Using modern coexistence theory to understand community disassembly

This paper introduces a framework based on Modern Coexistence Theory and community disassembly graphs to identify the timing and mechanistic causes of secondary extinctions across various species interactions, offering a flexible approach to understanding community disassembly.

The Gist

Imagine an ecosystem as a complex, bustling city. In this city, every species is a resident with a specific job, and they all rely on a delicate web of relationships to survive. Some neighbors compete for the same apartment (resources), some help each other move furniture (facilitation), and some are the police force keeping the rowdy gangs in check (predation).

This paper, titled "Using Modern Coexistence Theory to Understand Community Disassembly," asks a critical question: What happens when one resident suddenly moves out (goes extinct)?

Often, one person leaving doesn't just leave an empty apartment; it triggers a domino effect where other neighbors are forced to leave too. This is called a secondary extinction. The authors, Joe Brennan and Sebastian Schreiber, want to figure out when this happens and, more importantly, why.

Here is the breakdown of their approach using simple analogies:

1. The Problem: The "Static Map" vs. The "Live City"

Previous studies tried to predict these domino effects using static maps (like a subway map). They would look at the lines connecting stations and say, "If Station A closes, Station B loses its connection and must close."

• The Flaw: This only works for obvious dependencies (like a predator that only eats one specific prey). It misses the invisible, dynamic forces.
• The Real World: In a live city, if the police chief (a keystone predator) leaves, the gangs might start fighting over territory, causing innocent bystanders to flee. A static map wouldn't predict this because the police chief wasn't directly connected to the bystanders on the map. The authors argue we need to study the dynamics—the actual movement and interactions—to see the real danger.

2. The Solution: The "Community Disassembly Graph"

To solve this, the authors invented a new tool called the Community Disassembly Graph. Think of this as a choose-your-own-adventure book for a city.

• The Nodes (Pages): Each page represents a possible version of the city (e.g., "City with 5 species," "City with 4 species").
• The Edges (Arrows): The arrows show what happens if you remove one resident.
  • Black Arrow: You remove a resident, and the city settles down with the remaining neighbors. Everyone is happy.
  • Yellow Arrow: You remove a resident, and it causes a secondary extinction. Another neighbor is forced out immediately.
  • Red Arrow: The specific "disaster" event the authors are analyzing.

This graph allows them to trace every possible path of collapse, showing exactly which removals trigger a chain reaction.

3. The Detective Work: "Invasion Growth Rates"

Once they identify a disaster (a yellow or red arrow), they need to know why it happened. Did the remaining neighbor leave because they were hungry? Because they were bullied? Because they lost a friend?

They use a concept called Invasion Growth Rates (IGR).

• The Analogy: Imagine a new person trying to move into a neighborhood.
  • If their IGR is positive, they can move in, grow their family, and thrive. The neighborhood is stable.
  • If their IGR is negative, they can't survive there; they will eventually leave or die out.

The authors use a mathematical "decomposition" (like taking apart a clock to see which gear is broken) to break down the IGR. They ask:

• Was it the competition? (The neighbors were too aggressive).
• Was it the loss of help? (A friend who used to help them move was gone).
• Was it the predator's absence? (The police chief left, so the gangs took over).

4. Three Case Studies (The Experiments)

The authors tested their theory on three different "cities" to prove it works:

• Case 1: The Annual Plant City (Competition)

  • Scenario: Three plants compete for water.
  • The Twist: Plant A and Plant B are fierce rivals. Plant C is weak. However, because A and B are fighting each other so hard, they keep each other's populations down, leaving just enough water for Plant C to survive.
  • The Disaster: If Plant A dies, Plant B stops fighting and grows huge, choking out Plant C.
  • The Lesson: Sometimes, a "bad" neighbor (Plant A) is actually keeping the "worse" neighbor (Plant B) in check, saving the weak one.

