Mean species responses predict effects of environmental change on coexistence.

Environmental change research is plagued by the curse of dimensionality: the number of communities at risk and the number of environmental drivers are both large. This raises the pressing question if a general understanding of ecological effects is achievable. Here, we show evidence that this is indeed possible. Using theoretical and simulation-based evidence for bi- and tritrophic communities, we show that environmental change effects on coexistence are proportional to mean species responses and depend on how trophic levels on average interact prior to environmental change. We then benchmark our findings using relevant cases of environmental change, showing that means of temperature optima and of species sensitivities to pollution predict concomitant effects on coexistence. Finally, we demonstrate how to apply our theory to the analysis of field data, finding support for effects of land use change on coexistence in natural invertebrate communities.

Competitive Exclusion and Evolution: Convergence Almost Never Produces Ecologically Equivalent Species: (A Comment on McPeek, "Limiting Similarity? The Ecological Dynamics of Natural Selection among Resources and Consumers Caused by Both Apparent and Resource Competition").

In a recent modeling study ("Limiting Similarity? The Ecological Dynamics of Natural Selection among Resources and Consumers Caused by Both Apparent and Resource Competition") that appeared in the April 2019 issue of The American Naturalist, Mark A. McPeek argued that ecologically equivalent species may emerge via competition-induced trait convergence, in conflict with naive expectations based on the limiting similarity principle. Although the emphasis on the possibility of the convergence of competitors is very timely, here we show that the proposed mechanism will only lead to actual coexistence in the converged state for specially chosen fine-tuned parameter settings. It is therefore not a robust mechanism for the evolution of ecologically equivalent species. We conclude that invoking trait convergence as an explanation for the co-occurrence of seemingly fully equivalent species in nature would be premature.

Let's Train More Theoretical Ecologists - Here Is Why.

A tangled web of vicious circles, driven by cultural issues, has prevented ecology from growing strong theoretical roots. Now this hinders development of effective conservation policies. To overcome these barriers in view of urgent societal needs, we propose a global network of postgraduate theoretical training programs.

Evolution of heritable behavioural differences in a model of social division of labour.

The spectacular diversity of personality and behaviour of animals and humans has evoked many hypotheses intended to explain its developmental and evolutionary background. Although the list of the possible contributing mechanisms seems long, we propose that an underemphasised explanation is the division of labour creating negative frequency dependent selection. We use analytical and numerical models of social division of labour to show how selection can create consistent and heritable behavioural differences in a population, where randomly sampled individuals solve a collective task together. We assume that the collective task needs collaboration of individuals performing one of the two possible subtasks. The total benefit of the group is highest when the ratio of different subtasks is closest to 1. The probability of choosing one of the two costly subtasks and the costs assigned to them are under selection. By using adaptive dynamics we show that if a trade-off between the costs of the subtasks is strong enough, then evolution leads to coexistence of specialized individuals performing one of the subtasks with high probability and low cost. Our analytical results were verified and extended by numerical simulations.

Sensitivity analysis of coexistence in ecological communities: theory and application.

Sensitivity analysis, the study of how ecological variables of interest respond to changes in external conditions, is a theoretically well-developed and widely applied approach in population ecology. Though the application of sensitivity analysis to predicting the response of species-rich communities to disturbances also has a long history, derivation of a mathematical framework for understanding the factors leading to robust coexistence has only been a recent undertaking. Here we suggest that this development opens up a new perspective, providing advances ranging from the applied to the theoretical. First, it yields a framework to be applied in specific cases for assessing the extinction risk of community modules in the face of environmental change. Second, it can be used to determine trait combinations allowing for coexistence that is robust to environmental variation, and limits to diversity in the presence of environmental variation, for specific community types. Third, it offers general insights into the nature of communities that are robust to environmental variation. We apply recent community-level extensions of mathematical sensitivity analysis to example models for illustration. We discuss the advantages and limitations of the method, and some of the empirical questions the theoretical framework could help answer.

Fixed point sensitivity analysis of interacting structured populations.

