The remarkable discreteness of being

Abstract

Life is a discrete, stochastic phenomenon: for a biological organism, the time of the two most important events of its life (reproduction and death) is random and these events change the number of individuals of the species by single units. These facts can have surprising, counterintuitive consequences. I review here three examples where these facts play, or could play, important roles: the spatial distribution of species, the structuring of biodiversity and the (Darwinian) evolution of altruistic behaviour.

Appendix

1.1 Stochastic modelling

A deterministic behaviour is perfectly predictable: knowing that at time t a system (for example, a projectile) is in a state x (from example, its position), we know its position x′ + x at a time t + t′. This knowledge is modelled by a function x ′ = f(x, t, t ′) and completely characterizes the temporal dynamics of this particular system. For many laws, it is sufficient to know the function f for a very short (infinitesimal) time increment t′, dt. The evolution of the system for the small increment x′ = dx is then often obtained as dx = f (x, t)dt and is called a differential equation. It is enough to add these short increments to determine the state of the system for any long time.

A stochastic behaviour is only partially predictable: knowing the system is in a state x at time t, we cannot predict its state at a later time t + t′; but we can only give a probability P(x′) that it will be at x + x′ at this time. The function P(x′) can be experimentally measured by making a large number of measurement: let N similar systems at time t be in the state x, and we measure their state x + x′ at time t + t′, then make a normalized histogram of all this measurements, which is a function P(x ′ |x, t, t ′). Again, if we know this function for very short time t′, we can know the probability function for longer times by summing up the short-term evolution. This is, for example, how meteorology works: knowing the evolution of probabilities on time scales of seconds, we can predict that, for example, there is 70% chance that tomorrow will be rainy and 30% that it will be sunny.

For many stochastic systems (called Markov chains), it is enough to know the temporal evolution of probabilities for infinitesimal time increments t′ = dt, written as

$$ P\left(x^{\prime}\left|x,t, dt\right.\right)=W\left(x^{\prime}\left|x,t\right.\right) dt $$

where the function W is called the transition rate. Equations (2), (3) and (4), and (8) and (9) are examples of such transition rates. W ^±(n) are shorthand for W(±1|n), i.e. the probability (per unit of time) that the system (here number of individuals of a given type) will be in state n + 1 at time t + dt, knowing that it is in state n at time t. The knowledge of these short time transition rates allows for the determination of probabilities at long times, through an evolution equation called the Master equation when the states are discrete.

It not always easy to solve a Master equation. A crude approximation, called mean field, is to write a deterministic equation for the evolution of the mean increment 〈dx〉 during a short time dt, in the form of

$$ \left\langle dx\right\rangle =f\left(x,t\right) dt $$

and then use deterministic procedure to predict the mean state of the system at long times. The function f(x,t) is readily obtained from the transition rates W(x′|x,t). Mean field approximation can be useful or misleading, depending on the stochastic nature of the system.