I come from a country where long period of country-wide stability is a norm in history. Several dynasties in China last longer than 300 years with remarkable levels of economic prosperity. After fulfilling their personal/political ambitions, many emperors sought for elixir that would give them eternal youth, but, good luck with that. It’s an interesting question though, from an evolutionist point: why we cannot live forever?

There is obviously selective advantages for living longer. The length of sexually active period of an animal is generally correlated with the lifespan, which means the longer an animal lives, the more likely he will have a larger number of offsprings. But the longevity has cost too. For example, a robust immune system and good recovering/regeneration capability is probably necessary, and all these consume energy generated from metabolism. Thus instead of pursing a longer lifespan, many organisms choose a different evolutionary path by reproducing more and faster, such as most insects, and rodents. But aside from biology, statistics could also provide some insight about the lifespan of animals in nature.

An important thing to note is that even if an individual can live forever, he probably will not because of the unexpected adverse events, such as famines, floods and deadly diseases. A quick fact is that prior to the modern era, the average lifespan of human beings are below 30 years old, mainly because of the aforementioned factors. With that in mind, we can start the calculation of the expected lifespan given a natural lifespan and the incidence rate of adverse events.

For simplicity, assume the natural lifespan is always *L*. Further assume that the time until an adverse event is an exponential distribution, with parameter *λ*. In other words, an individual randomly dies according to a poisson process prior to the age of *L*, but never lives pass *L* because we assume it is the limite of natural lifespan. We thus have the expectation of the actual lifespan being:

Simple calculus reduces this to:

This is the actual average lifespan of an organism in the wild, despite the natural lifespan of *L*. Let’s put in some real numbers. Given that the pre-history average human lifespan is roughly 30 years old, I estimate for human, *λ *equals 1/30 years-1. We can plot the actual average lifespan (y-axis) vs. the natural lifespan (x-axis):

It’s obvious from the plot that the curve plateaued around 70-90 years old, meaning, even if human natural lifespan is beyond 90 years, it has basically no impact on the actual lifespan of pre-history humans. Coincidentally, the modern human lifespan is around this number. This might imply that evolution purposefully designed our natural lifespan, so that as cavemen we don’t die early because of the limit of natural lifespan, and we never grow too old, which is a waste of energy that can be used toward reproduction or whatever.

This story will obviously change in the modern era, because the unexpected death rate reduced thanks to agriculture and modern medicine. Unfortunately, few of us can live pass 90 years old since this was the optimal design for millions of years. Given enough time, we can probably acquire new mutations to live longer.

Let’s get back to the maths for a moment. The maximal value of expression (2) is 1/ *λ*, meaning that no matter how high the natural lifespan is, the actual average lifespan can never pass 1/ *λ*. If we are the designer of life, we probably won’t set the natural lifespan parameter too high because it will just be a waste of resources. Since statisticians love the number of 0.95, let’s just see what natural lifespan (L) will give us 95% of the maximal expectancy:

Here L is the natural lifespan and L0 is the average lifespan in the wild. This equation means, when the natural lifespan is three times the average lifespan in the wild, an organism will achieve 95% of the maximum life expectancy. We can define this as efficient (if you will). Interestingly, many animal species seem to satisfy this equation. For example, wolf has a life expectancy of 6-8 years in the wild but their natural life span is 20 years. Feral cat has a reported median (not mean though) age of 4.7 years, and their natural lifespan is around 14 years. Pre-history cavemen generally lives up to 30 years, and the modern human can typically live to 80~90 years old, if he is healthy. So a ratio of 3 seems to be the magical number!

There are probably more factors that can be incorporated into the lifespan model, such as the correlation of reproduction capability/cancer incidence rate to the natural lifespan, but it is amazing how such a simple mathematic model is already starting to work. If I have more time in the future, I’ll revisit this topic.

good job.

good model.