A Comparison of the "Reduced Losses" and "Increased Production" Models for Mussel Bed Dynamics.

Self-organised regular pattern formation is one of the foremost examples of the development of complexity in ecosystems. Despite the wide array of mechanistic models that have been proposed to understand pattern formation, there is limited general understanding of the feedback processes causing pattern formation in ecosystems, and how these affect ecosystem patterning and functioning. Here we propose a generalised model for pattern formation that integrates two types of within-patch feedback: amplification of growth and reduction of losses. Both of these mechanisms have been proposed as causing pattern formation in mussel beds in intertidal regions, where dense clusters of mussels form, separated by regions of bare sediment. We investigate how a relative change from one feedback to the other affects the stability of uniform steady states and the existence of spatial patterns. We conclude that there are important differences between the patterns generated by the two mechanisms, concerning both biomass distribution in the patterns and the resilience of the ecosystems to disturbances.

An integrodifference model for vegetation patterns in semi-arid environments with seasonality.

Vegetation patterns are a characteristic feature of semi-deserts occurring on all continents except Antarctica. In some semi-arid regions, the climate is characterised by seasonality, which yields a synchronisation of seed dispersal with the dry season or the beginning of the wet season. We reformulate the Klausmeier model, a reaction-advection-diffusion system that describes the plant-water dynamics in semi-arid environments, as an integrodifference model to account for the temporal separation of plant growth processes during the wet season and seed dispersal processes during the dry season. The model further accounts for nonlocal processes involved in the dispersal of seeds. Our analysis focusses on the onset of spatial patterns. The Klausmeier partial differential equations (PDE) model is linked to the integrodifference model in an appropriate limit, which yields a control parameter for the temporal separation of seed dispersal events. We find that the conditions for pattern onset in the integrodifference model are equivalent to those for the continuous PDE model and hence independent of the time between seed dispersal events. We thus conclude that in the context of seed dispersal, a PDE model provides a sufficiently accurate description, even if the environment is seasonal. This emphasises the validity of results that have previously been obtained for the PDE model. Further, we numerically investigate the effects of changes to seed dispersal behaviour on the onset of patterns. We find that long-range seed dispersal inhibits the formation of spatial patterns and that the seed dispersal kernel's decay at infinity is a significant regulator of patterning.

Long-distance seed dispersal affects the resilience of banded vegetation patterns in semi-deserts.

Landscape-scale vegetation stripes (tiger bush) observed on the gentle slopes of semi-arid regions are useful indicators of future ecosystem degradation and catastrophic shifts towards desert. Mathematical models like the Klausmeier model-a set of coupled partial differential equations describing vegetation and water densities in space and time-are central to understanding their formation and development. One assumption made for mathematical simplicity is the local dispersal of seeds via a diffusion term. In fact, a large amount of work focuses on fitting dispersal 'kernels', probability density functions for seed dispersal distance, to empirical data of different species and modes of dispersal. In this paper, we address this discrepancy by analysing an extended Klausmeier model that includes long-distance seed dispersal via a non-local convolution term in place of diffusion, and assessing its effect on the resilience of striped patterns. Many authors report a slow uphill migration of stripes; but others report no detectable migration speed. We show that long-distance seed dispersal permits the formation of patterns with a very slow (possibly undetectable) migration speed, and even stationary patterns which could explain the inconsistencies in the empirical data. In general, we show that the resilience of patterns to reduced rainfall may vary significantly depending on the rate of seed dispersal and the width of the dispersal kernel, and compare a selection of ecologically relevant kernels to examine the variation in pattern resilience.

Metastability as a Coexistence Mechanism in a Model for Dryland Vegetation Patterns.

