Coexistence in competitive communities
Interspecific competition
Lotka-Volterra competition equations:
\[\frac{dN_1}{dT} = r_1N_1(1-\alpha_{11}N_1 - \alpha_{12}N_2)\]
\[\frac{dN_2}{dT} = r_2N_2(1-\alpha_{21}N_1 - \alpha_{22}N_2)\]
Possible outcomes of two species competing:
- Both species can have stable coexistence
- Species 1 can win, and exclude Species 2
- Species 2 can win, and exclude Species 1
- “It depends” – whichever species comes first, wins in competition.
How to distinguish between the possible options?
Zero net-growth isocline (ZNGI) analysis
Key question: Under what conditions does each species reach equilibrium?
\(\frac{dN_1}{dt} = 0\) and \(\frac{dN_2}{dt} = 0\)
Species 1’s equilibrium analysis:
- \(\frac{dN_1}{dt} = 0\) at two extreme conditions:
- Species 1 is at its carrying capacity, and Species 2 is absent (\(N_1 = 1/\alpha_{11},~ N_2 = 0\))
- Species 2 is very abundant, and Species 1 is nearly absent (\(N_1 = 0,~ N_2 = 1/\alpha_{12}\))
- \(\frac{dN_2}{dt} = 0\) at two extreme conditions:
- Species 2 is at its carrying capacity, and Species 1 is absent (\(N_1 = 0,~ N_2 = 1/\alpha_{22}\))
- Species 1 is very abundant, and Species 2 is nearly absent (\(N_1 = 1/\alpha_{21}, ~ N_2 = 0\))
Visualizing the equilibrium points
Species 1’s equilibrium analysis:
- Species 1 is at its carrying capacity, and Species 2 is absent (\(N_1 = 1/\alpha_{11},~ N_2 = 0\))
- Species 2 is very abundant, and Species 1 is nearly absent (\(N_1 = 0,~ N_2 = 1/\alpha_{12}\))
Visualizing the equilibrium points
Species 2’s equilibrium analysis:
Species 2 is at its carrying capacity, and Species 1 is absent (\(N_1 = 0,~ N_2 = 1/\alpha_{22}\))
Species 1 is very abundant, and Species 2 is nearly absent (\(N_1 = 1/\alpha_{21}, ~ N_2 = 0\))
Jointly visualizing both species’ isoclines
Putting numbers to the variables
Consider the following interaction coefficients:
\[\alpha_{11} = 0.005;~ \alpha_{12} = 0.002;~ \alpha_{22} = 0.007;~ \alpha_{21} = 0.002\]
\[1/\alpha_{11} = 200;~ 1/\alpha_{12} = 500;~ 1/\alpha_{22} = 142;~ 1/\alpha_{21} = 500\]
Putting numbers to the variables
Consider the following interaction coefficients:
\[\alpha_{11} = 0.005;~ \alpha_{12} = 0.002;~ \alpha_{22} = 0.01;~ \alpha_{21} = 0.007\]
\[1/.005 = 200;~ 1/.002 = 500;~ 1/.01 = 100;~ 1/.007 = 142\]
Recall that the equilibrium points for the two species are:
- Species 1: (\(N_1 = 1/\alpha_{11},~ N_2 = 0\)) and (\(N_1 = 0,~ N_2 = 1/\alpha_{12}\))
- Species 2: (\(N_1 = 0,~ N_2 = 1/\alpha_{22}\)) and (\(N_1 = 1/\alpha_{21},~ N_2 = 0\))
Your turn
- Draw the isoclines for a pair of species whose interaction coefficients are as follows:
\[\alpha_{11} = 0.008;~ \alpha_{12} = 0.008;~ \alpha_{22} = 0.006;~ \alpha_{21} = 0.005\]
\[1/0.008 = 125;~ 1/0.008 = 125;~ 1/.006 = 166;~ 1/.005 = 200\]
Your turn
- Draw the isoclines for a pair of species whose interaction coefficients are as follows:
\[\alpha_{11} = 0.002;~ \alpha_{12} = 0.005;~ \alpha_{22} = 0.002;~ \alpha_{21} = 0.007\]
\[1/.002 = 500;~ 1/.005 = 200;~ 1/.002 = 500;~ 1/007 = 142\]
Recall that the equilibrium points for the two species are:
- Species 1: (\(N_1 = 1/\alpha_{11},~ N_2 = 0\)) and (\(N_1 = 0,~ N_2 = 1/\alpha_{12}\))
- Species 2: (\(N_1 = 0,~ N_2 = 1/\alpha_{22}\)) and (\(N_1 = 1/\alpha_{21},~ N_2 = 0\))
Coexistence under competition
Possible outcomes in Lotka-Volterra competition
- Both species can coexist
- If each species limits itself more than it limits the other
- More “intra-specific competition” than “inter-specific competition”
- Species 1 wins, while Species 2 is excluded
- Species 2 wins, while Species 1 is excluded
- “It depends”
We are often interested in understanding how it is that species can coexist
- Organisms need basically the same set of resources, yet in many systems we see lots of diversity
- Alternatively, this diversity is challenged in some systems, e.g. due to species invasions.
