A two-timescale model of plankton-oxygen dynamics predicts formation of oxygen minimum zones and global anoxia.

Decline of the dissolved oxygen in the ocean is a growing concern, as it may eventually lead to global anoxia, an elevated mortality of marine fauna and even a mass extinction. Deoxygenation of the ocean often results in the formation of oxygen minimum zones (OMZ): large domains where the abundance of oxygen is much lower than that in the surrounding ocean environment. Factors and processes resulting in the OMZ formation remain controversial. We consider a conceptual model of coupled plankton-oxygen dynamics that, apart from the plankton growth and the oxygen production by phytoplankton, also accounts for the difference in the timescales for phyto- and zooplankton (making it a "slow-fast system") and for the implicit effect of upper trophic levels resulting in density dependent (nonlinear) zooplankton mortality. The model is investigated using a combination of analytical techniques and numerical simulations. The slow-fast system is decomposed into its slow and fast subsystems. The critical manifold of the slow-fast system and its stability is then studied by analyzing the bifurcation structure of the fast subsystem. We obtain the canard cycles of the slow-fast system for a range of parameter values. However, the system does not allow for persistent relaxation oscillations; instead, the blowup of the canard cycle results in plankton extinction and oxygen depletion. For the spatially explicit model, the earlier works in this direction did not take into account the density dependent mortality rate of the zooplankton, and thus could exhibit Turing pattern. However, the inclusion of the density dependent mortality into the system can lead to stationary Turing patterns. The dynamics of the system is then studied near the Turing bifurcation threshold. We further consider the effect of the self-movement of the zooplankton along with the turbulent mixing. We show that an initial non-uniform perturbation can lead to the formation of an OMZ, which then grows in size and spreads over space. For a sufficiently large timescale separation, the spread of the OMZ can result in global anoxia.

Oscillations and Pattern Formation in a Slow-Fast Prey-Predator System.

We consider the properties of a slow-fast prey-predator system in time and space. We first argue that the simplicity of the prey-predator system is apparent rather than real and there are still many of its hidden properties that have been poorly studied or overlooked altogether. We further focus on the case where, in the slow-fast system, the prey growth is affected by a weak Allee effect. We first consider this system in the non-spatial case and make its comprehensive study using a variety of mathematical techniques. In particular, we show that the interplay between the Allee effect and the existence of multiple timescales may lead to a regime shift where small-amplitude oscillations in the population abundances abruptly change to large-amplitude oscillations. We then consider the spatially explicit slow-fast prey-predator system and reveal the effect of different timescales on the pattern formation. We show that a decrease in the timescale ratio may lead to another regime shift where the spatiotemporal pattern becomes spatially correlated, leading to large-amplitude oscillations in spatially average population densities and potential species extinction.

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.

Effect of complex landscape geometry on the invasive species spread: Invasion with stepping stones.

Spatial proliferation of invasive species often causes serious damage to agriculture, ecology and environment. Evaluation of the extent of the area potentially invadable by an alien species is an important problem. Landscape features that reduces dispersal space to narrow corridors can make some areas inaccessible to the invading species. On the other hand, the existence of stepping stones - small areas or 'patches' with better environmental conditions - is known to assist species spread. How an interplay between these factors can affect the invasion success remains unclear. In this paper, we address this question theoretically using a mechanistic model of population dynamics. Such models have been generally successful in predicting the rate and pattern of invasive spread; however, they usually consider the spread in an unbounded, uniform space hence ignoring the complex geometry of a real landscape. In contrast, here we consider a reaction-diffusion model in a domain of a complex shape combining corridors and stepping stones. We show that the invasion success depends on a subtle interplay between the stepping stone size, location and the strength of the Allee effect inside. In particular, for a stepping stone of a small size, there is only a narrow range of locations where it can unblock the otherwise impassable corridor.

Effect of density-dependent individual movement on emerging spatial population distribution: Brownian motion vs Levy flights.

Individual animal movement has been a focus of intense research and considerable controversy over the last two decades, however the understanding of wider ecological implications of various movement behaviours is lacking. In this paper, we consider this issue in the context of pattern formation. Using an individual-based modelling approach and computer simulations, we first show that density dependence ("auto-taxis") of the individual movement in a population of random walkers typically results in the formation of a strongly heterogeneous population distribution consisting of clearly defined animal clusters or patches. We then show that, when the movement takes place in a large spatial domain, the properties of the clusters are significantly different in the populations of Brownian and non-Brownian walkers. Whilst clusters tend to be stable in the case of Brownian motion, in the population of Levy walkers clusters are dynamical so that the number of clusters fluctuates in the course of time. We also show that the population dynamics of non-Brownian walkers exhibits two different time scales: a short time scale of the relaxation of the initial condition and a long time scale when one type of dynamics is replaced by another. Finally, we show that the distribution of sample values in the populations of Brownian and non-Brownian walkers is significantly different.

