alternative lotka-volterra competition
DESCRIPTION
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 ] - PowerPoint PPT PresentationTRANSCRIPT
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
Resources for 0 growth
dN / N dt = 0
R*
dN / N dt > 0dN / N dt < 0
R0
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 ]
Limitation by 1 resource
R
dN / N dt
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 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]
Competition for 1 resource
sp. 1
R
dN / N dt
R*1
-m1
0
R*2-m2
sp. 2
Dynamics of competition for 1 resource
t
N
R*1
R*2
R
R
sp. 1SP.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
R2 Ci1
Ci2Ci
consumes more R1
Essential resources
consumption vectors are parallel(essential)
R1
R2 Ci1
Ci2C1
Substitutable resources
consumption vectors are not parallel(substitutable)
R1
R2 Ci1
Ci2Ci
Switching resources
consumption vectors are perpendicularto isocline(switching)
R1
R2
C1
Renewal for 2 resources
supply vector: points at supplypoint S1,S2
R1
R2
S1,S2
U
Equilibrium: 1 sp. 2 resources
consumption vector equal &opposite supplyvector
R1
R2
Ci
Ci
Ci
U
S1,S2
UU
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
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