Community Ecology

Feedback from Activity 2

  • Nearly two-thirds of responses (29/46) mentioned confusion about Equbilibrium, Stability, Carrying capacities, or Allee effects.
  • Nearly one-third of responses (15/46) mentioned confusing about graphing in general.

These themes continue to be important throughout ecology, so let’s take some time to review

What does “Equilibrium” and “Stability” mean to you?

  • How would you quantify whether a system is at equilibrium?
  • How would you quantify stability?

Equlibrium in ecology

  • Some property of a system is constant over time (Dynamic equlibrium)
    • E.g. Number of fish in a lake
    • E.g. Amount of land that is covered in forest
    • E.g. “Position of a ball”
    • Where have you seen this version of equilibrium before?

Equilibrium in ecology

  • What would equilibrium look like on a graph?

Stability in ecology

  • An equilibrium may be stable or it may be unstable
  • The key question is:
    What happens to the system if it is at an equilibrium, but it is “pushed”?
    • A few possibilities:
      It returns to the original point
      It keeps moving in the direction it was pushed
      It stays at the point it was pushed to

Stability in ecology

  • An equilibrium may be stable or it may be unstable
  • The key question is:
    What happens to the system if it is at an equilibrium, but it is “pushed”?
    • A few possibilities:
      It returns to the original point \(\to\) Stable
      It keeps moving in the direction it was pushed\(\to\) Unstable
      It stays at the point it was pushed to \(\to\) Neutral
  • Each equilibrium point has an associated stability

Let us analyze the Logistic growth model:

\[\frac{dN}{dt} = rN*\bigg(1-\frac{N}{K}\bigg)\]

  • What are the equilibrium points?
    • i.e. “At what point does the population size stop shrinking or growing?”
    • We can set these by setting the equation equal to zero:

\[\frac{dN}{dt} = rN*\bigg(1-\frac{N}{K}\bigg) = 0\]

  • Two possibilities
    • First possibility: \(N = 0\)
    • Second possibility: \(N = K\)

\[\frac{dN}{dt} = rN*\bigg(1-\frac{N}{K}\bigg) = 0\]

  • Two equilibrium points
    • First equilibrium point: \(N = 0\)
    • Second equilibrium point: \(N = K\)

  • 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.

    • What happens if \(N = 0\), and the system gets “pushed” slightly?
    • What happens if \(N = K\), and the system gets “pushed” slightly?

Graphical representation of stability

\[\frac{dN}{dt} = rN*\bigg(1-\frac{N}{K}\bigg) = 0\]

  • Two equilibrium points
    • First equilibrium point: \(N = 0\)
    • Second equilibrium point: \(N = K\)
  • We can visualize population dynamics along a number line:


Stability of equilibria in Allee effects model

  • The logistic growth model assumes declining fitness with population size
  • But in nature, fitness may increase with population size

\[\frac{dN}{dt} = - rN \bigg( 1-\frac{N}{T} \bigg) \bigg( 1-\frac{N}{K} \bigg)\]

  • 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 \(T = 100\), \(K = 1000\)

Stability of equilibria w/ Allee effects

\[\frac{dN}{dt} = - rN \bigg( 1-\frac{N}{T} \bigg) \bigg( 1-\frac{N}{K} \bigg)\]

  • \(N = 0\), \(N = T\), \(N = K\): If this is true, the population isn’t growing or shrinking
  • Let’s take some examples: Assume \(T = 100\), \(K = 1000\)




  • 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 \(N = 10\)?

Review

  • What defines equilibrium in ecology? How is this different from stability?

  • Consider a population with logistic growth:

    • \(\frac{dN}{dt} = rN(1-\frac{N}{K})\)
    • How many equlibrium points? What is their stability?

  • Consider a population with Allee-growth:

    • \(\frac{dN}{dt} = - rN \bigg( 1-\frac{N}{T} \bigg) \bigg( 1-\frac{N}{K} \bigg)\)
    • How many equlibrium points? What is their stability?

Back to communities

Nature of species interactions

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

Why start with competition?

  • Thought to be ubiquitous: Living things generally require similar sets of things…
    • Plants need water, light, nutrients, space,etc.
    • Herbivores need plants to consume, space to reproduce, etc.
    • Predators need herbivores to consume, space to reproduce, etc.
  • Allows us to focus on organisms in one “guild” at a time
    • Guilds are groups of organisms with similar life histories that we expect interact strongly with one another.
    • e.g. Trees; herbaceous vegetation
    • e.g. Raptors (birds of prey); seed-eating birds

Approach to modeling competition

  • 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

\[\frac{dN_1}{dt} = \overbrace{r_1N_1}^{\substack{\text{growth}\\\text{without}\\ \text{competition}}} \overbrace{(1-\alpha_{11}{N_1})}^{\substack{\text{reduction due}\\ \text{to competition}}}\]

Approach to modeling 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:

  • 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 predict the outcome for any given pair of species?