• Case 2: The Grassland City (Facilitation)

  • Scenario: A group of grasses and clovers. One clover (T. repens) acts like a "social worker," helping another clover (T. pratense) survive the harsh competition.
  • The Disaster: When the "social worker" clover dies, the other clover loses its support system and is immediately crushed by the competition.
  • The Lesson: We often forget that species rely on help, not just on food. Losing a helper can be just as deadly as losing a food source.

• Case 3: The Diamond City (Predation)

  • Scenario: A resource (R) feeds two competitors (C1 and C2), and a top predator (P) eats both.
  • The Twist: The predator prefers to eat C1. This keeps C1's population low, giving C2 a chance to survive.
  • The Disaster: If the predator (P) dies, C1 explodes in numbers and eats all the resources, starving C2.
  • The Lesson: This is the classic "Keystone Predator" effect. The predator isn't just eating prey; it's acting as a referee that keeps the game fair for everyone.

The Big Takeaway

The authors show that extinction is rarely a simple chain reaction. It's a complex web of direct and indirect effects.

• Old Way: "Species A eats Species B, so if A dies, B dies." (Too simple).
• New Way: "Species A keeps Species B in check, which allows Species C to survive. If A dies, B takes over and kills C." (Accurate).

By using this "Community Disassembly Graph" and breaking down the math of why species fail to invade, we can better predict which species are the "keystones" holding the whole ecosystem together. It helps us understand that saving one species might be the only way to save the entire city from collapsing.

Technical Summary

1. Problem Statement

Ecological communities are complex networks where the extinction of a single species can trigger cascading losses of other species, known as secondary extinctions. While traditional approaches to studying community disassembly rely on static network analyses (e.g., food webs or mutualist networks), these methods have significant limitations:

• They typically model interactions as static links without accounting for changes in species densities or dynamics.
• They often fail to predict secondary extinctions driven by indirect effects, such as the loss of keystone predators, indirect facilitation, or complex competitive suppression.
• They struggle to disentangle the relative importance of multiple ecological drivers (e.g., competition vs. facilitation) when a secondary extinction occurs.

The authors argue that a dynamic framework is needed to not only identify when secondary extinctions occur but also to decompose why they happen by quantifying the specific ecological mechanisms that fail upon the loss of a species.

2. Methodology

The authors propose a framework integrating Modern Coexistence Theory (MCT) with community dynamics to analyze disassembly. The methodology consists of three core components:

A. The Community Disassembly Graph (CDG)

The CDG is a directed acyclic graph where:

• Vertices represent subsets of coexisting species (feasible attractors).
• Edges represent transitions between these communities caused by the removal of a single species.
• Mechanism: To determine the next state after removing species $i$ from community $S$, the authors utilize $-i$ communities. A community $T \subset S$ (excluding $i$) is the $-i$ community if all species in $S \setminus (T \cup \{i\})$ have negative Invasion Growth Rates (IGRs) at the equilibrium of $T$.
• Secondary Extinction Identification: If a species $j$ (where $j \neq i$) has a negative IGR in the resulting $-i$ community, a secondary extinction is predicted.

B. Invasion Growth Rate (IGR) Decomposition

To understand the mechanism of a secondary extinction, the authors decompose the IGR of the extinct species ($j$) using the method of Ellner et al. (2019).

• Comparison: They compare the IGR of species $j$ in two scenarios:
  1. The pre-extinction community (specifically the $-j$ community, where $j$ is missing but $i$ is present).
  2. The post-extinction community (the $-i$ community, where $i$ is gone and $j$ subsequently goes extinct).
• Decomposition Formula: The IGR $r_j$ is partitioned into:
  $$r_j(T, 1) = \Delta_0 + \sum \Delta_\ell + \sum \Delta_{\ell,m} + \Delta_{higher\_order}$$
  • $\Delta_0$: Baseline difference relative to competitors.
  • $\Delta_\ell$: Contribution of specific ecological factors (e.g., competition, facilitation, predation).
  • $\Delta_{\ell,m}$: Interactive effects between factors.
• Interpretation: By comparing the decomposition terms between the two communities, the authors identify which factors shifted from promoting persistence to causing exclusion.

