Predator-Prey Cycles in Arctic Tundra

Study Notes

A classic example involves the North American lynx (Lynx canadensis) that primarily preys on the snowshoe hare (Lepus americanus).
Lynx and hare populations exhibit a strong synchronization, with their numbers fluctuating in cyclical patterns.
The cycles are evident in data spanning roughly from 1850 to 1925.

In Arctic tundra regions, large animals (hare, muskrat, ruffed grouse) have a 9-year cycle.
Smaller animals (voles, mice, lemmings) have a 4-year cycle .
Corresponding predators (lynx, red fox for large; hawks, owls for smaller) also exhibit cyclical population fluctuations which match the prey.

Prey population increases, resulting in more food for predators, and subsequently, predator populations also increase (though with a time delay).
Increased predator populations lead to higher predation rates and a decrease in the prey population's size.
These two phenomena lead to recurring cycles.

This model charts the oscillatory relationship between predator and prey populations over time.
The time between peaks in the respective population curves is denoted as the period, roughly 10 year in the model presented.

A model for prey population change over time: dR/dt = rR - cRP.
- R is the prey population size. r is the prey's growth rate. c is the predator capture/efficiency rate. P is the predator population size.
A model for predator population change over time: dP/dt = acRP - dP.
- a is a measure of assimilation efficiency (how well prey is converted to predator growth). d is the constant predator death rate.
The models demonstrate interacting relationships between predator and prey numbers

Prey populations will increase when the prey's intrinsic growth rate (rR) exceeds the rate of consumption (cRP).
Mathematically, this translates to rR > cRP, which is equivalent to having a predator population less than the ratio r/c.

Populations reach equilibrium when the rate of change of each population is zero.
- For prey: dR/dt = rR - cRP = 0, so P = r/c.
- For predators: dP/dt = acRP - dP = 0, so R = d/ac.

Equilibrium isoclines are lines where populations neither grow nor contract due to the balance between predator and prey dynamics.
- Prey-isocline is a horizontal line where change in prey population is null.
- Predator-isocline is a vertical line where change in predator population is null.

This model shows how the populations change over time in a cyclic pattern, graphically demonstrating the interplay between prey and predator populations.

A rise in prey population is typically followed by a rise in the predator population.
The rise in predators causes a decline in prey populations.
The decline in prey results in fewer predators, leading prey populations to grow again, thus completing a cyclical pattern.