Organisms as resources and consumers
Part 2
Overarching questions in Consumer–resource dynamics
- What allows persistence of consumers and resources in the long term?
- Are there conditions in which consumers “over-consume” and engineer their own demise?
- How does perturbation of the system (e.g. removal of a “top predator”) affect the network as a whole?
Historical inspiration
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Resource dynamics (\(R\)):
\[\frac{dR}{dt} = \text{exponential growth - loss to consumer}\]
Consumer dynamics (\(C\):
\[\frac{dC}{dt} = \text{growth from consumption - energy loss}\]
\[\frac{dR}{dt} = rR - aRC\]
\[\frac{dC}{dt} = eaRC - mC\]
- This model features tightly coupled dynamics
- Changes in one population immediately cause feedbacks onto the other population
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- In-built “delay”: peaks in resources lead to peaks in consumers
Let’s think about the “equilibrium” in this model
Is this system at equilibrium?
Do the population dynamics always reach the same cycles?
Let’s perturb the system by slightly reducing \(R\)
The populations always cycle, but the amplitude of the cycle depends on the initial size of the populations
“Neutral oscillations” - a different type of equilibrium
Amplitude of cycles change, but average remains the same!
The oscillations happen around a central equilibrium point
Review of equilibrium stability
Equilibrium happens whenever \(\frac{dN}{dt} = 0\).
Stability of an equilibrium reflects what happens if a system is “pushed”
Stable equilibrum: System comes back to the original point
Unstable equilibrium: System keeps moving away from the original point, in the direction of the “push”
Neutral equilibrium: Once pushed, the system doesn’t return to the origina point, but doesn’t keep “moving away”
Challenge question: What is the equilibrium condition for this model?
\[\frac{dR}{dt} = rR - aRC\]
\[\frac{dC}{dt} = eaRC - mC\]
- Solve the \(\frac{dR}{dt}\) equilibrium with respect to \(C\) (i.e. \(C = ?\) for \(\frac{dR}{dt} = 0\)?)
- Solve the \(\frac{dC}{dt}\) equilibrium with respect to \(R\) (i.e. \(R = ?\) for \(\frac{dC}{dt} = 0\)?)
Challenge: Draw the Zero Net-Growth Isoclines for the predator–prey model.
Dicrostonyx torquatus - Northern collared lemming
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Mustela erminea - Stoat/“ermine”
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Organisms as consumers and resources
Recall the canonical model for consumer–resource dynamics:
\[\frac{dR}{dt} = rR - aRC\]
\[\frac{dC}{dt} = eaRC - mC\]
How do we find the equilibrium conditions of this model?
Resource equilibrium (\(\frac{dR}{dt} = 0\)) happens at \(C = \frac{r}{a}\)
Consumer equilibrium (\(\frac{dC}{dt} = 0\)) happens at \(R = \frac{m}{ea}\)
Trophic networks in ecological systems
What are the consequences of perturbing a whole network?
Let’s start by considering a tri-trophic interaction network.
“Enemy of my enemy is my friend”
But, trophic networks can get further complicated
Let’s draw a schematic of this trophic network
Consumer–Resource wrapup and mid-semester checkin
Importance of consumers in maintaining ecosystems
Mid-semester retrospective
Overview of the past eight weeks
Population ecology
- Questions that we asked:
- How can population sizes be estimated?
- Under what conditions do populations grow, shrink, or stay constant?
- \(\frac{dN}{dt} > 0, ~ \frac{dN}{dt} < 0, ~ \frac{dN}{dt} = 0\)
Population ecology
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- Estimating population sizes through Mark–recapture approaches
Population ecology
How do populations grow and shrink?
A simple model with no population structured; constant reproduction rate and death rate
Expand to structured populations (not all individuals have the same rates of reproduction/mortality)
What if birth/reproduction rates respond to population size?
Assumes closed populations - no migration
Overview of the past eight weeks
Community ecology
Overview of past eight weeks
Reductionism as an approach to study ecological communities
Often times, we see lots of “similar” species coexisting
Multiple herbacious plant species; multiple trees; multiple seed-eating birds, multiple predatory birds, etc.
Key question: What determines diversity of coexisting species that compete for shared resources?
Overview of past eight weeks
Key question: What determines diversity of coexisting species that compete for shared resources?
- Expand from \(\frac{dN}{dt}\) to \(\frac{dN_1}{dt}\) and \(\frac{dN_2}{dt}\)
- Introduced a new form of graphical analysis: state-space analysis
- Coexistence requires stronger competition within species than between species
Overview of past eight weeks
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- What about interactions among consumers and resources?
Consumer–resource interactions
Reductionist approach: start with one consumer and one resource
Analysis: Again, through isocline analysis
Insight: population cycling is likely – thus, oscillating equilibrium
Consumer–resource interactions
Consumer–resource interactions
What escapes the eye, however, is a much more insidious kind of extinction: the extinction of ecological interactions
Next steps in the course
Ecosystems ecology
- What governs the movement of energy and nutrients through ecological communities?
- How do human disruptions of ecological systems impact ecosystem processes?