What conditions enable coexistence?

Defining species coexistence

  • Sufficient to see two species in the same place at the same time?

Signature of stable coexistence:

  • System is at equilibrium (\(\frac{dN_1}{dt} = 0\) and \(\frac{dN_2}{dt} = 0\))
  • Both species are present (abundance > 0)
  • If the system is pushed a bit away from equilibrium, it will return to the same equilibrium

Under what conditions do both species co-exist?

Approach: Graphical analysis of the competition model

  • New type of visualization: phase space (AKA “state space”)
  • New type of analysis: null-cline analysis
    (AKA “zero net-growth isocline”)

We have already seen a phase space model







  • Now, the challenge is to extend this to two dimensions.

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

    • X-axis is abundance of species 1, Y-axis is abundance of species 2

  • 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 (\(N_1\)) not change?
    • At what points does the abundance of species 2 (\(N_2\)) not change?

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…

  • When species 1 is at its carrying capacity, and species 2 is not around
    • \(N_1 = \frac{1}{\alpha_{11}}, N_2 = 0\)
  • When there are so many individuals of Species 2 that Species 1 cannot begin to grow
    • When does that happen?
    • \(N_1\) is low (0), and \(N_2\) is… some high number

\[\frac{dN_1}{dT} = r_1N_1(1-\alpha_{11}N_1 - \alpha_{12}N_2)\]

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?

  • When species 2 is absent, and species 1 is at its carrying capacity
    • \(N_2 = 0, N_1 = \frac{1}{\alpha_{11}}\)
  • When there are so many individuals of Species 2 that Species 1 cannot begin to grow
    • When does that happen?
  • \(N_1 = 0, N_2 = \frac{1}{\alpha_{12}}\)

Summary of the null-cline analysis so far:

  • We set out to find cases where \(N_1\) doesn’t change, i.e. \(dN_1/dt = 0\)
  • We identified two extreme scenarios:
    • \(N_1 = 1/\alpha_{11}, N_2 = 0\)
    • \(N_1 = 0, N_2 = 1/\alpha_{12}\)
  • We can put these two extreme points on the state space graph.

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.

\[\frac{dN_1}{dT} = r_1N_1(1-\alpha_{11}N_1 - \alpha_{12}N_2)\]

  • This is the equation of a line!

\[N_2^* = \frac{1-\alpha_{11}N1}{\alpha_{12}} \]

\[N_2^* = \frac{1-\alpha_{11}N1}{\alpha_{12}} = \frac{1}{\alpha_{12}} + \frac{\alpha_{11}}{\alpha_{12}}N_1\]

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:

  • Key questions:
    At what points does the abundance of species 1 (\(N_1\)) not change?
    • \(N_1 = 1/\alpha_{11}, N_2 = 0\);
      \(N_1 = 0, N_2 = 1/\alpha_{12}\);
      \(N_2^ = \frac{1}{\alpha_{12}} - \frac{\alpha_{11}}{\alpha_{12}}N_1\)
  • At what points does the abundance of species 2 (\(N_2\)) not change?

At what points does the abundance of species 2 (N2) not change?

  • When species 1 is not around, and species 2 is at its carrying capacity
    • \(N_1 = 0, N_2 = 1/\alpha_{22}\)
  • When there are so many individuals of Species 1 that Species 2 cannot begin to grow

    • \(N_1\) is some high number… and \(N_2\) is low (0)

    • Following algebra, \(N_1 = 1/\alpha_{21}, N_2 = 0\)

  • We can add these two extremes to the state space plot

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:

  • Our goal is to identify conditions under which both species can coexist at equilibrium
  • Being ‘at equilibrium’ means \(dN_1/dt = dN_2/dt = 0\)
  • Null-cline analysis lets us find the conditions at which \(dN_1/dt = 0\), and the conditions at which \(dN_2/dt\)
  • There are a couple of ‘extreme’ cases (e.g. one species at carrying capacity, and the other absent), and a whole bunch of in-between cases that result in \(dN_1/dt = 0\) or \(dN_2/dt = 0\)

Test your recollection

  • Consider a pair of species that interact with the following strength:
    • \(\alpha_{11} = 0.01\), \(\alpha_{12} = 0.005\), \(\alpha_{22} = 0.02\), \(\alpha_{21} = 0.001\)
    • (\(\frac{1}{0.01} = 100,\ \frac{1}{0.01} = 200,\ \frac{1}{0.02} = 50,\ \frac{1}{0.001} = 1000\))
  • On separate graphs, draw the isoclines for species 1 and 2.