Island Biogeography and Neutral Theory

Island Biogeography

• Seeks to understand the processes governing species diversity on islands.
• Islands serve as simplified model systems for broader ecological principles.
• Applies to habitat fragments and inland lakes as well.

MacArthur and Wilson's Theory

• Explains the number of species on an island based on:
  • Immigration rate of new species
  • Extinction rate of existing species
• Equilibrium is reached when immigration and extinction rates are equal.
• Immigration rates decrease with increasing species richness due to:
  • Less potential for new immigrants.
  • Increased competition.
• Extinction rates increase with increasing species richness due to:
  • More competition.
  • Smaller average population size.
• Dynamic Equilibrium:
  • Number of species remains constant, but species identities change over time.

Factors Controlling Species Richness Variation

• Island Isolation:
  • Further distance leads to lower immigration rates and lower species richness.
• Island Area:
  • Larger area leads to lower extinction rates and higher species richness.
  • Larger islands have more resources, supporting larger populations.
  • Larger islands may have higher immigration rates (target area effect).

Mathematical Expressions

• $\frac{ds}{dt} = \lambda<em>s - \mu</em>s$
  • $s(t)$: Number of species on the island at time $t$.
  • $\lambda_s$: Immigration rate.
  • $\mu_s$: Extinction rate.
• Immigration rate as a function of island species richness:
  • $\lambda<em>s = \lambda</em>{max} (1 - \frac{s}{s_m})$
    • $\lambda_{max}$ is the immigration rate to an empty island
    • $s_m$ is the species richness of the mainland
• Extinction rate:
  • $\mu_s = ks$
    • $k$: Per species extinction rate.
• Area is added to the model through per species extinction rate:
  • $k = \frac{k'}{A}$
    • $k'$ is the original per species extinction rate
    • $A$ is island area
• Functional form for immigration as a function of isolation:
  • $\lambda = \frac{\lambda_{max}}{1 + aD}$
    • $d$ is distance
    • $a$ determines how fast immigration decays with distance

Speciation

• Incorporate in situ speciation into the model by adding a third term:
  • $\nu_s$ gives the number of species generated by speciation per unit time as a function of the current 's'
• Speciation can be modeled as:
  • Species dependent
  • Area dependent: Larger area leads to more individuals, and more mutations

Model Predictions

• Immigration maintains diversity of smaller islands.
• Speciation maintains diversity of larger islands.

Limitations

• Fails to explain the small island effect where species richness varies independently of area on very small islands.

Individual Based Models & Neutral Theory

• LACA Volterra competition model and Tillman's resource competition model or the Monarch model include only niches, but not the other three processes.
• Island biogeography theory include mainly speciation and dispersal.
• Neutral models include these three processes, but basically no niches.

Stochasticity

• Deterministic models give the same result every time, stochastic models give different results every time; they include randomness.
• Two types of stochasticity in ecology:
  • Demographic variance: Randomness in birth and death events.
  • Environmental variance: Environmental conditions vary over time.
• Demographic variance: Demographic stochasticity or demographic heterogeneity.
• Environmental variance is randomness arising from a temporarily varying and unpredictable environment.
• Demographic variants plays a more important role in small populations.
• Environmental variants can play an important role in both large and small populations.
• Tools for dealing with stochastic models are analytical solutions, analytical approximations, simulate realisations, or simulate the entire state space.

Individual Based Models

• Tracks individuals in the population.
• Almost always spatial.
• Individuals can be continuous or discrete.
• Space can be continuous or discrete.

Neutral Theory

• Assumes no differences between species (species equivalence).
• Species differences might not be that important to their dynamics.
• Model can act as a null model to point to cases where species differences are important.
• Allows exploration of effects of major processes besides niches.
• Simple neutral model with birth, death, and speciation (drift and speciation).
• Dynamics of the cartoon neutral model using a master equation. A master equation is a kind of dynamical system, but instead of tracking abundances directly, it tracks the probability of different abundance states over time.
• $p$ is the probability and not the abundance of of $n$. $p_n$ of a species having abundance $n$ rather than abundance itself because stochasticity is now inherent in the system.
• Functional forms of b and d:
  • Death: $d_n = \frac{n}{j}$
  • the equilibrium abundance distribution, this is the expected species richness in the community at equilibrium $s_{bar}$. Theta here is a quantity that often appears in the mathematics of neutral theory, and it's called the fundamental biodiversity number.
• The dynamic equilibrium species richness fluctuates over time around a mean value, which is maintained by a speciation extinction balance.