These themes continue to be important throughout ecology, so let’s take some time to review
What does “Equilibrium” and “Stability” mean to you?
Let us analyze the Logistic growth model:
\[\frac{dN}{dt} = rN*\bigg(1-\frac{N}{K}\bigg)\]
\[\frac{dN}{dt} = rN*\bigg(1-\frac{N}{K}\bigg) = 0\]
\[\frac{dN}{dt} = rN*\bigg(1-\frac{N}{K}\bigg) = 0\]
This means at if, at time \(t = 0\), \(N = 0\), then \(N = 0\) will continue – unless something happens!
This means at if, at time \(t = 0\), \(N = K\), then \(N = K\) will continue – unless something happens!
We can then ask if each equilibrium point is stable or not stable.
\[\frac{dN}{dt} = rN*\bigg(1-\frac{N}{K}\bigg) = 0\]
\[\frac{dN}{dt} = - rN \bigg( 1-\frac{N}{T} \bigg) \bigg( 1-\frac{N}{K} \bigg)\]
What are the equilibrium points?
(Cases where \(\frac{dN}{dt} = 0\))
\(N = 0\), \(N = T\), \(N = K\): If this is true, the population isn’t growing or shrinking
Let’s take some examples: Assume \(T = 100\), \(K = 1000\)
\[\frac{dN}{dt} = - rN \bigg( 1-\frac{N}{T} \bigg) \bigg( 1-\frac{N}{K} \bigg)\]
What happens if the system is pushed just a bit off?
e.g. What if \(N = 98\)?
Or what if \(N = 105\)?
Or \(N = 990\)?
Or \(N = 10\)?
What defines equilibrium in ecology? How is this different from stability?
Consider a population with logistic growth:
Consider a population with Allee-growth:
Sp 1 effect on Sp 2 |
Sp 2 effect on Sp 1 |
Shorthand |
---|---|---|
Benefit (+) | Benefit (+) | Mutualism |
Harm (-) | Harm (-) | Competition |
Benefit (+) | Harm (-) | Predation Herbivory Parasitism |
Neutral (0) | Benefit (+) | Commensalism |
Neutral (0) | Harm (-) | Ammensalism |
We already saw the effects of competition within a species: population growth slows down as the population gets big
Logistic growth dynamics consider a species whose individuals compete with each other
\[\frac{dN_1}{dt} = r_1N_1(1-\frac{N_1}{K_1})\]
\[\frac{dN_1}{dt} = r_1N_1(1-\alpha_{11}{N_1})\]
Where \(\alpha_{11}\) is the strength of competition within species.
We can now think extend this logic to a second species
\[\frac{dN_1}{dt} = \overbrace{r_1N_1}^{\substack{\text{growth}\\\text{without}\\ \text{competition}}} \overbrace{(1-\alpha_{11}{N_1})}^{\substack{\text{reduction due}\\ \text{to competition}}}\]
\(\alpha_{11}\) is the strength of competition within species,
\(\alpha_{12}\) is the impact of species \(2\) on species \(1\).
\[\frac{dN_1}{dT} = r_1N_1(1-\alpha_{11}N_1 - \alpha_{12}N_2)\]
\[\frac{dN_2}{dT} = r_2N_2(1-\alpha_{21}N_1 - \alpha_{22}N_2)\]
\(\alpha_{22}\) is the strength of competition within species,
\(\alpha_{21}\) is the impact of species \(1\) on species \(2\).
Population dynamics of two competing species
\[\frac{dN_1}{dT} = r_1N_1(1-\alpha_{11}N_1 - \alpha_{12}N_2)\]
What happens if species 1 is growing alone?
\[\frac{dN_1}{dT} = r_1N_1(1-\alpha_{11}N_1)\]
Growth to Species 1’s carrying capacity (\(N_1^* = \frac{1}{\alpha_{11}} = K_1\))
Similarly, if Species 2 is growing alone, it will grow to its carrying capacity \(N_2^* = \frac{1}{\alpha_{22}} = K_2\)
But what happens if both species are present in the system?
Possible outcomes of two species competing:
How to predict the outcome for any given pair of species?
What conditions enable coexistence?
Signature of stable coexistence:
Under what conditions do both species co-exist?
Approach: Graphical analysis of the competition model
Extending the state-space to two species
Instead of a number line (1-dimension), we need a graph with an X- and a Y-axis (2-dimensional)
The number line represented the abundance of our species; now each axis represents the abundance of one of our two species
Any point on the graph represents a possible state of the system
Lines on the graph show how a system changes through time (trajectory)
Null-cline analyses (AKA zero net growth isocline analysis)
Is there any point in this space that allows the system to not change over time? (i.e. to reach equilibrium)
We can approach this problem one axis at a time.
Key questions:
At what points does the abundance of species 1 (N1) not change?
\[\frac{dN_1}{dT} = r_1N_1(1-\alpha_{11}N_1 - \alpha_{12}N_2)\]
Two “extreme” cases…
\[\frac{dN_1}{dT} = r_1N_1(1-\alpha_{11}N_1 - \alpha_{12}N_2)\]
Solve for \(\frac{dN_1}{dT} = 0, N_1 = 0, N_2 > 0\) (on paper)
At what points does the abundance of species 1 (N1) not change?
Summary of the null-cline analysis so far:
Draw state space with Species 1 equilibrium points (on paper)
Growth of species 1 is also zero for intermediate combinations between these extremes
These intermediate combinations are defined by the line connecting the two extremes.
We can solve for the equation of this line.
\[\frac{dN_1}{dT} = r_1N_1(1-\alpha_{11}N_1 - \alpha_{12}N_2)\]
\[N_2^* = \frac{1-\alpha_{11}N1}{\alpha_{12}} \]
\[N_2^* = \frac{1-\alpha_{11}N1}{\alpha_{12}} = \frac{1}{\alpha_{12}} + \frac{\alpha_{11}}{\alpha_{12}}N_1\]
Growth of species 1 is also zero for intermediate combinations between these extremes
\[\frac{dN_1}{dT} = r_1N_1(1-\alpha_{11}N_1 - \alpha_{12}N_2)\]
\[N_2^* = \overbrace{\frac{1}{\alpha_{12}}}^{\text{y-intercept}} - \overbrace{\frac{\alpha_{11}}{\alpha_{12}}}^{\text{slope}}N_1\]
This is the equation of the null-cline for species 1 (AKA zero net-growth isocline, or ZNGI)
Draw state space with Species 1 equilibrium points, plus intermediate line (on paper)
Discussion point: What happens on either side of the null-cline?
Recall the key questions of null-cline analysis:
At what points does the abundance of species 2 (N2) not change?
\(N_1\) is some high number… and \(N_2\) is low (0)
Following algebra, \(N_1 = 1/\alpha_{21}, N_2 = 0\)
Growth of species 2 is also zero for intermediate combinations between these extremes
\[\frac{dN_2}{dT} = r_2N_2(1-\alpha_{21}N_1 - \alpha_{22}N_2)\]
\[N_2^* = \frac{1}{\alpha_{22}} - \frac{\alpha_{21}}{\alpha_{22}}N_1\]
This is the equation of the null-cline for species 2 (AKA zero net-growth isocline, or ZNGI)
Add null-cline to state space
What happens on either side of the nullcline?
Graphical analysis of the Lotka-Volterra competition model
Review:
Test your recollection