Nurgaliev's law

In population dynamics, Nurgaliev's law is an equation that describes the rate of change of the size of a population at a given time, in terms of the current population size. It is a deterministic ordinary differential equation in which the rate of change is expressed as a quadratic function of the population size.

Specification

Nurgaliev's law is expressed as

{dn \over dt} = an^2 - bn,

where 'n' is the size of a population, t is time measured in years, a is a half of the average probability of a birth of a male (the same for females) of a potential arbitrary parents pair within a year, b is an average probability of a death of a person within a year.

The first term is twice proportional to the half of population (number of males and number of females). The second term is responsible for death rate and has a clear and precise sense— death rates are constant in time but vary with position on the age scale (babies are at risk at birth, the middle aged are at risk of trauma, old men become ill). It is known to demographers, for example, that the probability of death within the first year of a life is precisely equal to similar probability for the 55th year of a life. Thus, in the given model, the average person dies under the same law as an unstable atomic nucleus decays.

Stability

The population has steady states at n = 0, b/a . The state with n=0 is stable whereas the state with n=b/a is unstable, meaning that the equation describes a population which crashes (tends to zero) when the death rate is greater than the birth rate and explodes (tends to infinity) when it is the birth rate that is greater.

Sources


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