Vegetation patterns are a ubiquitous feature of water-deprived ecosystems. Despite the competition for the same limiting resource, coexistence of several plant species is commonly observed. We propose a two-species reaction-diffusion model based on the single-species Klausmeier model, to analytically investigate the existence of states in which both species coexist. Ecologically, the study finds that coexistence is supported if there is a small difference in the plant species' average fitness, measured by the ratio of a species' capabilities to convert water into new biomass to its mortality rate. Mathematically, coexistence is not a stable solution of the system, but both spatially uniform and patterned coexistence states occur as metastable states. In this context, a metastable solution in which both species coexist corresponds to a long transient (exceeding [Formula: see text] years in dimensional parameters) to a stable one-species state. This behaviour is characterised by the small size of a positive eigenvalue which has the same order of magnitude as the average fitness difference between the two species. Two mechanisms causing the occurrence of metastable solutions are established: a spatially uniform unstable equilibrium and a stable one-species pattern which is unstable to the introduction of a competitor. We further discuss effects of asymmetric interspecific competition (e.g. shading) on the metastability property.

Large scale patterns in mussel beds: stripes or spots?

An aerial view of an intertidal mussel bed often reveals large scale striped patterns aligned perpendicular to the direction of the tide; dense bands of mussels alternate periodically with near bare sediment. Experimental work led to the formulation of a set of coupled partial differential equations modelling a mussel-algae interaction, which proved pivotal in explaining the phenomenon. The key class of model solutions to consider are one-dimensional periodic travelling waves (wavetrains) that encapsulate the abundance of peak and trough mussel densities observed in practice. These solutions may, or may not, be stable to small perturbations, and previous work has focused on determining the ecologically relevant (stable) wavetrain solutions in terms of model parameters. The aim of this paper is to extend this analysis to two space dimensions by considering the full stripe pattern solution in order to study the effect of transverse two-dimensional perturbations-a more true to life problem. Using numerical continuation techniques, we find that some striped patterns that were previously deemed stable via the consideration of the associated wavetrain solution, are in fact unstable to transverse two-dimensional perturbations; and numerical simulation of the model shows that they break up to form regular spotted patterns. In particular, we show that break up of stripes into spots is a consequence of low tidal flow rates. Our consideration of random algal movement via a dispersal term allows us to show that a higher algal dispersal rate facilitates the formation of stripes at lower flow rates, but also encourages their break up into spots. We identify a novel hysteresis effect in mussel beds that is a consequence of transverse perturbations.

Analysis of a model for banded vegetation patterns in semi-arid environments with nonlocal dispersal.

Vegetation patterns are a characteristic feature of semi-arid regions. On hillsides these patterns occur as stripes running parallel to the contours. The Klausmeier model, a coupled reaction-advection-diffusion system, is a deliberately simple model describing the phenomenon. In this paper, we replace the diffusion term describing plant dispersal by a more realistic nonlocal convolution integral to account for the possibility of long-range dispersal of seeds. Our analysis focuses on the rainfall level at which there is a transition between uniform vegetation and pattern formation. We obtain results, valid to leading order in the large parameter comparing the rate of water flow downhill to the rate of plant dispersal, for a negative exponential dispersal kernel. Our results indicate that both a wider dispersal of seeds and an increase in dispersal rate inhibit the formation of patterns. Assuming an evolutionary trade-off between these two quantities, mathematically motivated by the limiting behaviour of the convolution term, allows us to make comparisons to existing results for the original reaction-advection-diffusion system. These comparisons show that the nonlocal model always predicts a larger parameter region supporting pattern formation. We then numerically extend the results to other dispersal kernels, showing that the tendency to form patterns depends on the type of decay of the kernel.

Using wavelength and slope to infer the historical origin of semiarid vegetation bands.

