^{1}

^{2}

The authors have declared that no competing interests exist.

The important concept of equilibrium has always been controversial in ecology, but a new, more general concept, an asymptotic environmentally determined trajectory (AEDT), overcomes many concerns with equilibrium by realistically incorporating long-term climate change while retaining much of the predictive power of a stable equilibrium. A population or ecological community is predicted to approach its AEDT, which is a function of time reflecting environmental history and biology. The AEDT invokes familiar questions and predictions but in a more realistic context in which consideration of past environments and a future changing profoundly due to human influence becomes possible. Strong applications are also predicted in population genetics, evolution, earth sciences, and economics.

The concept of equilibrium has been the basis of prediction in ecology, as in many sciences, because the vicinity of equilibrium commonly defines the properties expected of a system. In conservation, equilibrium, as a formalization of the ancient concept of the balance of nature, has been imagined to define the essence of a system and to be treated with reverence [

Last century, dissatisfaction with the equilibrium concept led to the introduction of models of population dynamics in which the environment is a stochastic process [

AEDT: Asymptotic environmentally determined trajectory. A trajectory,

Attractor: In nonautonomous dynamics theory, an “attractor” consists of a set of sets {_{t}}, indexed by time _{t} consists of just 1 element, then _{t} = {

Backward convergence: In terms of the AEDT,

Equilibrium: The idea that

Forward convergence: In terms of the AEDT,

Stationary environment: The idea that the environment, when viewed over a sufficiently large interval of time, will have the same statistical properties (mean, variance, autocorrelation, and frequencies of events) independently of when that interval of time starts. This is the common assumption in models with variable environments.

Fortunately, a foundation for rising to this challenge exists. Although not generally known, for decades, mathematicians have been developing relevant concepts and machinery in the theory of nonautonomous dynamics [

Consider the Beverton-Holt model (

Illustration using the Beverton-Holt model. Red line: the AEDT,

The Beverton-Holt model of density-dependent population growth is a discrete-time model defined by the following difference equation for population density of a single-species,

With

This transformation to linearity gives an exact solution by iteration,

The AEDT can also be considered as a stochastic process, and in general, it is a nonstationary stochastic process described statistically by a nonstationary probability distribution. In particular, its mean and variance change over time. Nonstationary distributions have the potential to be highly complex, but a simple form applies to the Beverton-Holt model when the parameter ^{2}(

The results are simplest when expressed in terms of the reciprocal of the density, namely, ^{2}, … reflecting ^{2} by the formula

These mean and variance functions have very straightforward interpretations. The environmental mean and variance, ^{2}(

It is important to emphasize that the AEDT,

General discrete-time population dynamics can be represented in the form

This demonstration provides no formula for the AEDT and allows the possibility that, although trajectories converge on each other, the starting time,

As emphasized, the AEDT,

The AEDT concept applies not just to the Beverton-Holt model but generally in ecological models.

Lines of the same color but different intensity represent the same species with different initial conditions. Although starting at very different values, the effect of the initial conditions has all but disappeared by midway through the simulation. The environmental fluctuations driving this lottery simulation are lognormal, independent between species and over time, with a linear trend creating nonstationarity. Specifically, the ln_{j}(

As proposed previously [

In this model, success in competition for space depends on the ratio _{i}(_{i} for each species [_{1} = _{2}), convergence can be proved by transforming density to the log-odds scale:
_{i}, with different starting values can be shown to decrease monotonically over time, _{i}(_{j}(^{-1}(_{j}(_{i}(^{– 1}(_{i}(_{j}(_{i} require inequality between species in their responses to the environment (_{i}(_{j}(_{i}

The AEDT concept also resolves 1 of the discontents nearly universally seen with the traditional equilibrium idea. The system equations give convergence on a constant state, but a constant state cannot in fact be found in nature, which goes to the heart of the classic dispute between Nicholson and Andrewartha & Birch [

Traditionally, stability of an equilibrium was interpreted as demonstrating robustness to environmental fluctuations. Even though a population or community might be continually perturbed from equilibrium, it would always be returning: equilibrium would define a central tendency for population fluctuations. However, the lottery model (

Although proposed here to address nonstationary environments, the AEDT concept applies to the special case of a stationary environment too. Traditionally, analysis of a stochastic population model in a stationary environment sought a corresponding stationary probability distribution for population size, not a trajectory [

Traditional equilibrium analysis focuses on determining the existence of a stable equilibrium with specific properties. In coexistence analysis, for instance, it is an equilibrium where all species have positive densities [

The AEDT concept is introduced here specifically to replace the point equilibrium idea but has generalizations under the general heading of “nonautonomous attractors” to also replace multiple stable points, limit cycles, and strange attractors under nonstationary environmental conditions, along with bifurcation theory for transitions between them, in the new mathematical field of nonautonomous dynamics (

Although I have presented the AEDT concept for ecological models, it can be applied equally well in many areas of science that involve dynamics over time subject to nonstationary environments. A simple and obvious extension is to population genetics, where the dynamics of gene frequencies have many parallels to the dynamics of populations and are no less affected by nonstationary environmental change. Indeed, the lottery model (

Earth sciences are often intimately involved with the environmental change yet still make use of equilibrium concepts [

Hydrological theory provides a natural application of the AEDT beyond biology [

In ecology, population fluctuations and trends are universal, yet a standard equilibrium perspective relegates them to noise or temporary anomalies, not part of the essence of a system. With the AEDT, change is of the essence, reflecting environmental history and biology. Both forward and backward convergence imply that the reach of environmental history is limited, and the AEDT formulae in Boxes

Parts A-H.

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asymptotic environmentally determined trajectory