- So, how can competition promote coexistence?
Possible outcomes under competition:
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- We also saw “it depends” as an outcome – let’s hold off on that one for now.
- What gives rise to scenario 1 (coexistence) vs. scenarios 2 or 3 (competitive dominance)?
Making sense of competitive dominance
- Let’s start with competitive dominance, as it is conceptually more straightforward
- Dominance will happen whenever one isocline is “outside” the other, i.e. no intersection
- There can be no condition under which \(\frac{dN_1}{dt} = 0\) AND \(\frac{dN_2}{dt} = 0\), which means no equilibrium possible with both species
- So, the only equilibrium points are when one species is absent.
Why would one isocline be “outside” another?
Why would one isocline be “outside” another?
- Consider a case where species 2 “wins”:
Why would one isocline be “outside” another?
- Consider a case where species 2 “wins”:
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Why would one isocline be “outside” another?
- Consider a case where species 2 “wins”:
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Why would one isocline be “outside” another?
- Consider a case where species 2 “wins”:
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Species 1 is strongly limited by Species 1, AND
it is strongly limited by Species 2
What supports species coexistence?
Let’s follow a similar logic for understanding why competition could support coexistence
What supports species coexistence?
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What supports species coexistence?
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What supports species coexistence?
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Biologically, this means that for coexistence to occur, each species should limit itself more than it limits the other
- When would this happen?
- If species have distinct niches, competition might be stronger within a niche than between niches.
Conclusion:
Coexistence in competitive communities
Photo by Gaurav, of Tejon Ranch in southern California
Why study species coexistence?
- Ecological systems are often incredibly diverse
- But sometimes, they are not – as in the case of species invasion rapidly eroding biodiversity
- So, the question arises: Why can certain species coexist, but others cannot?
Photo by Gaurav, of Tejon Ranch in southern California
How to study coexistence?
- There’s often lots of species coexisting, but we can take a reductionist approach
- Model the dynamics of two species at a time
- Analyze whether the species can coexist at equilibrium
- Graphical analysis using zero-net growth isoclines on Phase Spaces
Photo by Gaurav, of Tejon Ranch in southern California
How to move beyond this reductionist approach
- Extending the competition model to many species within a guild
- Matrix math, but this extension is very much based on the same principles of equilibrium and stability
- How do interactions across different guilds affect coexistence?
- E.g. can predators affect coexistence of two prey species?
- E.g. can fungal mutualists affect the coexistence of tree species? (focus of Gaurav’s PhD research)
- How can we conduct experiments to quantify these effects?
Coexistence under competition
Competitive coexistence
Coexistence can happen when:
- Individuals limit members of their own species more than other species
- \(\alpha_{11} > \alpha_{21}\) and \(\alpha_{22} > \alpha_{12}\)
- This is expected when there is some degree of “niche separation” among species
- Coexistence is limited if one species experiences the brunt of competition and the other doesn’t
- If this is true, then one of the isoclines will be “inside” the other – no coexistence equilibrium exists.
What gives rise to niche separation?
- Classically, resource separation
- E.g. two herbivore species eat grass, but each has its preferred grass
What gives rise to niche separation?
- Time is also a resource
- E.g. two plants need the exact same set of resources, but one active in early-spring and other in late-spring
What gives rise to niche separation?
- Space is also a resource
- E.g. two birds eat similar insects, but primarily hunt in distinct microhabitats
Other mechanisms
- Feedbacks between plants and microbes in the soil
Current state
- Soil microbial feedbacks shape plant coexistence in all terrestrial ecosystems
- But in some cases they promote coexistence (\(\alpha_{11} > \alpha_{21}\) and \(\alpha_{22} > \alpha_{12}\)), and in other cases they promote exclusion
- Understanding how this happens, which microbes are driving the effects, and how this is affected by climate warming, drought, fires, nutrient enrichment remains an open question.
Next week
- Moving beyond pairwise competition, to think about other interactions in communities
- Consumer–resource (e.g. predator-prey, herbivore-prey) and disease dynamics
