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