Landscape-scale patterns of vegetation occur worldwide at interfaces between semiarid and arid climates. They are important as potential indicators of climate change and imminent regime shifts and are widely thought to arise from positive feedback between vegetation and infiltration of rainwater. On gentle slopes the typical pattern form is bands (stripes), oriented parallel to the contours, and their wavelength is probably the most accessible statistic for vegetation patterns. Recent field studies have found an inverse correlation between pattern wavelength and slope, in apparent contradiction with the predictions of mathematical models. Here I show that this "contradiction" is based on a flawed approach to calculating the wavelength in models. When pattern generation is considered in detail, the theory is fully consistent with empirical results. For realistic parameters, degradation of uniform vegetation generates patterns whose wavelength increases with slope, whereas colonization of bare ground gives the opposite trend. Therefore, the empirical finding of an inverse relationship can be used, in conjunction with climate records, to infer the historical origin of the patterns. Specifically, for the African Sahel my results suggest that banded vegetation originated by the colonization of bare ground during circa 1760-1790 or since circa 1850.

How does tidal flow affect pattern formation in mussel beds?

In the Wadden Sea, mussel beds self-organise into spatial patterns consisting of bands parallel to the shore. A leading explanation for this phenomenon is that mussel aggregation reduces losses from dislodgement and predation, because of the adherence of mussels to one another. Previous mathematical modelling has shown that this can lead to spatial patterning when it is coupled to the advection from the open sea of algae-the main food source for mussels in the Wadden Sea. A complicating factor in this process is that the advection of algae will actually oscillate with the tidal flow. This has been excluded from previous modelling studies, and the present paper concerns the implications of this oscillation for pattern formation. The authors initially consider piecewise constant ("square-tooth") oscillations in advection, which enables analytical investigation of the conditions for pattern formation. They then build on this to study the more realistic case of sinusoidal oscillations. Their analysis shows that future research on the details of pattern formation in mussel beds will require an in-depth understanding of how the tides affect long-range inhibition among mussels.

Spatio-temporal Models of Lymphangiogenesis in Wound Healing.

Several studies suggest that one possible cause of impaired wound healing is failed or insufficient lymphangiogenesis, that is the formation of new lymphatic capillaries. Although many mathematical models have been developed to describe the formation of blood capillaries (angiogenesis), very few have been proposed for the regeneration of the lymphatic network. Lymphangiogenesis is a markedly different process from angiogenesis, occurring at different times and in response to different chemical stimuli. Two main hypotheses have been proposed: (1) lymphatic capillaries sprout from existing interrupted ones at the edge of the wound in analogy to the blood angiogenesis case and (2) lymphatic endothelial cells first pool in the wound region following the lymph flow and then, once sufficiently populated, start to form a network. Here, we present two PDE models describing lymphangiogenesis according to these two different hypotheses. Further, we include the effect of advection due to interstitial flow and lymph flow coming from open capillaries. The variables represent different cell densities and growth factor concentrations, and where possible the parameters are estimated from biological data. The models are then solved numerically and the results are compared with the available biological literature.

When does colonisation of a semi-arid hillslope generate vegetation patterns?

Patterned vegetation occurs in many semi-arid regions of the world. Most previous studies have assumed that patterns form from a starting point of uniform vegetation, for example as a response to a decrease in mean annual rainfall. However an alternative possibility is that patterns are generated when bare ground is colonised. This paper investigates the conditions under which colonisation leads to patterning on sloping ground. The slope gradient plays an important role because of the downhill flow of rainwater. One long-established consequence of this is that patterns are organised into stripes running parallel to the contours; such patterns are known as banded vegetation or tiger bush. This paper shows that the slope also has an important effect on colonisation, since the uphill and downhill edges of an isolated vegetation patch have different dynamics. For the much-used Klausmeier model for semi-arid vegetation, the author shows that without a term representing water diffusion, colonisation always generates uniform vegetation rather than a pattern. However the combination of a sufficiently large water diffusion term and a sufficiently low slope gradient does lead to colonisation-induced patterning. The author goes on to consider colonisation in the Rietkerk model, which is also in widespread use: the same conclusions apply for this model provided that a small threshold is imposed on vegetation biomass, below which plant growth is set to zero. Since the two models are quite different mathematically, this suggests that the predictions are a consequence of the basic underlying assumption of water redistribution as the pattern generation mechanism.

