resources: both required 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 = FmaxR / [K1/2 + R ] • Fmax = maximal feeding rate • K1/2 = resource level for 1/2 maximal feeding • 1/2 saturation constant

  13. Fmax F R K1/2 Feeding rate • Holling type 2 Functional response • Michaelis-Menten enzyme kinetics • Monod microbial growth

  14. Modeling resource-based population growth • dN / N dt = p FmaxR / [K1/2 + 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* = K1/2m / [ pFmax - m ]

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

  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 K1/2 increases • increases as m increases • decreases as p or Fmax increase

  18. Two consumers competing for one resource • dNi / Ni dt = pi Fmax iR / [K1/2 i + R ] - mi • dR / dt = a ( S - R ) - S(dNi / dt + miNi) ci • each species has its own R* [ R*1 and R*2]

  19. dN / N dt sp. 1 sp. 2 R*1 0 R R*2 -m2 -m1 Competition for 1 resource

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

  21. Prediction for 2 species competing for 1 resource • The species with the lowerR* 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. R2 Ci1 Ci2 Ci R1 Consumption of 2 resources consumption vector: resultant of consumption of each resource consumes more R1

  25. R2 Ci1 Ci2 C1 R1 Essential resources consumption vectors are parallel (essential)

  26. R2 Ci1 Ci2 Ci R1 Substitutable resources consumption vectors are not parallel (substitutable)

  27. R2 C1 R1 Switching resources consumption vectors are perpendicular to isocline (switching)

  28. S1,S2 R2 U R1 Renewal for 2 resources supply vector: points at supply point S1,S2

  29. S1,S2 R2 U U Ci Ci Ci U R1 Equilibrium: 1 sp. 2 resources consumption vector equal & opposite supply vector

  30. Equilibrium • Equilibrium (R1,R2) falls on isocline • therefore, dN / N dt =0 • U and C vectors equal in magnitude, opposite direction • therefore dR1 / dt = 0 and dR2 / dt = 0

  31. S1,S2 S1,S2 S1,S2 R2    sp. 2 sp. 1 R1 Competition for 2 resources  sp. 1 always excludes sp. 2  sp. 2 cannot survive  neither spp. can survive

  32. S1,S2 S1,S2 S1,S2 S1,S2 R2     sp. 2 sp. 2 sp. 1 sp. 1 R1 Competition for 2 resources  neither spp. can survive  sp. 2 cannot survive  sp. 1 always excludes sp. 2  coexistence

  33. Equilibrium • sp. 1 • needs less R1(limited by R2) • consumes more R2 • sp. 2 • needs less R2(limited by R1) • consumes more R1 • consumes more of the resource limiting to itself

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