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? (\(\frac{dR}{dt}\) and \(\frac{dC}{dt}\))

Resource species dynamics

• The resource species has plenty of resources available and can grow exponentially…
• … but it is suppressed by the consumer.

\[\frac{dR}{dt} = \text{exponential growth - loss to consumer}\]

\[\frac{dR}{dt} = rR - aRC\]

• \(R\) 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)

\[\frac{dC}{dt} = \text{growth from consumption - energy loss}\]

\[\frac{dC}{dt} = eaRC - mC\]

• \(C\) 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

\[\frac{dR}{dt} = rR - aRC\]

\[\frac{dC}{dt} = eaRC - mC\]

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

\[\frac{dR}{dt} = rR - aRC\]

\[\frac{dC}{dt} = eaRC - mC\]

Some more takeaways:

• If the consumer attacks the resource at a very high rate (\(\uparrow a\)), resource populations will be suppressed, and consumer population grows more rapidly
• If the consumer is very efficient at getting energy from the resource (\(\uparrow e\)), the consumer populations grow more rapidly
• If consumer has high maintenance cost (\(\uparrow m\)), 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?

\[\frac{dR}{dt} = rR - aRC\]

\[\frac{dC}{dt} = eaRC - mC\]

• 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 \(\frac{dC}{dt} = 0\)?)