BI 559 Lecture 10: Multi-species Pt. II
Protoss
Zerg
resources
Starcraft 2 (?)
Today:
From last time
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2
Lotka-Volterra model
Growth rate of each species’ population
Per-capita growth rate: how fast each species doubles
Carrying capacity for each species: its maximum density under the current nutrient conditions
Competition coefficients: how strongly each species inhibits the other
There is a stable equilibtrium with non-zero densities of species 1 and species 2!
“coexistence”
Stable points: where the growth rates of each species are 0!
Coexisting species
Phase plot
Time-axis growth curve solutions
Model:
Phase plane:
Data:
When do we get stable coexistence?
Need two conditions:
and
If mutual inhibition is “low”, we get coexistence!
Let’s look at the other possibilities!
One species dominates
Phase plot
Time-axis growth curve solutions
When does one species dominate?
Need two conditions:
and
Third scenario:
Species 1 can dominate or species 2 can dominate. It depends on the initial cell densities!
“bistability”
Need two conditions:
and
If both species are competitive we get bistability!
To summarize
coexistence
bistability
single-species dominance
Is Lotka-Volterra a good model for 2-species competition and growth?
Today’s paper
Can the Lotka-Volterra model allow us to understand multi-species growth under changing conditions?
Can the Lotka-Volterra model predict multiple phases of bacterial growth?
Abreu et al.’s question:
If species compete
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2
but there is also an external agent killing both,
External agent
how does that effect who dominates or if they coexist?
vs
?
Why ask this question?
Species competing in a gut environment:
broad-spectrum antibiotic
Neither species is able to develop high density as it normally would. How does this affect mutual inhibition?
Why ask this question?
Species competing in an ocean environment:
Changing ocean temperature
How can we add such a global mortality effect into the LV model?
What does the model predict?
Abreu et al.’s hypothesis
Low competition → coexistence
Both species competitive → bistability
One species competitive → dominance
They hypothesize that in high-mortality conditions (e.g. antibiotics), fast growers will tend to dominate more!
How do you incorporate this into the model and then test it experimentally? We’ll step through both.
How do you use the Lotka-Volterra model to interrogate this question?
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2
A little math to get the equations into the Abreu et al. form:
(you’ll see why they did this on the next slide)
Why normalize the population density variable?
It makes for a very simple interpretation of the competition coefficients:
Stable coexistence if:
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1
Now stable coexistence if:
Abreu et al.’s new stability conditions:
coexistence
bistability
single-species dominance
Abreu et al. model to investigate competition in a harsh environment
Now they do something very clever. They rearrange their equations (this is a lot of math) to define new competition parameters that aren’t constants anymore. . . .
Abreu et al. model to investigate competition in a harsh environment
In this new model, competition is not a constant, but depends on:
Abreu et al. model to investigate competition in a harsh environment
Abreu et al.’s experiment
5 soil bacteria species:
Pseudomonas putida (Pp)
(sciencephoto.com)
Enterobacter aerogenes (Ea)
(microbewiki)
Pseudomonas aurantiaca (Pa)
Pseudomonas citronellolis (Pc)
(credit Dennis Kunkel)
Pseudomonas veronii (Pv)
Abreu et al.’s experiment
5 soil bacteria species:
(couldn’t find PC colony image)
Distinct colony morphologies allow plating of mixed cultures and counting colony-forming units of each species.
Abreu et al.’s experiment
Grown in a defined medium:
Gives a wide range of per-capita growth rates to probe the predictions of their LV model!
Abreu et al.’s experiment
E. aerogenes
P. veronii
Implement mortality by diluting culture
E. aerogenes
P. veronii
time
At regular time intervals, remove a small fraction of the mixed culture and dilute into fresh medium and discard the rest.
time
Abreu et al. model to investigate competition in a harsh environment
Measured for each strain on its own
Measured in constant condition experiments
Controlled experimentally
Controlled or measured!
What does the model predict?
Can experimentally control this.
Can measure this!
A very elegant way of plotting this prediction
Bistability
Winner depends on initial fraction
Coexistence
Fast wins
Slow wins
Each point on this plot represents an entire 2-species system.
Its position corresponds to what stable point the L-V model predicts.
A phase diagram of phase diagrams!
The model predicts specific transitions
These two conditions can be realized with their strain combinations.
This can be manipulated by increasing the dilution (i.e. death/mortality) rate!
Then they can measure whether they observe these transitions.
Each point on this plot represents an entire 2-species system.
Its position corresponds to what stable point the L-V model predicts.
A phase diagram of phase diagrams!
What do they observe experimentally?
What do they observe experimentally?
How did Lotka-Volterra do?
One species wins:
Coexistence:
Bistability:
What have we learned?
coexistence
bistability
single-species dominance
The Lotka-Volterra model predicts three kinds of 2-species systems
With the correct extension to the model and a clever experimental system, we can understand the complex dynamics of species competing in environments with harsh mortality!
In environments with high mortality, it is better to be a fast grower!
It’s a simple model, but maybe not bad!