alternative lotka-volterra competition absolute competition coefficients dn i / n i dt = r i [1 – ...
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Alternative Lotka-Volterra competition
• Absolute competition coefficients
dNi / Nidt = ri [1 – bii Ni - bij Nj]equivalent to:
dNi / Nidt = ri [Ki - Ni - aj Nj] / Ki
= ri [Ki/Ki - Ni/Ki - ajNj/Ki]
= ri [1- (1/Ki)Ni – (aj/Ki)Nj]
Absolute Lotka-Volterra
N1
0
1/b21
1/b22
dN2 / N
2dt = 0
1/b11dN
1 / N1 dt = 0
1/b12
Stable coexistence
N2
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)
Absolute Lotka-Volterra
N1
0
1/b21
1/b22
dN2 / N
2dt = 0
1/b11dN
1 / N1 dt = 0
1/b12
Stable coexistence
N2
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
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
R2
R1
Zero growthisocline
dN / N dt < 0
dN / N dt > 0Prey for most animals
Switching resources: One at a time
R2
R1
Zero growthisocline
dN / N dt < 0
dN / N dt > 0Nutritionallysubstitutable
Constraints onconsumption
Essential resources: both required
R2
R1
Zero growthisocline
dN / N dt < 0
dN / N dt > 0Soil nutrientsfor 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 = FmaxR / [K1/2 + R ]
– Fmax = maximal feeding rate– K1/2 = resource level for 1/2 maximal feeding
• 1/2 saturation constant
Feeding rate
R
F
Fmax
K1/2
• Holling type 2 Functional response
• Michaelis-Menten enzyme kinetics
• Monod microbial growth
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
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 ]
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
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]
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. 40-49• Chase & Leibold, pp. 62-63
Consumption of 2 resources
consumption vector: resultantof consumption of each resource
R1
R2Ci1
Ci2Ci
consumes more R1
Equilibrium: 1 sp. 2 resources
consumption vector equal &opposite supplyvector
R1
R2
Ci
Ci
Ci
U
S1,S2
U
U
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
Competition for 2 resources
R1
R2
sp. 1
S1,S2
S1,S2
S1,S2
sp. 2
sp. 1 alwaysexcludes sp. 2
sp. 2 cannotsurvive
neither spp.can survive
Competition for 2 resources
R1
R2
S1,S2
S1,S2
S1,S2 neither spp.can survive
sp. 2 cannotsurvive
sp. 1 alwaysexcludes sp. 2
S1,S2
coexistence
sp. 1
sp. 2
sp. 2
sp. 1