By the conservation law of population, the rate of increase of population (varphi ^{l}_{t}(t)u^{l-} varphi ^{l}(t),t)) is equ^{l-} varphiux (-hat {a}^{l} u^{l-}_{x}(varphi ^{l}(t),t) ), where (u^{l-}(varphis^{l}(t),t)) and (u^{l+}(varphi ^{l}(t),t)) requalentothe limits oflux^{l}) from left and right for the space variable, respectively.
A primary law of population ecology is the Malthusian growth model which states, "a population will grow (or decline) exponentially as long as the environment experienced by all individuals in the population remains constant".
In particular, a major finding in this field has been the discovery that many diverse socio-economic properties of cities, from personal income to gasoline consumption, follow a power law function of population size with scaling exponents that fall into distinct universality classes [4 8].
Simple models of terrestrial ecosystems with a limited number of components are an efficient tool to study the main laws of functioning of populations, including microbial ones, and their communities, as components of natural ecosystems, under variable environmental conditions.
It is in the aggregate of the city that these stochastic micro-processes add up to population size dependent stable averages, expressing the general effects of urbanization in terms of non-trivial scaling laws as functions of population size [15].
The law of motion population is N t+1=(1+n)N t.
The rich theory of diffusion approximations for branching processes, valid when m is near unity and the number of ancestors is large, may be viewed as descendants of Fisher's derivation of the exponential limit law of the population number in a branching process having a Poisson offspring law with m a little more than unity, and conditioned on non-extinction.
According to the empirical regularity called Taylor's law, the variance of population density in samples of populations is a power of the mean population density.
With the discovery of gold in 1851 (when the Port Phillip District became the new Colony of Victoria), and the resulting influx of population, law and order became more difficult to maintain.
