Competition Notes

Competition Overview

Introduction to Competition

• Interspecific Competition: Interaction between different species where each is harmed while using the same limiting resource.

• Intraspecific Competition: Occurs between individuals of a single species.

• Population growth rates can be described using differential equations:

  • Species 1: dN1/dt =

  • Where:

    • $N_1$ = Population size of species 1

    • $t$ = Time

    • $r_1$ = Per capita growth rate of species 1

    • $K_1$ = Carrying capacity of species 1

  • Species 2: $\frac{dN<em>2}{dt} = r</em>2 N<em>2 \left(\frac{K</em>2 - N<em>2}{K</em>2}\right)$

  • Where:

    • $N_2$ = Population size of species 2

    • $t$ = Time

    • $r_2$ = Per capita growth rate of species 2

    • $K_2$ = Carrying capacity of species 2

Competitive Exclusion Principle

• Lotka-Volterra Competition Model: Describes the effects of competition using coefficients:

  • $\alpha$ (effect of species 2 on species 1)

  • $\beta$ (effect of species 1 on species 2)

Competition Coefficients

• Interpretation of Coefficients:

  • $\alpha = 1$: Species equally impact each other's growth.

  • \alpha < 1: Species 2 has a lesser impact on species 1 than species 1 does on itself.

  • \alpha > 1: Species 2 has a greater impact on species 1 than itself.

Equilibrium in Competitive Dynamics

• Equilibrium conditions are found when:

  • $\frac{dN<em>1}{dt} = 0$ and $\frac{dN</em>2}{dt} = 0$

• Isoclines:

  • For species 1, when:

  • $N<em>1 = 0, \ N</em>2 = \frac{K_1}{\alpha}$

  • $N<em>2 = 0, \ N</em>1 = K_1$

  • For species 2, when:

  • $N<em>2 = 0, \ N</em>1 = \frac{K_2}{\beta}$

  • $N<em>1 = 0, \ N</em>2 = K_2$

Outcomes of Competition

• Competitive Exclusion:

  • When isoclines do not intersect, one species outcompetes the other.

  • Leads to extinction of one species.

• Coexistence:

  • Only occurs under specific conditions governed by the values of $\alpha$, $\beta$, $K<em>1$, and $K</em>2$:

  • If $\alpha$, $\beta$ close to 1, species exhibit similar competitive strength.

  • For stable coexistence:

    • \frac{b}{a} < \frac{K1}{K2} < 1

Examples of Coexistence

• Example with $\alpha = \beta = 0.95$: Narrow range of carrying capacities results in coexistence.

• Example with $\alpha = \beta = 0.1$: A broader range of carrying capacities allows coexistence as interspecific competition is much weaker.

Critical Conclusions

• The Competitive Exclusion Principle explains that two species using the same limiting resource cannot coexist indefinitely.

• Niche differentiation is essential as two species cannot occupy the same niche simultaneously, reinforcing the importance of resource partitioning in species interactions.