MacArthur and Wilson's Theory of Island Biogeography is a simple model of the number of species that live on an island. Species arrive on the island as immigrants from a continental species pool and species are lost from the island when they go locally extinct. The equilibrium species richness is the number of species on the island when new species colonize the island at the same rate that existing species go extinct. This occurs when the colonization and extinction curves intersect (the dashed line in the figure).

Two main factors influence colonization and extinction rates: distance from the mainland and island size. More distant islands are harder to reach than closer islands. As a result, colonization rate decreases with increasing distance from the mainland. Larger islands have more resources than smaller islands, and thus support more species. Therefore, extinction rate decreases with increasing island size.

Use the controls on the left, below, to explore how changing the distance and size of an island affects the number of species on the island at equilibrium.

This model applies to any island-like habitat in which patches of suitable habitat occur within an inhospitable matrix, such as forest fragments. What kind of forest fragments would you expect to have the most species?

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Model

The Lotka-Volterra model is a simple model of the population growth of two competing species. In the absence of competitors, each species grows according the logistic equation, eventually reaching its species-specific carrying capacity. Competition between the two species has four possible outcomes: 1) species 1 wins and species 2 goes extinct, 2) species 2 wins and species 1 goes extinct, 3) the two species coexist at equilibrium, or 4) the outcome depends on the initial population sizes of the two species.

Use the controls on the left, below, to explore how competition between sparrows (species 1) and squirrels (species 2) affects their population dynamics. Use the sliders to adjust the values of the competition coefficients (alphas), carrying capacities, and initial population sizes for both species.

What makes squirrels out-compete sparrows? What makes sparrows out-compete squirrels? And what allows the two species to stably coexist?

Note: the subscripts identify the species (e.g., K₁ is the carrying capacity of species 1). For the competition coefficients, alpha₁₂ quantifies the impact of species 2 on species 1, and alpha₂₁ quantifies the impact of species 1 on species 2.

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Model

In real populations, not all individuals have the same probability of giving birth or dying. Instead, fecundity and survivorship depend on age. A life table summarizes age-specific fecundity and survivorship rates. We can use these rates to predict how population size will change over time.

Use the life table below to explore how age-specific survivorship and fecundity affect population growth. Input new values in the life table, and click on the Update Plots button to see the resulting survivorship, fecundity, and population growth curves.

What makes a population grow? What makes a population decline? Given what you learned in lecture 18, what numbers correspond to an r versus K life history strategy?

Note: The number of survivors in each age class must always be a positive, non-zero number. The number of offspring born to each age class can be zero, but must not be a negative number. Also, make sure that the number of surviving individuals stays the same or declines with age; individuals that were dead at age 2 cannot be alive at age 3, so the number of survivors cannot increase with age.

Life Table

Age

Survivors

Offspring

l_x

m_x

x

Total individuals surviving to age x

Total female offspring born to all females of age x

Probability of surviving to age x

Number of female offspring per female of age x

0

1

2

3

4

5

Model