Regime shifts and ecological catastrophes in a model of plankton-oxygen dynamics under the climate change.

It is estimated that more than a half of the total atmospheric oxygen is produced in the oceans due to the photosynthetic activity of phytoplankton. Any significant decrease in the net oxygen production by phytoplankton is therefore likely to result in the depletion of atmospheric oxygen and in a global mass mortality of animals and humans. In its turn, the rate of oxygen production is known to depend on water temperature and hence can be affected by the global warming. We address this problem theoretically by considering a model of a coupled plankton-oxygen dynamics where the rate of oxygen production slowly changes with time to account for the ocean warming. We show that, when the temperature rises sufficiently high, a regime shift happens: the sustainable oxygen production becomes impossible and the system's dynamics leads to fast oxygen depletion and plankton extinction. We also consider a scenario when, after a certain period of increase, the temperature is set on a new higher yet apparently safe value, i.e. before the oxygen depletion disaster happens. We show that in this case the system dynamics may exhibit a long-term quasi-sustainable dynamics that can still result in an ecological disaster (oxygen depletion and mass extinctions) but only after a considerable period of time. Finally, we discuss the early warning signals of the approaching regime shift resulting in the disaster.

Fortune favours the brave: Movement responses shape demographic dynamics in strongly competing populations.

Animal movement is a key mechanism for shaping population dynamics. The effect of interactions between competing animals on a population's survival has been studied for many decades. However, interactions also affect an animal's subsequent movement decisions. Despite this, the indirect effect of these decisions on animal survival is much less well-understood. Here, we incorporate movement responses to foreign animals into a model of two competing populations, where inter-specific competition is greater than intra-specific competition. When movement is diffusive, the travelling wave moves from the stronger population to the weaker. However, by incorporating behaviourally induced directed movement towards the stronger population, the weaker one can slow the travelling wave down, even reversing its direction. Hence movement responses can switch the predictions of traditional mechanistic models. Furthermore, when environmental heterogeneity is combined with aggressive movement strategies, it is possible for spatially segregated co-existence to emerge. In this situation, the spatial patterns of the competing populations have the unusual feature that they are slightly out-of-phase with the environmental patterns. Finally, incorporating dynamic movement responses can also enable stable co-existence in a homogeneous environment, giving a new mechanism for spatially segregated co-existence.

Delay driven spatiotemporal chaos in single species population dynamics models.

Questions surrounding the prevalence of complex population dynamics form one of the central themes in ecology. Limit cycles and spatiotemporal chaos are examples that have been widely recognised theoretically, although their importance and applicability to natural populations remains debatable. The ecological processes underlying such dynamics are thought to be numerous, though there seems to be consent as to delayed density dependence being one of the main driving forces. Indeed, time delay is a common feature of many ecological systems and can significantly influence population dynamics. In general, time delays may arise from inter- and intra-specific trophic interactions or population structure, however in the context of single species populations they are linked to more intrinsic biological phenomena such as gestation or resource regeneration. In this paper, we consider theoretically the spatiotemporal dynamics of a single species population using two different mathematical formulations. Firstly, we revisit the diffusive logistic equation in which the per capita growth is a function of some specified delayed argument. We then modify the model by incorporating a spatial convolution which results in a biologically more viable integro-differential model. Using the combination of analytical and numerical techniques, we investigate the effect of time delay on pattern formation. In particular, we show that for sufficiently large values of time delay the system's dynamics are indicative to spatiotemporal chaos. The chaotic dynamics arising in the wake of a travelling population front can be preceded by either a plateau corresponding to dynamical stabilisation of the unstable equilibrium or by periodic oscillations.

How animals move along? Exactly solvable model of superdiffusive spread resulting from animal's decision making.

Patterns of individual animal movement have been a focus of considerable attention recently. Of particular interest is a question how different macroscopic properties of animal dispersal result from the stochastic processes occurring on the microscale of the individual behavior. In this paper, we perform a comprehensive analytical study of a model where the animal changes the movement velocity as a result of its behavioral response to environmental stochasticity. The stochasticity is assumed to manifest itself through certain signals, and the animal modifies its velocity as a response to the signals. We consider two different cases, i.e. where the change in the velocity is or is not correlated to its current value. We show that in both cases the early, transient stage of the animal movement is super-diffusive, i.e. ballistic. The large-time asymptotic behavior appears to be diffusive in the uncorrelated case but super-ballistic in the correlated case. We also calculate analytically the dispersal kernel of the movement and show that, whilst it converge to a normal distribution in the large-time limit, it possesses a fatter tail during the transient stage, i.e. at early and intermediate time. Since the transients are known to be highly relevant in ecology, our findings may indicate that the fat tails and superdiffusive spread that are sometimes observed in the movement data may be a feature of the transitional dynamics rather than an inherent property of the animal movement.

