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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:

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slide1

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
slide2

Example:

r1 = 0.12

K1 = 170

a = 0.9

r2 = 0.09

K2 = 170

b = 0.5

slide3

Hierarchy of explanation

Changes in population size

(population dynamics)

births

deaths

migrations

Energy balance:

food availability

maintenance cost

cost of reproduction

Risk factors:

predator encounters

disease exposure

physical conditions

Behavior:

dispersal

foraging

group dynamics

slide4

Empirical models:

The observations required to estimate parameters are the very same that the model predicts (parameterization = calibrating, fitting).

observation

Population changes through time

MODEL

prediction

parameter estimation

slide5

Mechanistic models:

Some of the observations required to estimate parameters are at least one step removed from the level of prediction.

observation

Population changes through time

MODEL

parameter estimation

prediction

slide6

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.

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.

loss

Species A

growth

Biomass growth or loss rate

loss

Resource level

slide7

Tilman’s resource ratio model of plant competition

loss

loss

Species A

Species B

growth

Biomass growth or loss rate

loss

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)

Resource level

Resource level

slide8

Species could be competing for two resources:

loss

loss

Species A

Species A

growth

Biomass growth or loss rate

loss

Resource 1 level at fixed value of Resource 2

Resource 2 level at fixed value of Resource 1

slide9

Species depend on different resources in different ways:

The zero-net-growth-isoclines (ZNG’s)

R2

R2

R2

R2*

R1

R1

R1

R1*

Resources are perfectly essential

Resources are complementary

Resources are perfectly

substitutable

slide10

Adding resource dynamics

R2

Resource consumption vector

Resource

supply point; what resources would be without uptake

Resource supply

vector

R1

slide11

At equilibrium, both consumers and resources must be unchanging.

Thus, resource supply = resource demand:

R2

This is where consumer and resource are at equilibrium

Resource consumption vector

Resource

supply point; what resources would be without uptake

Resource supply

vector

R1

slide12

Prediction: if different habitats have different resource supply points, resource levels at equilibrium will be different.

R2

Resource consumption vector

Resource

supply point; what resources would be without uptake

Resource supply

vector

R1

slide17

Habitat determines if coexistence is possible.

B wins, because it

can draw R1 to levels intolerable to A.

R2

B

A

R1

R1

slide18

Habitat determines if coexistence is possible.

A wins, because it

can draw R2 to levels intolerable to B.

R2

B

A

R1

R1

slide19

Habitat determines if coexistence is possible.

A wins, because it

can draw R2 to levels intolerable to B.

R2

Species A & B coexist

Species B wins

Species A wins

B

A

Both species die

R1

R1

slide20

Tilman’s model still predicts the four outcomes of competition that the Lotka-Volterra model does,

and one more: no species lives

A always wins

B always wins

A

B

B

A

A & B can coexist

in some habitats

A & B can coexist

in some habitats

B always wins

A always wins

B

B

A

A

slide21

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.