**Tags**

inequality, nonlinear, nonlinearity, paleoecology, population dynamics, population growth, systems paleoecology

*WHAT IS ECOLOGICAL STABILITY ? In 2019 I posed this question informally to colleagues, using Twitter, a professional workshop that I lead, and a conference. Respondents on Twitter consisted mostly of ecological scientists, but the workshop included paleontologists, biologists, physicists, applied mathematicians, and an array of social scientists, including sociologists, anthropologists, economists, archaeologists, political scientists, historians and others. And this happened…*

Previous posts in this series

*There is always inequality in life — *John F. Kennedy

John Lanchester, a British novelist and journalist, expresses a view that is becoming increasingly widespread in our increasingly stressed human socio-economic system: “*Inequality is not a law of nature*“. I disagree, but let me explain why before you form a judgement. As a scientist, one of my responsibilities, and one for which I have been trained, is to identify and explain laws of nature. In an earlier post I defined a natural law as “*…a statement, based on multiple observations or experiments, that describes a natural phenomenon. Laws do not have absolute certainty and can be refuted by future observations*.” In that sense, inequality would indeed seem to be a law given its persistence and pervasiveness. My disagreement with Lanchester, however, is based solely on the idea that laws do not have absolute certainty, and are not immutable. Laws are not necessarily fundamental, but instead arise from the unfolding of fundamental relationships and interactions over time (and consequently, in space). Lanchester goes on to state, “[inequality] *is a consequence of political and economic arrangements, and those arrangements can be changed.*” which is perfectly consistent with the nature of natural laws.

But where does inequality come from in biological systems? It can originate in the differences of rates between interacting processes, the context-dependent expressions of genes, the plasticity of behaviours, and so forth. One important net result of these variations is nonlinearity, a condition where the proportional relationship between an input and an output changes with the size of the input. We have seen nonlinearity already of course: exponential growth, and the logistic function where population size increases rapidly when it is small, but only very slowly when near carrying capacity. Those models incorporate nonlinear relationships to describe how we think populations grow. Remove them and your model of population growth is reduced to a simpler, linear, more boring, and less accurate description of real populations. Nonlinearity is more fundamental, however, than a mere ingredient for enhancing model accuracy, because inequality is an inescapable feature of the natural world. If that realization creates some discomfort, perhaps it’s because we commonly equate “fairness” or “equality” with equilibrium, a “balance of Nature”. There is no balance in nature, and that sort of static stability is neither necessary nor capable of explaining the complexity of ecological phenomena. In coming sections we will explore many types and consequences of nonlinearity in ecological systems, how those lead to complex ecological systems, why I (and many others) believe that those systems are often far from equilibrium, and how nonlinearity, broader concepts of equilibrium, and complexity, all generate and explain many aspects of ecological systems. Before I go there though, I’ll discuss a concise nonlinear concept, one that not only captures the essence of why understanding nonlinearity and its implications is rewarding, but also has broad implications in biology.

**Jensen’s Inequality**

The previous post discussed the potentially misleading outcome of treating population dynamics in mean environments when environmental variation is omitted. Another issue related to interpretations or forecasts involve environmental averages, and arises when the relationship between λ (growth rate independent of population size) and an environmental driver is nonlinear. We assumed in the previous example that the relationship was a simple linear one, e.g. higher temperatures drive a constant proportional increase of birth rate. But metabolic, physiologic and other phenotypic traits often respond nonlinearly to controlling or input factors based on nonlinear phenotypic relationships (e.g. surface area to volume ratios), or differences of response timescales of various organic systems, among other factors. In such cases, the mean performance in a variable environment is not the same as performance at the environmental mean. This is known in mathematics as Jensen’s inequality, and is one consequence of nonlinear averaging or, more generally, making linear approximations of nonlinear curves or surfaces (see Denny, 2017, for a very accessible review).

For example, examine the relationship between water temperature (T) and λ in species belonging to the marine copepod genus *Arcatia* (Fig. 1) (Huntley and Lopez, 1992; Drake, 2005). The relationship is exponential, and within the range of observed temperatures, incremental increases of temperature result in proportionally greater increases of population growth rates at higher temperatures. Now consider the case of two populations, one inhabiting a region where daily temperatures vary little, with a mean temperature of 20°C. The other population experiences daily temperature fluctuations between 15°C and 25°C, and also experiences a daily mean temperature of 20°C. The growth rate of the first population is the expected λ given the dependency on temperature, but λ of the second population is the mean λ experienced over the temperature range. Because the relationship between λ and T is a nonlinear concave up function, the average growth rate under variable conditions is greater than the growth rate at the average daily temperature:

Eq. 1: [(average growth rate) equals (average of function of temperature variation)] is greater than [(average growth rate) equals (function of average temperature)

Populations will grow faster for the copepods living under variable temperatures than for those living at the mean of that variability. How much greater depends on the shape of the function, and the range of

environmental variation. The opposite is true if the relationship is a concave downward function.

**If the relationship between population growth rate and an environmental factor is nonlinear, then the average growth rate under variable conditions does not equal the growth rate under average conditions.**

As with environmental variation, these mathematical considerations take on increased significance under current global climate change conditions where both environmental means and variances are shifting (Drake, 2005; Pickett et al., 2015).

**Vocabulary***Law* — A scientific law is a statement, based on multiple observations or experiments, that describes a natural phenomenon. Laws do not have absolute certainty and can be refuted by future observations.*Nonlinear* — A nonlinear system is a system in which the change of the output is not proportional to the change of the input (Wikipedia).

**References**

- Denny, M. (2017). The fallacy of the average: on the ubiquity, utility and continuing novelty of Jensen’s inequality.
*Journal of Experimental Biology*, 220(2):139–146. - Drake, J. M. (2005). Population effects of increased climate variation.
*Proceedings of the Royal Society B: Biological Sciences*, 272(1574):1823–1827. - Huntley, M. E. and Lopez, M. D. (1992). Temperature-dependent production of marine copepods: a global synthesis.
*The American Naturalist, 140(2):201–242.*

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