flowchart BT Resource --> Mid-consumer Mid-consumer --> Top-consumer
General insight: more coexistence under stronger within-species limitation than between-species limitation
General insight: Relative interaction strengths shape competitive outcomes
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?
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 \(\frac{dC}{dt}\))
\[\frac{dR}{dt} = \text{exponential growth - loss to consumer}\]
\[\frac{dR}{dt} = rR - aRC\]
\[\frac{dC}{dt} = \text{growth from consumption - energy loss}\]
\[\frac{dC}{dt} = eaRC - mC\]
\[\frac{dR}{dt} = rR - aRC\]
\[\frac{dC}{dt} = eaRC - mC\]
Some “obvious” takeaways:
\[\frac{dR}{dt} = rR - aRC\]
\[\frac{dC}{dt} = eaRC - mC\]
Some more takeaways:
Given the consumer–resource dynamics equations:
\[\frac{dR}{dt} = rR - aRC\]
\[\frac{dC}{dt} = eaRC - mC\]
Where does equilibrium occur?
\[\frac{dR}{dt} = rR - aRC = 0\]
\[rR = aRC\]
\[C^* = \frac{r}{a}\]
Interpretation: There is a set abundace of the consumer \(C\) that would cause the resource \(R\) to be at equilibrium.
\[\frac{dC}{dt} = eaRC - mC = 0\]
\[eaRC = mC\]
\[R = \frac{m}{ea}\]
Interpration: There is a set abundance of the resource \(R\) that would cause the consumer to reach equilibrium
This suggests that populations should cycle. Do they?
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)
Simplified model of consumer–resource dynamics, with a number of assumptions:
Simplified model of consumer–resource dynamics:
\[\frac{dR}{dt} = rR - aRC\]
\[\frac{dC}{dt} = eaRC - mC\]
Set \(\frac{dR}{dt} = 0\) and \(\frac{dC}{dt} = 0\)
\[\frac{dR}{dt} = rR - aRC\]
\[\frac{dC}{dt} = eaRC - mC\]
Resource equilibrium occurs when \(C = \frac{r}{a}\),
Consumer equilibrium occurs when \(R = \frac{m}{ea}\)
This suggests that consumer and resource populations should be fluctuating
The population abundances do fluctuate in the model dynamics…
… and there’s also evidence that this happens in nature
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)
The classical model makes some assumptions
But, we now know enough about simple models to see how we can make different pieces fit together
e.g. Consider a pair of resource species (e.g. two grasses that are consumed by a rabbit; two fish species that are consumed by eagles) that also compete with one another
What would a model of their interactions look like?
Let’s start by considering a tri-trophic interaction network.
flowchart BT Resource --> Mid-consumer Mid-consumer --> Top-consumer
Origins of this idea
Nuclear testing at Amchitka island; see here for more details
The nuclear testing affected otters…
James A. Estes, a University of Arizona doctoral candidate in biology under Atomic Energy Commission contract for an otter study made the estimate of 900 to 1,100 dead otters. (From New York Times, December 1971)
And Jim Estes continued studying these islands, and its otters, through a long career.
Estes chronicles his career in the 2016 memoir “Serendipity : An Ecologist’s Quest to Understand Nature”; ebook available through LSU Library
Trophic cascades as important processes structuring ecosystems around the world