Organisms as resources and consumers
Keep track of the (potential) consumer–resource interactions:
Overarching questions
- 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 context
- In the 1920s, Humberto D’Ancona was working on the statics of fish caught and sold at markets in Fiume (an Italian port city)
- Interested in understanding trends in the relative number of big “predator” fish (sharks, skates, rays, etc.) vs. smaller “prey” fish
- Interesting patterns the data:
- The rate of large (prey) fish being caught was higher during World War 1 than before or after.
- Did the war have something to do with the fish population dynamics?
- Turned to his soon-to-be father in law Vito Volterra for insight. (Volterra was a celebrated mathematician.)
- Main question: Why are percentages of predators vs. prey fluctuating over time?
Illustrated with a phase-plane:
It turns out Volterra was not alone in coming up with this idea.
Now, their shared idea has shaped ecology for the past 100 years, and is called the “Lotka-Volterra model”.
What was the idea that both of these people shared?
Consider the dynamics of a “consumer” species C and a “resource” species R
The resource species has plenty of resources available and can grow exponentially…
… but it is suppressed by the consumer.
The consumer is a specialist (only eats the “resource” species)
(If there is no resource around, the consumer population dies out)
The consumer loses energy at some rate (maintenance cost)
How should the abundances of consumers and resources change over time? (dRdt and dCdt)
Resource species dynamics
- The resource species has plenty of resources available and can grow exponentially…
- … but it is suppressed by the consumer.
dRdt=exponential growth - loss to consumer
dRdt=rR−aRC
- R is the abundance of the resource species
- r is the intrinsic growth rate of the resource
- a is the attack rate of the consumer on the resource
- C is the abundance of the consumer species
Consumer species dynamics
- The consumer is a specialist (only eats the “resource” species)
- The consumer loses energy at some rate (maintenance cost)
dCdt=growth from consumption - energy loss
dCdt=eaRC−mC
- C is the abundance of the consumer species
- a is the attack rate of the consumer on the resource
- e is the efficiency with which consumer uses the energy it gets from resource
- R is the abundance of the resource species
- m is the maintenance cost of the consumer species
Lotka-Volterra consumer-resource system
dRdt=rR−aRC
dCdt=eaRC−mC
Some “obvious” takeaways:
- If consumer species is absent (C=0), the resource species grows exponentially
- If the resource species is absent (R=0), the consumer species dies out
Lotka-Volterra consumer-resource system
dRdt=rR−aRC
dCdt=eaRC−mC
Some more takeaways:
- If the consumer attacks the resource at a very high rate (↑a), resource populations will be suppressed, and consumer population grows more rapidly
- If the consumer is very efficient at getting energy from the resource (↑e), the consumer populations grow more rapidly
- If consumer has high maintenance cost (↑m), consumer populations will be suppressed
Population coupling
- This model features tightly coupled dynamics
- Changes in one population immediately cause feedbacks onto the other population
- In-built “delay”: peaks in resources lead to peaks in consumers
Relevance to Volterra’s inspiration
Without any additional complexity, this model can provide an explanation for changes in proportion of prey:predator caught and sold at fish markets
Discussion: Is this system at equilibrium?
Do the population dynamics always reach the same cycles?
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
Challenge question: What is the equilibrium condition for this model?
dRdt=rR−aRC
dCdt=eaRC−mC
- Solve the dRdt equilibrium with respect to C (i.e. C=? for dRdt=0?)
- Solve the dCdt equilibrium with respect to R (i.e. R=? for dCdt=0?)
Organisms as resources and consumers