On time scale invariance of random walks in confined space.

Animal movement is often modelled on an individual level using simulated random walks. In such applications it is preferable that the properties of these random walks remain consistent when the choice of time is changed (time scale invariance). While this property is well understood in unbounded space, it has not been studied in detail for random walks in a confined domain. In this work we undertake an investigation of time scale invariance of the drift and diffusion rates of Brownian random walks subject to one of four simple boundary conditions. We find that time scale invariance is lost when the boundary condition is non-conservative, that is when movement (or individuals) is discarded due to boundary encounters. Where possible analytical results are used to describe the limits of the time scaling process, numerical results are then used to characterise the intermediate behaviour.

Mathematical Modelling of Plankton-Oxygen Dynamics Under the Climate Change.

Ocean dynamics is known to have a strong effect on the global climate change and on the composition of the atmosphere. In particular, it is estimated that about 70% of the atmospheric oxygen is produced in the oceans due to the photosynthetic activity of phytoplankton. However, the rate of oxygen production depends on water temperature and hence can be affected by the global warming. In this paper, we address this issue theoretically by considering a model of a coupled plankton-oxygen dynamics where the rate of oxygen production slowly changes with time to account for the ocean warming. We show that a sustainable oxygen production is only possible in an intermediate range of the production rate. If, in the course of time, the oxygen production rate becomes too low or too high, the system's dynamics changes abruptly, resulting in the oxygen depletion and plankton extinction. Our results indicate that the depletion of atmospheric oxygen on global scale (which, if happens, obviously can kill most of life on Earth) is another possible catastrophic consequence of the global warming, a global ecological disaster that has been overlooked.

Patchy Invasion of Stage-Structured Alien Species with Short-Distance and Long-Distance Dispersal.

Understanding of spatiotemporal patterns arising in invasive species spread is necessary for successful management and control of harmful species, and mathematical modeling is widely recognized as a powerful research tool to achieve this goal. The conventional view of the typical invasion pattern as a continuous population traveling front has been recently challenged by both empirical and theoretical results revealing more complicated, alternative scenarios. In particular, the so-called patchy invasion has been a focus of considerable interest; however, its theoretical study was restricted to the case where the invasive species spreads by predominantly short-distance dispersal. Meanwhile, there is considerable evidence that the long-distance dispersal is not an exotic phenomenon but a strategy that is used by many species. In this paper, we consider how the patchy invasion can be modified by the effect of the long-distance dispersal and the effect of the fat tails of the dispersal kernels.

Variation in individual walking behavior creates the impression of a Levy flight.

Many animal paths have an intricate statistical pattern that manifests itself as a power law-like tail in the distribution of movement lengths. Such distributions occur if individuals move according to a Lévy flight (a mode of dispersal in which the distance moved follows a power law), or if there is variation between individuals such that some individuals move much farther than others. Distinguishing between these two mechanisms requires large quantities of data, which are not available for most species studied. Here, we analyze paths of black bean aphids (Aphis fabae Scopoli) and show that individual animals move in a predominantly diffusive manner, but that, because of variation at population level, they collectively appear to display superdiffusive characteristics, often interpreted as being characteristic for a Lévy flight.

Synchronized dynamics of Tipula paludosa metapopulation in a southwestern Scotland agroecosystem: linking pattern to process.

Synchronization of population fluctuations at disjoint habitats has been observed in many studies, but its mechanisms often remain obscure. Synchronization may appear as a result of either interhabitat dispersal or regionally correlated environmental stochastic factors, the latter being known as the Moran effect. In this article, we consider the population dynamics of a common agricultural pest insect, Tipula paludosa, on a fragmented habitat by analyzing data derived from a multiannual survey of its abundance in 38 agricultural fields in southwestern Scotland. We use cross-correlation coefficients and show that there is a considerable synchronization between different populations across the whole area. The correlation strength exhibits an intermittent behavior, such that close populations can be virtually uncorrelated, but populations separated by distances up to approximately 150 km can have a cross-correlation coefficient close to one. To distinguish between the effects of stochasticity and dispersal, we then calculate a time-lagged cross-correlation coefficient and show that it possesses considerably different properties to the nonlagged one. In particular, the time-lagged correlation coefficient shows a clear directional dependence. The distribution of the time-lagged correlations with respect to the bearing between the populations has a striking similarity to the distribution of wind velocities, which we regard as evidence of long-distance wind-assisted dispersal.

Dynamical modelling of street protests using the Yellow Vest Movement and Khabarovsk as case studies.

