Real Mathematics – Special Numbers #3

Lynx – Snowshoe Hare War

In North America there is a predator-prey relationship between lynx and snowshoe hare. Snowshoe hare is the primary food of the lynx which effects the populations of both species directly. For instance if lynx population is high in a forest, it is safe to assume that snowshoe hare population is (or at least once was) high in the same forest. And if lynx population decreases, one can conclude that one of its main reasons is the decrease in snowshoe hare population.

Ecologists have been conducting researches to understand the behaviors of this population change between lynx and snowshoe hare. Within time they found interesting results:

  • Whenever lynx population sees its highest point, snowshoe hare population hits its lowest point.
  • As its primary food supply almost runs out, population of lynx starts to decrease.
  • Decrease for lynx population directly effects the population of snowshoe hare as it increases rapidly.
  • Around the time when population of snowshoe hare hits its heights, lynx population bounces back from its lowest point.
  • More snowshoe hare meant more food for lynx and this leads lynx population to rise as snowshoe hare population to fall down once again.
Figure_45_06_01
Comparison of  the lynx (blue) population with the snowshoe hare (red) population between 1845-1935.

This is a cycle. Ecologists came up with a conclusion for the cycle between these two species. Their results showed them that one cycle lasts 8 to 11 years. It is still unknown what derives this time period. Ecology is filled with numerous secrets such as this one. Scientists showed that predator-prey relationship has a direct effect for the changes in the populations of lynx and snowshoe hare. Although this is not the only reason behind the cycle as climate, human factor and other predators also have an effect.

Periodic Living Beings

Population cycles are crucial for all the species living on Earth. Living beings had evolved so that they can maintain their lives and they managed it with evolving defense mechanisms. Some species don’t have enough strength so they had to come up with new defense mechanisms which are not physical. For instance some animals hide for long periods of time and only come out to breed. Hiding is their defense against predators. These animals could be called as “periodic living beings”.

Assume that the animal A comes out only once in every 10 years to breed for a couple of weeks.

Q: If A’s predator B is also a periodic living being, how long its period should be in order to catch A’s coming out?

If A comes out every 10 years, predator’s period should be a factor to A’s period so that B could catch A as soon as possible.

Factors of 10 are 1, 2, 5 and 10.

  • If predator’s period is 1 year, then (10/1=10) predator catches A in its 10th cycle.
  • If predator’s period is 2 years, then (10/2=5) predator catches A in its 5th cycle.
  • If predator’s period is 5 years, then (10/5=2) predator catches A in its 2nd cycle.
  • If predator’s period is 10 years, then (10/10=1) predator catches A in its 1st cycle.

There is still more bad news to come for A. These four numbers are not the only cycles a predator can have in order to catch A. For example if a predator’s cycle is 6 years, then it would catch A in their least common multiple. For 6 and 10, least common multiple is 30. Thus these two species meet once in every 30 years.

In the end, 10 years of cycle is not suitable for a living being.

Q: Is there a periodic living being on nature?

Cicadas

There are different types of cicada which only come out once in every 7, 13 and 17 years just for breeding. This is why cicadas are known as “periodic living beings”.

indir (10)

Their cycles are 7, 13 and 17. These three numbers are not ordinary numbers: They are primes.

Prime numbers are special in their own way as they don’t have factors besides 1 and themselves.

It is still a mystery how these cicadas evolved into these specific primes. But we can speculate over this fact. A physicist named Mario Markus claimed that cicadas emerge in these prime numbers so that they will survive. This is a bold but interesting assumption.

Example: Let’s choose a kind of cicada that has 13 year-cycles. If its predator has a cycle of 5 years, there two creatures could only meet once in every (13*5=65) 65 years. Since 13 is a prime number, predator and prey can meet only at the multiples of 13.

M. Serkan Kalaycıoğlu