C. Case Studies

The framework is applied to three distinct models:

1. Competition: An empirically parameterized annual plant model (Van Dyke et al., 2022) involving Acmispon wrangelianus, Hordeum murinum, and Plantago erecta.
2. Facilitation: A Lotka-Volterra grassland model (Geijzendorffer et al., 2011) involving competitive and facilitative interactions among five grass/legume species.
3. Predation: A "Diamond" model of keystone predation with a resource, two competitors, and a top predator.

3. Key Contributions

• Extension of MCT: The paper extends MCT concepts (IGRs and $-i$ communities), traditionally used for analyzing coexistence, to the study of community disassembly and secondary extinctions.
• Dynamic vs. Static: It demonstrates that dynamic approaches capture secondary extinctions driven by indirect effects (e.g., competitive release, loss of top-down control) that static network models miss.
• Mechanistic Decomposition: It provides a rigorous mathematical method to decompose the causes of extinction, distinguishing between direct effects (e.g., direct competition) and indirect effects (e.g., a competitor suppressing a superior competitor).
• Computational Framework: It outlines a numerical approach to constructing CDGs and calculating IGRs for complex systems, acknowledging computational limits for highly diverse communities.

4. Key Results

The application of the framework to the three case studies yielded specific insights:

• Competition (Annual Plants):

  • Finding: The extinction of A. wrangelianus caused the secondary extinction of P. erecta.
  • Mechanism: In the full community, A. wrangelianus competed with H. murinum, suppressing H. murinum's density. This suppression allowed the inferior competitor P. erecta to persist. When A. wrangelianus was removed, H. murinum was released from competition, grew to high density, and competitively excluded P. erecta.
  • Static Model Failure: A static network would not predict this, as A. wrangelianus is not a direct resource or mutualist for P. erecta.

• Facilitation (Grassland):

  • Finding: The extinction of T. repens caused the secondary extinction of T. pratense.
  • Mechanism: T. repens provided direct facilitation to T. pratense. Furthermore, in the full community, facilitation among competitors of T. pratense helped balance competitive pressures. Upon removal of T. repens, T. pratense lost direct facilitation and faced intensified indirect competitive pressure from the remaining species, leading to exclusion.
  • Insight: The IGR decomposition revealed that the loss of facilitation combined with stronger direct competition drove the extinction.

• Predation (Diamond Model):

  • Finding: The extinction of the top predator P caused the secondary extinction of consumer C2.
  • Mechanism: The predator preferentially consumed the superior competitor C1. This "keystone predation" suppressed C1, allowing C2 to coexist. Without the predator, C1 utilized the resource more efficiently and competitively excluded C2.
  • Decomposition: The analysis showed that the predator's preference was the primary positive factor enabling C2's persistence; its removal eliminated the top-down control necessary for coexistence.

5. Significance and Future Directions

• Predictive Power: The framework offers a flexible, interpretable tool for predicting community responses to species loss, crucial for conservation biology and managing biodiversity under anthropogenic change.
• Beyond Coexistence: It validates IGRs as a "common currency" for ecological research, applicable to both assembly (coexistence) and disassembly (extinction) processes.
• Limitations & Challenges:
  • Computational Cost: Constructing full CDGs for species-rich communities is computationally intensive ($2^n$ states). The authors suggest focusing on specific $-i$ communities for targeted robustness analysis.
  • Complex Dynamics: The current framework assumes convergence to a stable attractor. It requires extension to handle heteroclinic cycles (e.g., rock-paper-scissors dynamics) where trajectories oscillate between states.
  • Gradual Decline: The model assumes instantaneous extinction. It may be less informative for gradual environmental changes where species decline slowly, potentially triggering different indirect effects or alternative stable states.

In conclusion, Brennan and Schreiber provide a robust theoretical and computational bridge between Modern Coexistence Theory and community disassembly, enabling ecologists to move beyond static network topology to understand the dynamic, mechanistic drivers of biodiversity loss.