Mutualism at the leading edge: insights into the eco-evolutionary dynamics of host-symbiont communities during range expansion.

The evolution of mutualism between host and symbiont communities plays an essential role in maintaining ecosystem function and should therefore have a profound effect on their range expansion dynamics. In particular, the presence of mutualistic symbionts at the leading edge of a host-symbiont community should enhance its propagation in space. We develop a theoretical framework that captures the eco-evolutionary dynamics of host-symbiont communities, to investigate how the evolution of resource exchange may shape community structure during range expansion. We consider a community with symbionts that are mutualistic or parasitic to various degrees, where parasitic symbionts receive the same amount of resource from the host as mutualistic symbionts, but at a lower cost. The selective advantage of parasitic symbionts over mutualistic ones is increased with resource availability (i.e. with host density), promoting mutualism at the range edges, where host density is low, and parasitism at the population core, where host density is higher. This spatial selection also influences the speed of spread. We find that the host growth rate (which depends on the average benefit provided by the symbionts) is maximal at the range edges, where symbionts are more mutualistic, and that host-symbiont communities with high symbiont density at their core (e.g. resulting from more mutualistic hosts) spread faster into new territories. These results indicate that the expansion of host-symbiont communities is pulled by the hosts but pushed by the symbionts, in a unique push-pull dynamic where both the host and symbionts are active and tightly-linked players.

Stochastic resonance in climate reddening increases the risk of cyclic ecosystem extinction via phase-tipping.

Human activity is leading to changes in the mean and variability of climatic parameters in most locations around the world. The changing mean has received considerable attention from scientists and climate policy makers. However, recent work indicates that the changing variability, that is, the amplitude and the temporal autocorrelation of deviations from the mean, may have greater and more imminent impact on ecosystems. In this paper, we demonstrate that changes in climate variability alone could drive cyclic predator-prey ecosystems to extinction via so-called phase-tipping (P-tipping), a new type of instability that occurs only from certain phases of the predator-prey cycle. We construct a mathematical model of a variable climate and couple it to two self-oscillating paradigmatic predator-prey models. Most importantly, we combine realistic parameter values for the Canada lynx and snowshoe hare with actual climate data from the boreal forest. In this way, we demonstrate that critically important species in the boreal forest have increased likelihood of P-tipping to extinction under predicted changes in climate variability, and are most vulnerable during stages of the cycle when the predator population is near its maximum. Furthermore, our analysis reveals that stochastic resonance is the underlying mechanism for the increased likelihood of P-tipping to extinction.

Memory-guided foraging and landscape design interact to determine ecosystem services.

Many studies examine how the landscape affects memory-informed movement patterns, but very few examine how memory-informed foragers influence the landscape. This reverse relationship is an important factor in preventing the continued decline of many ecosystem services. We investigate this question in the context of crop pollination services by wild bees, a critical ecosystem service that is in steep decline. Many studies suggest that adding wild flower patches near crops can result in higher crop pollination services, but specific advice pertaining to the optimal location and density of these wild flower patches is lacking, as well as any estimate of the expected change in crop pollination services. In this work, we seek to understand what is the optimal placement of a flower patch relative to a single crop field, during crop bloom and considering spatial factors alone. We develop an individual based model of memory-based foraging by bumble bees to simulate bee movement from a single nest while the crop is in bloom, and measure the resulting crop pollination services. We consider a single crop field enhanced with a wild flower patch in a variable location, and measure crop flower visitation over the course of a single day. We analyze the pollination intensity and spatial distribution of flower visits to determine optimal wild flower patch placement for an isolated crop field. We find that the spatial arrangement of crop and wild flower patch have a significant effect on the number of crop flower visits, and that these effects arise from the memory-informed foraging pattern. The most effective planting locations are either in the centre of the crop field or on the far side of the crop field, away from the single bumble bee nest.

Differential equation model for central-place foragers with memory: implications for bumble bee crop pollination.

Bumble bees provide valuable pollination services to crops around the world. However, their populations are declining in intensively farmed landscapes. Understanding the dispersal behaviour of these bees is a key step in determining how agricultural landscapes can best be enhanced for bumble bee survival. Here we develop a partial integro-differential equation model to predict the spatial distribution of foraging bumble bees in dynamic heterogeneous landscapes. In our model, the foraging population is divided into two subpopulations, one engaged in an intensive search mode (modeled by diffusion) and the other engaged in an extensive search mode (modeled by advection). Our model considers the effects of resource-dependent switching rates between movement modes, resource depletion, central-place foraging behaviour, and memory. We use our model to investigate how crop pollination services are affected by wildflower enhancements. We find that planting wildflowers such that the crop is located in between the wildflowers and the nest site can benefit crop pollination in two different scenarios. If the bees do not have a strong preference for wildflowers, a small or low density wildflower patch is beneficial. If, on the other hand, the bees strongly prefer the wildflowers, then a large or high density wildflower patch is beneficial. The increase of the crop pollination services in the later scenario is of remarkable magnitude.

