Processes governing species richness in communities exposed to temporal environmental stochasticity: A review and synthesis of modelling approaches.

Research into the processes governing species richness has often assumed that the environment is fixed, whereas realistic environments are often characterised by random fluctuations over time. This temporal environmental stochasticity (TES) changes the demographic rates of species populations, with cascading effects on community dynamics and species richness. Theoretical and applied studies have used process-based mathematical models to determine how TES affects species richness, but under a variety of frameworks. Here, we critically review such studies to synthesise their findings and draw general conclusions. We first provide a broad mathematical framework encompassing the different ways in which TES has been modelled. We then review studies that have analysed models with TES under the assumption of negligible interspecific interactions, such that a community is conceptualised as the sum of independent species populations. These analyses have highlighted how TES can reduce species richness by increasing the frequency at which a species becomes rare and therefore prone to extinction. Next, we review studies that have relaxed the assumption of negligible interspecific interactions. To simplify the corresponding models and make them analytically tractable, such studies have used mean-field theory to derive fixed parameters representing the typical strength of interspecific interactions under TES. The resulting analyses have highlighted community-level effects that determine how TES affects species richness, for species that compete for a common limiting resource. With short temporal correlations of environmental conditions, a non-linear averaging effect of interspecific competition strength over time gives an increase in species richness. In contrast, with long temporal correlations of environmental conditions, strong selection favouring the fittest species between changes in environmental conditions results in a decrease in species richness. We compare such results with those from invasion analysis, which examines invasion growth rates (IGRs) instead of species richness directly. Qualitative differences sometimes arise because the IGR is the expected growth rate of a species when it is rare, which does not capture the variation around this mean or the probability of the species becoming rare. Our review elucidates key processes that have been found to mediate the negative and positive effects of TES on species richness, and by doing so highlights key areas for future research.

Quantifying invasibility.

Invasibility, the chance of a population to grow from rarity and become established, plays a fundamental role in population genetics, ecology, epidemiology and evolution. For many decades, the mean growth rate of a species when it is rare has been employed as an invasion criterion. Recent studies show that the mean growth rate fails as a quantitative metric for invasibility, with its magnitude sometimes even increasing while the invasibility decreases. Here we provide two novel formulae, based on the diffusion approximation and a large-deviations (Wentzel-Kramers-Brillouin) approach, for the chance of invasion given the mean growth and its variance. The first formula has the virtue of simplicity, while the second one holds over a wider parameter range. The efficacy of the formulae, including their accompanying data analysis technique, is demonstrated using synthetic time series generated from canonical models and parameterised with empirical data.

How temporal environmental stochasticity affects species richness: Destabilization, neutralization and the storage effect.

Temporal environmental stochasticity (TES), along with the variations of demographic rates associated with it, is ubiquitous in nature. Here we study the effect of TES on the species richness of diverse communities. In such communities the biodiversity at equilibrium reflects the balance between the rate at which new types are added (via migration, mutation or speciation) and the rate of extinction. We analyze a few generic models in which the speciation rate is fixed and TES affects the rate of extinction, and identify three different mechanisms. First, TES increases abundance variations and shortens extinction times, thus decreasing the species richness (destabilizing effect). Second, TES blurs the time-independent fitness differences between species, making the dynamics more symmetric and thereby increasing the diversity (neutralizing effect). Third, the storage effect allows TES to facilitate the invasion of inferior species, again contributing to the species richness. The stabilizing effect of storage declines significantly in diverse communities and it can overcome the destabilizing effect of TES only when environmental fluctuations are rapid enough.

Invasion growth rate and its relevance to persistence: a response to Technical Comment by Ellner et al.

Ellner et al. (2020) state that identifying the mechanisms producing positive invasion growth rates (IGR) is useful in characterising species persistence. We agree about the importance of the sign of IGR as a binary indicator of persistence, but question whether its magnitude provides much information once the sign is given.

Taming the diffusion approximation through a controlling-factor WKB method.

The diffusion approximation (DA) is widely used in the analysis of stochastic population dynamics, from population genetics to ecology and evolution. The DA is an uncontrolled approximation that assumes the smoothness of the calculated quantity over the relevant state space and fails when this property is not satisfied. This failure becomes severe in situations where the direction of selection switches sign. Here we employ the WKB (Wentzel-Kramers-Brillouin) large-deviations method, which requires only the logarithm of a given quantity to be smooth over its state space. Combining the WKB scheme with asymptotic matching techniques, we show how to derive the diffusion approximation in a controlled manner and how to produce better approximations, applicable for much wider regimes of parameters. We also introduce a scalable (independent of population size) WKB-based numerical technique. The method is applied to a central problem in population genetics and evolution, finding the chance of ultimate fixation in a zero-sum, two-types competition.

Mean growth rate when rare is not a reliable metric for persistence of species.