Sensitivity analysis of structured populations is a useful tool in population ecology. Historically, methodological development of sensitivity analysis has focused on the sensitivity of eigenvalues in linear matrix models, and on single populations. More recently there have been extensions to the sensitivity of nonlinear models, and to communities of interacting populations. Here we derive a fully general mathematical expression for the sensitivity of equilibrium abundances in communities of interacting structured populations. Our method yields the response of an arbitrary function of the stage class abundances to perturbations of any model parameters. As a demonstration, we apply this sensitivity analysis to a two-species model of ontogenetic niche shift where each species has two stage classes, juveniles and adults. In the context of this model, we demonstrate that our theory is quite robust to violating two of its technical assumptions: the assumption that the community is at a point equilibrium and the assumption of infinitesimally small parameter perturbations. Our results on the sensitivity of a community are also interpreted in a niche theoretical context: we determine how the niche of a structured population is composed of the niches of the individual states, and how the sensitivity of the community depends on niche segregation.

Coexistence in a fluctuating environment by the effect of relative nonlinearity: a minimal model.

The minimal model of the "relative nonlinearity" type fluctuation-maintained coexistence is investigated. The competing populations are affected by an environmental white noise. With quadratic density dependence, the long-term growth rates of the populations are determined by the average and the variance of the (fluctuating) total density. At most two species can coexist on these two "regulating" variables; competitive exclusion would ensue in a constant environment. A numerical study of the expected time until extinction of any of the two species reveals that the criterion of mutual invasibility predicts the parameter range of long-term coexistence correctly in the limit of zero extinction threshold. However, any extinction threshold consistent with a realistic population size will allow only short-term coexistence. Therefore, our simulations question the biological relevance of mutual invasibility, as a sufficient condition of coexistence, for large density fluctuations. We calculate the average and the variance of the fluctuating density of the coexisting populations analytically via the moment-closure approximation; the results are reasonably close to the simulated behavior. Based on this treatment, robustness of coexistence is studied in the limit of infinite population size. We interpret the results of this analysis in the context of necessity of niche segregation with respect to the regulating variables using a framework theory published earlier.

Limiting similarity and niche theory for structured populations.

We develop the theory of limiting similarity and niche for structured populations with finite number of individual states (i-state). In line with a previously published theory for unstructured populations, the niche of a species is specified by the impact and sensitivity niche vectors. They describe the population's impact on and sensitivity towards the variables involved in the population regulation. Robust coexistence requires sufficient segregation of the impact, as well as of the sensitivity niche vectors. Connection between the population-level impact and sensitivity and the impact/sensitivity of the specific i-states is developed. Each i-state contributes to the impact of the population proportional to its frequency in the population. Sensitivity of the population is composed of the sensitivity of the rates of demographic transitions, weighted by the frequency and by the reproductive value of the initial and final i-states of the transition, respectively. Coexistence in a multi-patch environment is studied. This analysis is interpreted as spatial niche segregation.

When the exception becomes the rule: the disappearance of limiting similarity in the Lotka-Volterra model.

We investigate the transition between limiting similarity and coexistence of a continuum in the competitive Lotka-Volterra model. It is known that there exist exceptional cases in which, contrary to the limiting similarity expectation, all phenotypes coexist along a trait axis. Earlier studies established that the distance between surviving phenotypes is in the magnitude of the niche width 2sigma provided that the carrying capacity curve differs from the exceptional one significantly enough. In this paper we studied the outcome of competition for small perturbations of the exceptional (Gaussian) carrying capacity. We found that the average distance between the surviving phenotypes goes to zero when the perturbation vanishes. The number of coexisting species in equilibrium is proportional to the negative logarithm of the perturbation. Nevertheless, the niche width provides a good order of magnitude for the distance between survivors if the perturbations are larger than 10%. Therefore, we conclude that limiting similarity is a good framework of biological thinking despite the lack of an absolute lower bound of similarity.

An analytically tractable model for competitive speciation.

Several recent models have shown that frequency-dependent disruptive selection created by intraspecific competition can lead to the evolution of assortative mating and, thus, to competitive sympatric speciation. However, since most of these results rely on limited numerical analyses, their generality has been debated. Here, we consider one of the standard models (the so-called Roughgarden model) with a simplified genetics where the selected trait is determined by a single diallelic locus. This model is sufficiently complex to maintain key properties of the general multilocus case but simple enough to allow for comprehensive analytical treatment by means of invasion fitness arguments. Depending on (1) the strength and (2) the shape of stabilizing selection, (3) the strength and (4) the shape of pairwise competition, (5) the shape of the mating function, and (6) whether assortative mating leads to sexual selection, we find five different evolutionary regimes. In one of these regimes, complete reproductive isolation can evolve through arbitrarily small steps in the strength of assortative mating. Our approach provides a mechanistic understanding of several phenomena that have been found in previous models. The results demonstrate how even in a simple model, the evolutionary outcome depends in a complex way on ecological and genetic parameters.