Seasonal forcing in a host-macroparasite system.

Seasonal forcing represents a pervasive source of environmental variability in natural systems. Whilst it is reasonably well understood in interacting populations and host-microparasite systems, it has not been studied in detail for host-macroparasite systems. In this paper we analyse the effect of seasonal forcing in a general host-macroparasite system with explicit inclusion of the parasite larval stage and seasonal forcing applied to the birth rate of the host. We emphasise the importance of the period of the limit cycles in the unforced system on the resulting dynamics in the forced system. In particular, when subject to seasonal forcing host-macroparasite systems are capable of multi-year cycles, multiple solution behaviour, quasi-periodicity and chaos. The host-macroparasite systems show a larger potential for multiple solution behaviour and a wider range of periodic solutions compared to similar interacting population and microparasite systems. By examining the system for parameters that represent red grouse and the macroparasite nematode Trichostrongylus tenuis we highlight how seasonality could be an important factor in explaining the wide range of seemingly uncorrelated cycle periods observed in grouse abundance in England and Scotland.

A mathematical model for lymphangiogenesis in normal and diabetic wounds.

Several studies suggest that one possible cause of impaired wound healing is failed or insufficient lymphangiogenesis, that is the formation of new lymphatic capillaries. Although many mathematical models have been developed to describe the formation of blood capillaries (angiogenesis) very few have been proposed for the regeneration of the lymphatic network. Moreover, lymphangiogenesis is markedly distinct from angiogenesis, occurring at different times and in a different manner. Here a model of five ordinary differential equations is presented to describe the formation of lymphatic capillaries following a skin wound. The variables represent different cell densities and growth factor concentrations, and where possible the parameters are estimated from experimental and clinical data. The system is then solved numerically and the results are compared with the available biological literature. Finally, a parameter sensitivity analysis of the model is taken as a starting point for suggesting new therapeutic approaches targeting the enhancement of lymphangiogenesis in diabetic wounds. The work provides a deeper understanding of the phenomenon in question, clarifying the main factors involved. In particular, the balance between TGF-ß and VEGF levels, rather than their absolute values, is identified as crucial to effective lymphangiogenesis. In addition, the results indicate lowering the macrophage-mediated activation of TGF-ß and increasing the basal lymphatic endothelial cell growth rate, inter alia, as potential treatments. It is hoped the findings of this paper may be considered in the development of future experiments investigating novel lymphangiogenic therapies.

A mathematical biologist's guide to absolute and convective instability.

Mathematical models have been highly successful at reproducing the complex spatiotemporal phenomena seen in many biological systems. However, the ability to numerically simulate such phenomena currently far outstrips detailed mathematical understanding. This paper reviews the theory of absolute and convective instability, which has the potential to redress this inbalance in some cases. In spatiotemporal systems, unstable steady states subdivide into two categories. Those that are absolutely unstable are not relevant in applications except as generators of spatial or spatiotemporal patterns, but convectively unstable steady states can occur as persistent features of solutions. The authors explain the concepts of absolute and convective instability, and also the related concepts of remnant and transient instability. They give examples of their use in explaining qualitative transitions in solution behaviour. They then describe how to distinguish different types of instability, focussing on the relatively new approach of the absolute spectrum. They also discuss the use of the theory for making quantitative predictions on how spatiotemporal solutions change with model parameters. The discussion is illustrated throughout by numerical simulations of a model for river-based predator-prey systems.

Absolute stability and dynamical stabilisation in predator-prey systems.

Many ecological systems exhibit multi-year cycles. In such systems, invasions have a complicated spatiotemporal structure. In particular, it is common for unstable steady states to exist as long-term transients behind the invasion front, a phenomenon known as dynamical stabilisation. We combine absolute stability theory and computation to predict how the width of the stabilised region depends on parameter values. We develop our calculations in the context of a model for a cyclic predator-prey system, in which the invasion front and spatiotemporal oscillations of predators and prey are separated by a region in which the coexistence steady state is dynamically stabilised.