The coexistence of many species within ecological communities poses a long-standing theoretical puzzle. Modern coexistence theory (MCT) and related techniques explore this phenomenon by examining the chance of a species population growing from rarity in the presence of all other species. The mean growth rate when rare, E [ r ] , is used in MCT as a metric that measures persistence properties (like invasibility or time to extinction) of a population. Here we critique this reliance on E [ r ] and show that it fails to capture the effect of temporal random abundance variations on persistence properties. The problem becomes particularly severe when an increase in the amplitude of stochastic temporal environmental variations leads to an increase in E [ r ] , since at the same time it enhances random abundance fluctuations and the two effects are inherently intertwined. In this case, the chance of invasion and the mean extinction time of a population may even go down as E [ r ] increases.

Optimal motion of triangular magnetocapillary swimmers.

A system of ferromagnetic particles trapped at a liquid-liquid interface and subjected to a set of magnetic fields (magnetocapillary swimmers) is studied numerically using a hybrid method combining the pseudopotential lattice Boltzmann method and the discrete element method. After investigating the equilibrium properties of a single, two, and three particles at the interface, we demonstrate a controlled motion of the swimmer formed by three particles. It shows a sharp dependence of the average center-of-mass speed on the frequency of the time-dependent external magnetic field. Inspired by experiments on magnetocapillary microswimmers, we interpret the obtained maxima of the swimmer speed by the optimal frequency centered around the characteristic relaxation time of a spherical particle. It is also shown that the frequency corresponding to the maximum speed grows and the maximum average speed decreases with increasing interparticle distances at moderate swimmer sizes. The findings of our lattice Boltzmann simulations are supported by bead-spring model calculations.

Effect of body deformability on microswimming.

In this work we consider the following question: given a mechanical microswimming mechanism, does increased deformability of the swimmer body hinder or promote the motility of the swimmer? To answer this we run immersed-boundary-lattice-Boltzmann simulations of a microswimmer composed of deformable beads connected with springs. We find that the same deformations in the beads can result in different effects on the swimming velocity, namely an enhancement or a reduction, depending on the other parameters. To understand this we determine analytically the velocity of the swimmer, starting from the forces driving the motion and assuming that the deformations in the beads are known as functions of time and are much smaller than the beads themselves. We find that to the lowest order, only the driving frequency mode of the surface deformations contributes to the swimming velocity, and comparison to the simulations shows that both the velocity-promoting and velocity-hindering effects of bead deformability are reproduced correctly by the theory in the limit of small bead deformations. For the case of active deformations we show that there are critical values of the spring constant - which for a general swimmer corresponds to its main elastic degree of freedom - which decide whether the body deformability is beneficial for motion or not.

Lattice Boltzmann simulations of the bead-spring microswimmer with a responsive stroke-from an individual to swarms.

Propulsion at low Reynolds numbers is often studied by defining artificial microswimmers which exhibit a particular stroke. The disadvantage of such an approach is that the stroke does not adjust to the environment, in particular the fluid flow, which can diminish the effect of hydrodynamic interactions. To overcome this limitation, we simulate a microswimmer consisting of three beads connected by springs and dampers, using the self-developed waLBerla and [Formula: see text] framework based on the lattice Boltzmann method and the discrete element method. In our approach, the swimming stroke of a swimmer emerges as a balance of the drag, the driving and the elastic internal forces. We validate the simulations by comparing the obtained swimming velocity to the velocity found analytically using a perturbative method where the bead oscillations are taken to be small. Including higher-order terms in the hydrodynamic interactions between the beads improves the agreement to the simulations in parts of the parameter space. Encouraged by the agreement between the theory and the simulations and aided by the massively parallel capabilities of the waLBerla-[Formula: see text] framework, we simulate more than ten thousand such swimmers together, thus presenting the first fully resolved simulations of large swarms with active responsive components.

Forces and shapes as determinants of micro-swimming: effect on synchronisation and the utilisation of drag.

In this analytical study we demonstrate the richness of behaviour exhibited by bead-spring micro-swimmers, both in terms of known yet not fully explained effects such as synchronisation, and hitherto undiscovered phenomena such as the existence of two transport regimes where the swimmer shape has fundamentally different effects on the velocity. For this purpose we employ a micro-swimmer model composed of three arbitrarily-shaped rigid beads connected linearly by two springs. By analysing this swimmer in terms of the forces on the different beads, we determine the optimal kinematic parameters for sinusoidal driving, and also explain the pusher/puller nature of the swimmer. Moreover, we show that the phase difference between the swimmer's arms automatically attains values which maximise the swimming speed for a large region of the parameter space. Apart from this, we determine precisely the optimal bead shapes that maximise the velocity when the beads are constrained to be ellipsoids of a constant volume or surface area. On doing so, we discover the surprising existence of the aforementioned transport regimes in micro-swimming, where the motion is dominated by either a reduction of the drag force opposing the beads, or by the hydrodynamic interaction amongst them. Under some conditions, these regimes lead to counter-intuitive effects such as the most streamlined shapes forming locally the slowest swimmers.