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This article discusses the question of how elasticity of the system is intertwined with external stochastic disturbances. The speed at which a displaced system returns to its equilibrium is a measure of density dependence in population dynamics. Population dynamics in random environments, linearized around the equilibrium point, can be represented by a Langevin equation, where populations fluctuate under locally stable (not periodic or chaotic) dynamics. I consider a Langevin model in discrete time, driven by time-correlated random forces, and examine uncertainty in locating the population equilibrium. There exists a time scale such that for times shorter than this scale the dynamics can be approximately described by a random walk; it is difficult to know whether the system is heading toward the equilibrium point. Density dependence is a concept that emerges from a proper coarse-graining procedure applied for time-series analysis of population data. The analysis is illustrated using time-series data from fisheries in the North Atlantic, where fish populations are buffeted by stochastic harvesting in a random environment.

The nature of the negative feedback relationship between population growth rate and abundance is at the heart of population ecology. That said, statistical detection of density dependence using ecological time-series can be problematic. When plotting data on the form of the dependence of population growth rate on abundance, ecologists have been confounded by considerable noise around each relationship [

A relevant serious concern is the issue of noise color: how does the presence of serial correlation in the external stochastic forcing affect the density dependence in a population? Solow [

This article investigates the analysis of ecological time-series when the aim is to measure density dependence in the demographic processes. The equilibration time [

The population renewal process

with a growth-survival factor

All density dependence is assumed to be exerted by the adult population S [

In a stationary state (with equilibrium quantities denoted by the “

with constant coefficient

with

The population process with multiple decay-rate constants

with the ratio of the mean square successive difference

with the autocorrelation function of series

where the correction (the second term on the right-hand-side) is due to memory effects of external perturbations. From the recursion,

Equations (4) and (7) yield the quadratic equation for the elasticity

Thus, the (linear) response of a system to external perturbations is expressed in terms of fluctuation properties of the system in equilibrium. The elasticity and the variance of population fluctuations cannot be independent, but they are related to each other in the equilibrium system [

The average relaxation time of population fluctuations defines the total equilibration time

After the time

i.e. the expectation of change in population size given

Therefore, the density dependence D and the variance of population fluctuations are related to each other through the relation

Equation (2) with (8) yields

(where each term is standardized), implying that, if a population is governed by slowly damped dynamics

with

When one performs some observations for L years, the uncertainty in locating the population equilibrium,

with

By virtue of Equation (9), a time-equilibrium uncertainty relation is obtained:

where _{c} provides another measure of the time scale of population dynamics, and reads

_{c} is designated the complementary time. It is worth pointing out that the complementary time is always positive for

Let C_{L} be the degree of certainty, with which the supposed equilibrium lies within the bounds

for normally distributed population fluctuations (with the true (population) standard deviation_{L}. The larger the probability C_{L}, the more accurately the density dependence D is measured from the population time-series.

In order to calculate the probability C_{L}, I analyze the time-series at different resolutions by constructing a coarse-grained time-series. Let

The coarse-grained time-series are considered to be serially independent. Accordingly, the uncertainty of population equilibrium (the standard deviation of the mean) is calculated to be

and the statistic

has a standard Gaussian distribution. In reality, we only have the sample standard deviation. So, replacing _{L}, i.e. the integral of Student’s t-PDF between

(precisely the left-hand-side involves the remainder term of order_{L}. Equation (12) with

_{c} years, the negative relationship is visible with a probability of 78.4% or more. When monitoring an equilibrium population (in the dynamic balance) for L years, the probability of the true equilibrium lying outside the bounds

Empirical analysis [

I now apply the theory for measuring density dependence to time series of North Atlantic commercial species, which are the same as the fish stocks analyzed in [

For a sufficiently long

where

After averaged over sizes

In this section, I discuss the effects of multiple noise sources and noise color on measuring density dependence.

Recently, Ives et al. [

Consider the system perturbed by multiple noise sources, where external perturbations

with

Since the characteristic equation for the pth-order AR component is

Equation (2), transformed into the ARMA(3,1) form, gives

where

with

where the variance

I have shown that apparently density-independent influences, which arise from colored environmental variations, modulate the population elasticity. In the following let us further analyze the impact of color of environmental variation on the total density dependence in a population. Here, for mathematical simplicity, the effects of recruitment fluctuations

Equations (4), (5) and (7) describe the interaction between endogenous dynamics of the population and external disturbances. I demonstrate, in

The total density dependence D decreases when redness of noise color is increased (an increase in the autocorrelation is denoted “increased redness”), and

(with a mean-zero iid random variable

consequently, the amplitude jitter in the population oscillation exhibits a random-walk property and is non-stationary (regardless of the value

Analyzing a stochastic process with slowly damped dynamics, I have derived the expression of complementarity between uncertainty in locating the population equilibrium and length of monitoring. The complementarity relation imposes a limit on the degree of certainty to which a measurement of density dependence is known. The complementarity is an essential criterion for disentangling density-dependent signals from external noise in the population process. Looking at a long time-scale makes the negative feedback on population fluctuations visible. It is difficult to know whether the system is heading toward the equilibrium point in the time-series of length less than the complementary time. This implies that density dependence is a concept that emerges from applying a proper coarse-graining procedure for time-series analysis.

The approach taken in this article is based on linear approximation of a nonlinear stochastic model. A linear approximation applies for “small” perturbations from an equilibrium point, where the “smallness” should be measured relative to the range of the uncertainty of population equilibrium. Because a large uncertainty exists in determining the population equilibrium from short observation series in ecology, it is probable that the system spends most of its time “near” the equilibrium point; a linear approximation would be usually enough accurate.