5/23/11

The 13-year Cicada CyclePeriodic Functions• The highlighted section

of the map shows the affected regions.

• To see the cicada invasion up close, click on this link to watch a video.

The 13-year Cicada Cycle

Periodic Functions• This graph shows the

life cycle of the cicadas in 13-year increments, starting in 1998.

• Note the form of the sine function, which yields a 13-year half-period.(The full period of the curve is 26, so we are using half-periods to represent a cicada generation.)

The 13-year Cicada Cycle

Periodic Functions• To simplify, use the

sin2(x) version of the graph, which has a more easily identifiable period.

The 13-year Cicada Cycle

Periodic Functions• Animals that prey on

other animals often rely on a predictable pattern to the behavior of the prey. For example, bears know that every year salmon will swim upstream along rivers, providing a reliable meal.

The 13-year Cicada Cycle

Periodic Functions• Birds are natural predators

of cicadas. Every year a new generation of birds is born.

• But it takes 13 generations before the bird reproductive cycle coincides with the cicada cycle. As a result, these cicadas are not a reliable food source for newly hatched birds.

The advantages of a long reproductive cycle.

The 13-year Cicada Cycle

Periodic Functions• The most common bird

species that prey on cicadas are shown in this table, along with their average life spans.

The 13-year Cicada Cycle

Periodic Functions• The graphs of these bird life

cycles (several are shown here) reveal that they don’t overlap the cicada life cycle. So, even for adult birds, the 13-year cicadas are not a reliable food source.

• The is that 13 is a prime number.

The advantages of a prime number reproductive cycle.

The 13-year Cicada Cycle

Periodic Functions• The advantage of a prime

number life cycle is that it overlaps with the predator’s life cycle at the product of the two life cycles. In practical terms the cicada’s life cycle makes it immune to predators.

The advantages of a prime number reproductive cycle.

The 13-year Cicada Cycle

Periodic Functions• One additional advantage

that these cicadas have is that in the 13th year, a huge swarm of these insects are hatched, overwhelming the population of predators.

• Enough cicadas survive to reproduce the next generation. This is similar to the situation with the salmon. Enough survive bear predation.