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 ] = r i [1- (1/K i )N i – (a j /K i )N j ]
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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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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