Adaptive dynamics for physiologically structured population models.

We develop a systematic toolbox for analyzing the adaptive dynamics of multidimensional traits in physiologically structured population models with point equilibria (sensu Dieckmann et al. in Theor. Popul. Biol. 63:309-338, 2003). Firstly, we show how the canonical equation of adaptive dynamics (Dieckmann and Law in J. Math. Biol. 34:579-612, 1996), an approximation for the rate of evolutionary change in characters under directional selection, can be extended so as to apply to general physiologically structured population models with multiple birth states. Secondly, we show that the invasion fitness function (up to and including second order terms, in the distances of the trait vectors to the singularity) for a community of N coexisting types near an evolutionarily singular point has a rational form, which is model-independent in the following sense: the form depends on the strategies of the residents and the invader, and on the second order partial derivatives of the one-resident fitness function at the singular point. This normal form holds for Lotka-Volterra models as well as for physiologically structured population models with multiple birth states, in discrete as well as continuous time and can thus be considered universal for the evolutionary dynamics in the neighbourhood of singular points. Only in the case of one-dimensional trait spaces or when N = 1 can the normal form be reduced to a Taylor polynomial. Lastly we show, in the form of a stylized recipe, how these results can be combined into a systematic approach for the analysis of the (large) class of evolutionary models that satisfy the above restrictions.

Competitive exclusion and limiting similarity: a unified theory.

Robustness of coexistence against changes of parameters is investigated in a model-independent manner by analyzing the feedback loop of population regulation. We define coexistence as a fixed point of the community dynamics with no population having zero size. It is demonstrated that the parameter range allowing coexistence shrinks and disappears when the Jacobian of the dynamics decreases to zero. A general notion of regulating factors/variables is introduced. For each population, its impact and sensitivity niches are defined as the differential impact on, and the differential sensitivity towards, the regulating variables, respectively. Either the similarity of the impact niches or the similarity of the sensitivity niches results in a small Jacobian and in a reduced likelihood of coexistence. For the case of a resource continuum, this result reduces to the usual "limited niche overlap" picture for both kinds of niche. As an extension of these ideas to the coexistence of infinitely many species, we demonstrate that Roughgarden's example for coexistence of a continuum of populations is structurally unstable.

Link between population dynamics and dynamics of Darwinian evolution.

We provide the link between population dynamics and the dynamics of Darwinian evolution via studying the joint population dynamics of similar populations. Similarity implies that the relative dynamics of the populations is slow compared to, and decoupled from, their aggregated dynamics. The relative dynamics is simple, and captured by a Taylor expansion in the difference between the populations. The emerging evolution is directional, except at the singular points of the evolutionary state space. Here "evolutionary branching" may occur. The diversification of life forms thus is demonstrated to be a natural consequence of the Darwinian process.

On the impossibility of coexistence of infinitely many strategies.

We investigate the possibility of coexistence of pure, inherited strategies belonging to a large set of potential strategies. We prove that under biologically relevant conditions every model allowing for coexistence of infinitely many strategies is structurally unstable. In particular, this is the case when the "interaction operator" which determines how the growth rate of a strategy depends on the strategy distribution of the population is compact. The interaction operator is not assumed to be linear. We investigate a Lotka-Volterra competition model with a linear interaction operator of convolution type separately because the convolution operator is not compact. For this model, we exclude the possibility of robust coexistence supported on the whole real line, or even on a set containing a limit point. Moreover, we exclude coexistence of an infinite set of equidistant strategies when the total population size is finite. On the other hand, for infinite populations it is possible to have robust coexistence in this case. These results are in line with the ecological concept of "limiting similarity" of coexisting species. We conclude that the mathematical structure of the ecological coexistence problem itself dictates the discreteness of the species.