Towards Building a Sustainable Future: Positioning Ecological Modelling for Impact in Ecosystems Management.

As many ecosystems worldwide are in peril, efforts to manage them sustainably require scientific advice. While numerous researchers around the world use a great variety of models to understand ecological dynamics and their responses to disturbances, only a small fraction of these models are ever used to inform ecosystem management. There seems to be a perception that ecological models are not useful for management, even though mathematical models are indispensable in many other fields. We were curious about this mismatch, its roots, and potential ways to overcome it. We searched the literature on recommendations and best practices for how to make ecological models useful to the management of ecosystems and we searched for 'success stories' from the past. We selected and examined several cases where models were instrumental in ecosystem management. We documented their success and asked whether and to what extent they followed recommended best practices. We found that there is not a unique way to conduct a research project that is useful in management decisions. While research is more likely to have impact when conducted with many stakeholders involved and specific to a situation for which data are available, there are great examples of small groups or individuals conducting highly influential research even in the absence of detailed data. We put the question of modelling for ecosystem management into a socio-economic and national context and give our perspectives on how the discipline could move forward.

Co-inoculation with arbuscular mycorrhizal fungi differing in carbon sink strength induces a synergistic effect in plant growth.

Arbuscular mycorrhizal (AM) fungi play a key role in determining ecosystem functionality. Understanding how diversity in the fungal community affects plant productivity is therefore an important question in ecology. Current research has focused on understanding the role of functional complementarity in the fungal community when the host plant faces multiple stress factors. Fewer studies, however, have investigated how variation in traits affecting nutrient exchange can impact the plant growth dynamics, even in the absence of environmental stressors. Combining experimental data and a mathematical model based on ordinary differential equations, we investigate the role played by carbon sink strength on plant productivity. We simulate and measure plant growth over time when the plant is associated with two fungal isolates with different carbon sink strength, and when the plant is in pairwise association with each of the isolates alone. Overall, our theoretical as well as our experimental results show that co-inoculation with fungi with different carbon sink strength can induce positive non-additive effects (or synergistic effects) in plant productivity. Fungi with high carbon sink strength are able to quickly establish a fungal community and increase the nutrient supply to the plant, with a consequent positive impact on plant growth rate. On the other side, fungi with low carbon sink strength inflict lower carbon costs to the host plant, and support maximal plant productivity once plant biomass is large. As AM fungi are widely used as organic fertilizers worldwide, our findings have important implications for restoration ecology and agricultural management.

Phase tipping: how cyclic ecosystems respond to contemporary climate.

We identify the phase of a cycle as a new critical factor for tipping points (critical transitions) in cyclic systems subject to time-varying external conditions. As an example, we consider how contemporary climate variability induces tipping from a predator-prey cycle to extinction in two paradigmatic predator-prey models with an Allee effect. Our analysis of these examples uncovers a counterintuitive behaviour, which we call phase tipping or P-tipping, where tipping to extinction occurs only from certain phases of the cycle. To explain this behaviour, we combine global dynamics with set theory and introduce the concept of partial basin instability for attracting limit cycles. This concept provides a general framework to analyse and identify easily testable criteria for the occurrence of phase tipping in externally forced systems, and can be extended to more complicated attractors.

Diversity within mutualist guilds promotes coexistence and reduces the risk of invasion from an alien mutualist.

Biodiversity is an important component of healthy ecosystems, and thus understanding the mechanisms behind species coexistence is critical in ecology and conservation biology. In particular, few studies have focused on the dynamics resulting from the co-occurrence of mutualistic and competitive interactions within a group of species. Here we build a mathematical model to study the dynamics of a guild of competitors who are also engaged in mutualistic interactions with a common partner. We show that coexistence as well as competitive exclusion can occur depending on the competition strength and on strength of the mutualistic interactions, and we formulate concrete criteria for predicting invasion success of an alien mutualist based on propagule pressure, alien traits (such as its resource exchange ability) and composition of the recipient community. We find that intra guild diversity promotes the coexistence of species that would otherwise competitively exclude each other, and makes a guild less vulnerable to invasion. Our results can serve as a useful framework to predict the consequences of species manipulation in mutualistic communities.

The Timing and Nature of Behavioural Responses Affect the Course of an Epidemic.

