A simple model of two-species competition

Last time in class, we discussed reductionist and holistic approaches to wrestling with the complexity of ecological communities.

  • We will begin with the reductionist approach in this class, which has been a longstanding tradition across subdisciplines of biology.

Reductionist approach:

Ecological communities as composites of pairwise interactions

  • What are the ways in which two species can interact?

Nature of species interactions

Sp 1 effect
on Sp 2		 Sp 2 effect
on Sp 1		 Shorthand
Benefit (+) Benefit (+)
Harm (–) Harm (–)
Benefit (+) Harm (–)

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

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

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

To further complicate matters, the same pair of species can have different interactions under different conditions

Starting with competition

(But why start here?)

  • We will spend considerable time learning about competition’s effects on diversity
    • “Social” reasons:
      • Historically one of the most well-studied interactions
      • Tools to study competition have been applied to other types of interactions
    • “Biological” reasons:
      • Competition is frequently observed
      • Plenty of evidence that it shapes biodiversity as a whole

Building from population growth…

  • In the population ecology modules, we explored two “main” model structures

  • Exponential growth: \(\frac{dN}{dt} = rN\)

    • Stage-structured populations also grow exponentially, but the rate of growth \(r\) is more complicated

  • Logistic growth: \(\frac{dN}{dt} = rN(1-\frac{N}{K})\)

    • No matter initial population size, populations reach carrying capacity and equilibrate there

  • We also saw how facilitation can lead to allee effects

Incorporating species interactions

  • To study species competition, we will begin with the logistic growth model as our “baseline”:

\[\frac{dN_1}{dt} = r_1N_1(1-\frac{N_1}{K_1})\]

  • Note that because we are soon expanding to two species, we’ve added the subscript 1 to all variables and paramters (\(r_1\) is species 1’s growth rate, \(K_1\) is species 1’s carrying capacity, etc.)

  • Alternative expression:

    • If we set \(\alpha_{11} = \frac{1}{K_1}\), we can rewrite the logistic growth equation:

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

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

  • \(\alpha_{11}\) is the strength of within-population competition (also called intraspecific competition)

  • When there is competition between species, the growth rate of species 1 (\(\frac{dN_1}{dt}\)) is also affected by species 2

\[\frac{dN_1}{dt} = r_1N_1(1-\alpha_{11}{N_1} - \text{competition from species 2})\]

\[\frac{dN_2}{dt} = r_2N_2(1-\alpha_{22}{N_2} - \text{competition from species 1})\]

\[\frac{dN_1}{dt} = r_2N_1(1-\alpha_{11}{N_1} - \text{competition from species 2})\]

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

\[\frac{dN_1}{dt} = r_1N_1(1-\alpha_{11}{N_1} - \alpha_{21}N_1)\]