Organisms as resources and consumers

From self reflections

  • How do ecologists actually use simple models?
  • What are ways in which these reductionist approaches help us make sense of the whole?

General insight: more coexistence under stronger within-species limitation than between-species limitation

  • Is there more biodiversity in forests where within-species limitation is stronger?

Figure from Hulsman et al. 2024 in Nature, “Latitudinal patterns in stabilizing density dependence of forest communities”

General insight: Relative interaction strengths shape competitive outcomes

  • Can we understand pathogen co-infections based on coexistence theory?

Figure from Park et al. in Science, 2024, “Predicting pathogen mutual invasibility and co-circulation”

We address this gap by developing a pathogen invasion theory (PIT) based on modern ecological coexistence theory and testing the resulting framework against empirical systems. […] PIT unifies existing ideas about pathogen co-circulation, offering a quantitative framework for predicting the emergence of novel pathogen strains

In the previous unit, we explored how a reductionist approach can be applied to the study of species competition:

What can we learn from a similar approach to consumer–resource dynamics?

Building a simple consumer–resource system

  • 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

Equlibrium in C-R systems

Given the consumer–resource dynamics equations:

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

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

Where does equilibrium occur?

  • How can we illustrate this in state-space?

Equilibrium analysis of Resource

\[\frac{dR}{dt} = rR - aRC = 0\]

\[rR = aRC\]

\[C^* = \frac{r}{a}\]

Interpretation: There is a set abundace of the consumer \(C\) that would cause the resource \(R\) to be at equilibrium.

  • If the resource grows slowly (low \(r\)), this happens at smaller abundances of the consumer
  • And/or if the consumer attacks the resource at a high rate (high \(a\)), this happens at smaller abundances of the consumer

Equlibrium analysis of consumer

\[\frac{dC}{dt} = eaRC - mC = 0\]

\[eaRC = mC\]

\[R = \frac{m}{ea}\]

Interpration: There is a set abundance of the resource \(R\) that would cause the consumer to reach equilibrium

  • If the consumer has a high maintenance cost \(m\), this happens at a high value of \(R\)
  • If the consumer has high attack rate and/or a high conversion efficency, it equlibrates at lower abundances of the resource.

Putting the isoclines together

This suggests that populations should cycle. Do they?

Population cycling in the model

Population cycling in nature

Image from “Population cycles: generalities, exceptions and remaining mysteries”, Meyers 2018 in Proc. Royal Soc. B.

Image from Blasius et al. “Long-term cyclic persistence in an experimental predator–prey system”, 2020, in Nature. These authors were studying a lab population of an aquatic rotifer (consumer) and a grean algae (resource)