Lecture 9: Multi-species growth Pt. I
Today:�
Next lecture: explore a paper that has tested predictions of this model
Growth of one species
~ exponential
Growth of one species
~ exponential
Our model:
population growth rate
population density → positive growth!
Limited nutrients → negative growth
Growth of one species
Maybe a clearer way to write it:
Positive and negative terms cancel → no growth!
We can observe growth dynamics directly from the logistic equation
µ
population density (N)
→ population grows!
→ population shrinks!
Growth of one species
v
v
v
Few cells, many nutrients
Many cells, few nutrients
What if there is another species consuming the same nutrients???
?
Does the blue species reach a lower maximum density?
Does the blue species die out?
What about the gold species?
Simplest example of “microbial ecology”
Why should we care about multi-species growth?
The real environment is dense with many, many species.
We need a starting point for understanding their interactions.
Today we’ll look at the simplest experimental system and mathematical model: 2 species.
oral microbiome
Let’s do a simplified experiment
Paramecium bursaria
Paramecium caudatam
P. bursaria alone
P. caudatam alone
P. bursaria mixed w/P. caudatam
P. caudatam mixed w/P. bursaria
*Georgy Gause, Verifications Experimentales de la Theorie Mathematique de la Lutte pour la Vie (1935)
P. bursaria alone
P. caudatam alone
P. bursaria mixed w/P. caudatam
P. caudatam mixed w/P. bursaria
*Georgy Gause, Verifications Experimentales de la Theorie Mathematique de la Lutte pour la Vie (1935)
P. bursaria alone
P. caudatam alone
P. bursaria mixed w/P. caudatam
P. caudatam mixed w/P. bursaria
*Georgy Gause, Verifications Experimentales de la Theorie Mathematique de la Lutte pour la Vie (1935)
Each species inhibits the other’s growth and they coexist at a different maximum density than on their own!
Can we make a model like our logistic growth equation to describe these growth dynamics?
Remember growth of one species
nutrient limitation (- contribution)
What if another species is present that competes for nutrients?
?
competition for nutrients!
One species competing with another:
Population growth rate of species 1.
Exponential growth of species 1 (+).
Nutrient limitation for species 1 at high cell density (-).
Competition with species 2 for nutrients (-).
But what about species 2?
It has its own equation!
This inhibition is often from one species’ by-products that are toxic to the other
Growth of two species
2
Growth rate of each species
Exponential growth for each species
Nutrient limitation for each species
Competition with the other species
Different values of the parameters will predict different growth dynamics of this model multispecies system!
Growth of two species
2
Growth of two species
1
2
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
Growth of two species
1
2
This system of differential equations models the growth of two species competing for the same set of nutrients.
This model is called the competitive Lotka-Volterra model.
Let’s step through it slowly to see what it predicts.
First we’ll review the single species growth model.
Single species growth
µ
population density (N)
→ population grows!
→ population shrinks!
This axis illustrates population density.
A vector on the axis shows whether the population will grow or shrink at that density.
With a second species, we need two axes!!
Both species low
Both species high
You can represent the growth dynamics of a multispecies system as a path on this plane.
Let’s look at our original example.
P. bursaria mixed w/P. caudatam
P. caudatam mixed w/P. bursaria
P. caudatam
P. bursaria
Initially both species are low
Both grow, but caudatam grows more
The density of both species fluctuates about a steady level
etc.
This growth data can be visualized as a trajectory in this phase plane!
The Lotka-Volterra model tells us how trajectories flow through this plane!
Bear with me while we do some math.
We’re trying to figure out what this model says about how the species will grow.
“Phase plane”
?
?
?
The steady states:
This defines a line in the phase plane!
This defines a line in the phase plane!
= 0 along line
positive
stays the same
stays the same
more negative
What about the left side?
= 0 along line
positive
stays the same
stays the same
less negative
The model tells us how species 1 will grow depending on its own density and species 2’s density!
What about species 2?
positive
stays the same
stays the same
more negative
Below the line?
How does the Lotka-Volterra model predict that the densities of two competing species will evolve in time?
What happens when we put both together?
How does the Lotka-Volterra model predict that the densities of two competing species will evolve in time?
Unstable steady state
Let’s return to our original data
Starts out with a low number of both species, and then they grow toward a stable coexistence.
Actually solving the Lotka-Volterra equations:
Phase plane: