Coexistence in competitive communities

Logistics

• Reflections 1 and 2 graded in Moodle

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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:

\[\alpha_{11} = 0.005;~ \alpha_{12} = 0.002;~ \alpha_{22} = 0.007;~ \alpha_{21} = 0.002\]

\[1/\alpha_{11} = 200;~ 1/\alpha_{12} = 500;~ 1/\alpha_{22} = 142;~ 1/\alpha_{21} = 500\]

Coexistence in competitive communities

Putting numbers to the variables

Consider the following interaction coefficients:

\[\alpha_{11} = 0.005;~ \alpha_{12} = 0.002;~ \alpha_{22} = 0.01;~ \alpha_{21} = 0.007\]

\[1/\alpha_{11} = 200;~ 1/\alpha_{12} = 500;~ 1/\alpha_{22} = 100;~ \alpha_{21} = 0.007 = 142\]

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:

\[\alpha_{11} = 0.008;~ \alpha_{12} = 0.008;~ \alpha_{22} = 0.006;~ \alpha_{21} = 0.005\]

Your turn

• Draw the isoclines for a pair of species whose interaction coefficients are as follows:

\[\alpha_{11} = 0.002;~ \alpha_{12} = 0.005;~ \alpha_{22} = 0.002;~ \alpha_{21} = 0.007\]

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 (\(N_1 = 1/\alpha_{21},~ N_2 = 0\))

Possible outcomes in Lotka-Volterra competition

• Both species can coexist
  • 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”

Semester Project Discussion

In groups of 4, discuss the following:

• What medium do you hope to pursue for your semester project?
• What are some examples of work in this format that you find inspiring?
• What excites you most about this format?
• What do you see as the most challenging barrier to your completing this project?

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 biodviersity
• 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

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

\[\frac{dN_2}{dt} = r_2N_2(1-\alpha_{21}N_1 - \alpha_{22}N_2)\]

• Isocline analysis depends only on the values of the four \(\alpha\) parameters.
• What does each \(\alpha_{ij}\) mean?

What supports species coexistence?

• Individuals competes with conspecifics more strongly than with heterospecifics

\[\alpha_{11} > \alpha_{21} ~~~ \text{and} ~~~ \alpha_{22} > \alpha_{12}\] ##

If this is true:

\[\alpha_{11} > \alpha_{21} ~~~ \text{and} ~~~ \alpha_{22} > \alpha_{12}\]

Then, the following is true:

\[\frac{1}{\alpha_{11}} < \frac{1}{\alpha_{21}} ~~~ \text{and} ~~~ \frac{1}{\alpha_{22}} < \frac{1}{\alpha_{12}}\]

\[\frac{1}{\alpha_{11}} < \frac{1}{\alpha_{21}} ~~~ \text{and} ~~~ \frac{1}{\alpha_{22}} < \frac{1}{\alpha_{12}}\]

In phase-space graphs…

• If coexistence is not possible, three possible outcomes:
  • Species 1 always wins
  • Species 2 always wins
  • Who wins depends on where the population starts

Next week

• Moving on from competitive interactions, to consumer–resource interactions
  • Predator-prey
  • Plant-herbivore
  • Host-parasite

Semester Project Discussion

In groups of 4, discuss the following:

• What medium do you hope to pursue for your semester project?
• What are some examples of work in this format that you find inspiring?
• What excites you most about this format?
• What do you see as the most challenging barrier to your completing this project?