During an epidemic, the interplay of disease and opinion dynamics can lead to outcomes that are different from those predicted based on disease dynamics alone. Opinions and the behaviours they elicit are complex, so modelling them requires a measure of abstraction and simplification. Here, we develop a differential equation model that couples SIR-type disease dynamics with opinion dynamics. We assume a spectrum of opinions that change based on current levels of infection as well as interactions that to some extent amplify the opinions of like-minded individuals. Susceptibility to infection is based on the level of prophylaxis (disease avoidance) that an opinion engenders. In this setting, we observe how the severity of an epidemic is influenced by the distribution of opinions at disease introduction, the relative rates of opinion and disease dynamics, and the amount of opinion amplification. Some insight is gained by considering how the effective reproduction number is influenced by the combination of opinion and disease dynamics.

Random fluctuations around a stable limit cycle in a stochastic system with parametric forcing.

Many real populations exhibit stochastic behaviour that appears to have some periodicity. In terms of populations, these time series can occur as limit cycles that arise through seasonal variation of parameters such as, e.g., disease transmission rate. The general mathematical context is that of a stochastic differential system with periodic parametric forcing whose solution is a stochastically perturbed limit cycle. Earlier work identified the power spectral density (PSD) features of these fluctuations by computation of the autocorrelation function of the stochastic process and its transform. Here, we present an alternative analysis which shows that the structure of the fluctuations around the limit cycle is analogous to that of fluctuations about a fixed point. Furthermore, we show that these fluctuations can be expressed, approximately, as a factorization which reveals the combined frequencies of the limit cycle and the stochastic perturbation. This result, based on a new limit theorem near a Hopf point, yields an understanding of the previously found features of the PSD. Further insights are obtained from the corresponding stochastic equations for phase and amplitude.

Spatial opinion dynamics and the effects of two types of mixing.

Spatially situated opinions that can be held with different degrees of conviction lead to spatiotemporal patterns such as clustering (homophily), polarization, and deadlock. Our goal is to understand how sensitive these patterns are to changes in the local nature of interactions. We introduce two different mixing mechanisms, spatial relocation and nonlocal interaction ("telephoning"), to an earlier fully spatial model (no mixing). Interestingly, the mechanisms that create deadlock in the fully spatial model have the opposite effect when there is a sufficient amount of mixing. With telephoning, not only is polarization and deadlock broken up, but consensus is hastened. The effects of mixing by relocation are even more pronounced. Further insight into these dynamics is obtained for selected parameter regimes via comparison to the mean-field differential equations.

Moving forward in circles: challenges and opportunities in modelling population cycles.

Population cycling is a widespread phenomenon, observed across a multitude of taxa in both laboratory and natural conditions. Historically, the theory associated with population cycles was tightly linked to pairwise consumer-resource interactions and studied via deterministic models, but current empirical and theoretical research reveals a much richer basis for ecological cycles. Stochasticity and seasonality can modulate or create cyclic behaviour in non-intuitive ways, the high-dimensionality in ecological systems can profoundly influence cycling, and so can demographic structure and eco-evolutionary dynamics. An inclusive theory for population cycles, ranging from ecosystem-level to demographic modelling, grounded in observational or experimental data, is therefore necessary to better understand observed cyclical patterns. In turn, by gaining better insight into the drivers of population cycles, we can begin to understand the causes of cycle gain and loss, how biodiversity interacts with population cycling, and how to effectively manage wildly fluctuating populations, all of which are growing domains of ecological research.

A Lévy-flight diffusion model to predict transgenic pollen dispersal.

The containment of genetically modified (GM) pollen is an issue of significant concern for many countries. For crops that are bee-pollinated, model predictions of outcrossing rates depend on the movement hypothesis used for the pollinators. Previous work studying pollen spread by honeybees, the most important pollinator worldwide, was based on the assumption that honeybee movement can be well approximated by Brownian motion. A number of recent studies, however, suggest that pollinating insects such as bees perform Lévy flights in their search for food. Such flight patterns yield much larger rates of spread, and so the Brownian motion assumption might significantly underestimate the risk associated with GM pollen outcrossing in conventional crops. In this work, we propose a mechanistic model for pollen dispersal in which the bees perform truncated Lévy flights. This assumption leads to a fractional-order diffusion model for pollen that can be tuned to model motion ranging from pure Brownian to pure Lévy. We parametrize our new model by taking the same pollen dispersal dataset used in Brownian motion modelling studies. By numerically solving the model equations, we show that the isolation distances required to keep outcrossing levels below a certain threshold are substantially increased by comparison with the original predictions, suggesting that isolation distances may need to be much larger than originally thought.