Social protests, in particular in the form of street protests, are a frequent phenomenon of modern world often making a significant disruptive effect on the society. Understanding the factors that can affect their duration and intensity is therefore an important problem. In this paper, we consider a mathematical model of protests dynamics describing how the number of protesters change with time. We apply the model to two events such as the Yellow Vest Movement 2018-2019 in France and Khabarovsk protests 2019-2020 in Russia. We show that in both cases our model provides a good description of the protests dynamics. We consider how the model parameters can be estimated by solving the inverse problem based on the available data on protesters number at different time. The analysis of parameter sensitivity then allows for determining which factor(s) may have the strongest effect on the protests dynamics.

Knowledge gaps and missing links in understanding mass extinctions: Can mathematical modeling help?

Extinction of species, and even clades, is a normal part of the macroevolutionary process. However, several times in Earth history the rate of species and clade extinctions increased dramatically compared to the observed "background" extinction rate. Such episodes are global, short-lived, and associated with substantial environmental changes, especially to the carbon cycle. Consequently, these events are dubbed "mass extinctions" (MEs). Investigations surrounding the circumstances causing and/or contributing to mass extinctions are on-going, but consensus has not yet been reached, particularly as to common ME triggers or periodicities. In part this reflects the incomplete nature of the fossil and geologic record, which - although providing significant information about the taxa and paleoenvironmental context of MEs - is spatiotemporally discontinuous and preserved at relatively low resolution. Mathematical models provide an important opportunity to potentially compensate for missing linkages in data availability and resolution. Mathematical models may provide a means to connect ecosystem scale processes (i.e., the extinction of individual organisms) to global scale processes (i.e., extinction of whole species and clades). Such a view would substantially improve our understanding not only of how MEs precipitate, but also how biological and paleobiological sciences may inform each other. Here we provide suggestions for how to integrate mathematical models into ME research, starting with a change of focus from ME triggers to organismal kill mechanisms since these are much more standard across time and spatial scales. We conclude that the advantage of integrating mathematical models with standard geological, geochemical, and ecological methods is great and researchers should work towards better utilization of these methods in ME investigations.

A predictive model and a field study on heterogeneous slug distribution in arable fields arising from density dependent movement.

Factors and processes determining heterogeneous ('patchy') population distributions in natural environments have long been a major focus in ecology. Existing theoretical approaches proved to be successful in explaining vegetation patterns. In the case of animal populations, existing theories are at most conceptual: they may suggest a qualitative explanation but largely fail to explain patchiness quantitatively. We aim to bridge this knowledge gap. We present a new mechanism of self-organized formation of a patchy spatial population distribution. A factor that was under-appreciated by pattern formation theories is animal sociability, which may result in density dependent movement behaviour. Our approach was inspired by a recent project on movement and distribution of slugs in arable fields. The project discovered a strongly heterogeneous slug distribution and a specific density dependent individual movement. In this paper, we bring these two findings together. We develop a model of density dependent animal movement to account for the switch in the movement behaviour when the local population density exceeds a certain threshold. The model is fully parameterized using the field data. We then show that the model produces spatial patterns with properties closely resembling those observed in the field, in particular to exhibit similar values of the aggregation index.

Spatial ecology across scales.

The international conference 'Models in population dynamics and ecology 2010: animal movement, dispersal and spatial ecology' took place at the University of Leicester, UK, on 1-3 September 2010, focusing on mathematical approaches to spatial population dynamics and emphasizing cross-scale issues. Exciting new developments in scaling up from individual level movement to descriptions of this movement at the macroscopic level highlighted the importance of mechanistic approaches, with different descriptions at the microscopic level leading to different ecological outcomes. At higher levels of organization, different macroscopic descriptions of movement also led to different properties at the ecosystem and larger scales. New developments from Levy flight descriptions to the incorporation of new methods from physics and elsewhere are revitalizing research in spatial ecology, which will both increase understanding of fundamental ecological processes and lead to tools for better management.

Pattern formation, long-term transients, and the Turing-Hopf bifurcation in a space- and time-discrete predator-prey system.

Understanding of population dynamics in a fragmented habitat is an issue of considerable importance. A natural modelling framework for these systems is spatially discrete. In this paper, we consider a predator-prey system that is discrete both in space and time, and is described by a Coupled Map Lattice (CML). The prey growth is assumed to be affected by a weak Allee effect and the predator dynamics includes intra-specific competition. We first reveal the bifurcation structure of the corresponding non-spatial system. We then obtain the conditions of diffusive instability on the lattice. In order to reveal the properties of the emerging patterns, we perform extensive numerical simulations. We pay a special attention to the system properties in a vicinity of the Turing-Hopf bifurcation, which is widely regarded as a mechanism of pattern formation and spatiotemporal chaos in space-continuous systems. Counter-intuitively, we obtain that the spatial patterns arising in the CML are more typically stationary, even when the local dynamics is oscillatory. We also obtain that, for some parameter values, the system's dynamics is dominated by long-term transients, so that the asymptotical stationary pattern arises as a sudden transition between two different patterns. Finally, we argue that our findings may have important ecological implications.