From Lotka-Volterra to mechanism:
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From Lotka-Volterra to mechanism: Simple models have advantages: capturing essential features of dynamical systems with minimal mathematical effort tractable, relatively easy to analyze in full can be parameterized from observation However, they have limited utility: Parameter values are difficult to predict a prior from knowledge of the system
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From Lotka-Volterra to mechanism: . Simple models have advantages: capturing essential features of dynamical systems with minimal mathematical effort tractable, relatively easy to analyze in full can be parameterized from observation However, they have limited utility: - PowerPoint PPT Presentation
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From Lotka-Volterra to mechanism:
Simple models have advantages:
capturing essential features of dynamical systems with minimal mathematical effort
tractable, relatively easy to analyze in full
can be parameterized from observation
However, they have limited utility:
Parameter values are difficult to predict a prior from knowledge of the system
Example:
0 5 10 15 20 250
20
40
60
80
100
120
140
160
Series1Series3
r1 = 0.12K1 = 170a = 0.9 r2 = 0.09K2 = 170b = 0.5
Changes in population size(population dynamics)
Hierarchy of explanation
births migrationsdeaths
Energy balance:food availability
maintenance costcost of reproduction
Risk factors:predator encounters
disease exposurephysical conditions
Behavior:dispersalforaging
group dynamics
Empirical models:
The observations required to estimate parameters are the very same that the model predicts (parameterization = calibrating, fitting).
Population changes
through time
observation
MODEL
parameter estimation
prediction
Mechanistic models:
Some of the observations required to estimate parameters are at least one step removed from the level of prediction.
Population changes
through time
observation
MODELparameter estimation prediction
Tilman’s resource ratio model of plant competition
Observations used to parameterize the model describe resource uptake by plants. Hence, this is a mechanistic model.
loss
Resource level
Biom
ass g
row
th o
r los
s rat
e
growth
loss
Species A
*AR
*AR is the minimal amount
Of resource species A requires to persist in an environment;
If RA is supplied at a certain rate, the species should increase until the resource concentration reaches exactly this value.
Tilman’s resource ratio model of plant competition
loss
Resource level
Biom
ass g
row
th o
r los
s rat
e
growth
loss
Species A
*AR
loss
Resource level
Species B
*BR
When two species are competing for a single limiting resource, the species with the lower equilibrial resource requirement should completely replace the other (B outcompetes A)
Species could be competing for two resources:
loss
Resource 1 level at fixed value of
Resource 2
Biom
ass g
row
th o
r los
s rat
e
growth
loss
Species A
*,1 AR
loss
Resource 2 level at fixed value of
Resource 1
Species A
*,2 AR
Species depend on different resources in different ways:The zero-net-growth-isoclines (ZNG’s)
R1
R2
Resources are perfectly substitutable
R1
R2
Resources are complementary
R1
R2
Resources are perfectly essential
R2*
R1*
Adding resource dynamics
R1
R2
Resource consumption vector
Resourcesupply point; what resources would be without uptake
Resource supplyvector
At equilibrium, both consumers and resources must be unchanging.
Thus, resource supply = resource demand:
R1
R2
Resource consumption vector
Resourcesupply point; what resources would be without uptake
Resource supplyvector
This is where consumer and resource are at equilibrium
Prediction: if different habitats have different resource supply points, resource levels at equilibrium will be
different.
R1
R2
Resource consumption vector
Resourcesupply point; what resources would be without uptake
Resource supplyvector
Species with different resource requirements affect resource levels differently:
R1
R2
R1
What if two species with different resource requirements inhabit the same habitat?
R1
R2
R1
What if two species with different resource requirements inhabit the same habitat?
R1
R2
R1
A two-species equilibrium must be located on both species’ ZNG’s
R1
R2
R1
A and B coexist
A
B
Habitat determines if coexistence is possible.
R1
R2
R1
B wins, because itcan draw R1 to levels intolerable to A.
A
B
R1
R2
R1
A wins, because itcan draw R2 to levels intolerable to B.
A
B
Habitat determines if coexistence is possible.
R1
R2
R1
A wins, because itcan draw R2 to levels intolerable to B.
A
B
Spec
ies B
win
sSpecies A wins
Species A & B coexist
Habitat determines if coexistence is possible.
Both species die
Tilman’s model still predicts the four outcomes of competition that the Lotka-Volterra model does,
and one more: no species lives
AB
A always wins
AB
B always wins
AB
A & B can coexistin some habitats
AB
A & B can coexistin some habitats
B alw
ays w
ins
A always wins
Summary:
What do we gain from Tilman’s more mechanistic model?
Resource requirements for growth can be tested independently of competition.
New predictions: the effect of habitat on species interaction.
Previously overlooked outcomes: both species can fail.
There are predictions we can test and which can fail.
Because the model is process based, we can more easily expand the model to add
more realism.
