رشد جمعیت به صورت فرآیند تصادفی غیر خطی

The evolution of the probability density of a biological population is described using nonlinear stochastic differential equations for the growth process and the related Fokker-Planck equations for the time-dependent probability densities. It is shown that the effect of the initial conditions disappears rapidly from the evolution of the mean of the process. But the behaviour of the variance depends on the initial condition. It may monotonically increase, reaching its maximum in the steady state, or have a rather complicated evolution reaching the maximum near the point where growth rates (not population size) is maximal. The variance then decreases to its steady-state value. This observation has implications for risk assessments associated with growing populations, such as microbial populations, which cause food poisoning if the population size reaches a critical level.

A more general approach is to recognise that the fluctuations are an intrinsic part of the growth process and include them into the formulation of the equations. This approach does not involve any unreasonable assumptions of independence of the growth path and previous fluctuations. These are naturally included in dynamics. This paper investigates the situation when the deterministic growth equations are replaced by their stochastic forms, noting particularly the relationship between the mean and the variance of the process as growth proceeds.

Growth equations are assumed to provide an adequate description of biological growth. Dependencies between the mean and the variance have been observed, particularly in cases of animal growth [4]. These results show that the relationship between the mean and variance may be more complex when nonlinear growth processes are involved. In particular it should be expected that circumstances will exist where the variance of the process will increase to a level above the steady-state variance before falling back to the steady-state variance. Intuitively, one might expect that the process would reach maximum variance at the steady state, and in this case it would be expected that a measurement of the variance made at the steady state would establish an upper bound. This analysis makes clear the danger of such an approach if, for example, the goal was to measure the risk to food safety or to water quality through micro-organism growth. Note that the behaviour described is a specific feature of nonlinear growth. Both the mean and variance of growth processes that can be described by linear differential equations (for example, limited or decreasing exponential growth) with a random growth rate, always monotonically evolve to their steady-state values. This work also suggests that the statistical fitting of growth equations to data would be enhanced by using a model which allowed for the variance to change with time.