How do variations in seasonality affect population cycles?

Seasonality is an important component in many population systems, and factors such as latitude, altitude and proximity to the coastline affect the extent of the seasonal fluctuations. In this paper, we ask how changes in seasonal fluctuations impact on the population cycles. We use the Fennoscandian vole system as a case study, focusing on variations in the length of the breeding season. We use a predator-prey model that includes generalist and specialist predation alongside seasonal forcing. Using a combination of bifurcation analysis and direct simulations, we consider the effects of varying both the level of generalist predation and the length of the breeding season; these are the main changes that occur over a latitudinal gradient in Fennoscandia. We predict that varying the breeding season length leads to changes in the period of the multi-year cycles, with a higher period for shorter breeding season lengths. This concurs with the gradient of periodicity found in Fennoscandia. The Fennoscandian vole system is only one of many populations that are affected by geographical and temporal changes in seasonality; thus our results highlight the importance of considering these changes in other population systems.

Seasonal forcing and multi-year cycles in interacting populations: lessons from a predator-prey model.

Many natural systems are subject to seasonal environmental change. As a consequence many species exhibit seasonal changes in their life history parameters--such as a peak in the birth rate in spring. It is important to understand how this seasonal forcing affects the population dynamics. The main way in which seasonal models have been studied is through a two dimensional bifurcation approach. We augment this bifurcation approach with extensive simulation in order to understand the potential solution behaviours for a predator-prey system with a seasonally forced prey growth rate. We consider separately how forcing influences the system when the unforced dynamics have either monotonic decay to the coexistence steady state, or oscillatory decay, or stable limit cycles. The range of behaviour the system can exhibit includes multi-year cycles of different periodicities, parameter ranges with coexisting multi-year cycles of the same or different period as well as quasi-periodicity and chaos. We show that the level of oscillation in the unforced system has a large effect on the range of behaviour when the system is seasonally forced. We discuss how the methods could be extended to understand the dynamics of a wide range of ecological and epidemiological systems that are subject to seasonal changes.

Impact of different destocking strategies on the resilience of dry rangelands.

Half of the world's livestock live in (semi-)arid regions, where a large proportion of people rely on animal husbandry for their survival. However, overgrazing can lead to land degradation and subsequent socio-economic crises. Sustainable management of dry rangeland requires suitable stocking strategies and has been the subject of intense debate in the last decades. Our goal is to understand how variations in stocking strategies affect the resilience of dry rangelands. We describe rangeland dynamics through a simple mathematical model consisting of a system of coupled differential equations. In our model, livestock density is limited only by forage availability, which is itself limited by water availability. We model processes typical of dryland vegetation as a strong Allee effect, leading to bistability between a vegetated and a degraded state, even in the absence of herbivores. We study analytically the impact of varying the stocking density and the destocking adaptivity on the resilience of the system to the effects of drought. By using dynamical systems theory, we look at how different measures of resilience are affected by variations in destocking strategies. We find that the following: (1) Increasing stocking density decreases resilience, giving rise to an expected trade-off between productivity and resilience. (2) There exists a maximal sustainable livestock density above which the system can only be degraded. This carrying capacity is common to all strategies. (3) Higher adaptivity of the destocking rate to available forage makes the system more resilient: the more adaptive a system is, the bigger the losses of vegetation it can recover from, without affecting the long-term level of productivity. The first two results emphasize the need for suitable dry rangeland management strategies, to prevent degradation resulting from the conflict between profitability and sustainability. The third point offers a theoretical suggestion for such a strategy.

Delayed induced silica defences in grasses and their potential for destabilising herbivore population dynamics.

Some grass species mount a defensive response to grazing by increasing their rate of uptake of silica from the soil and depositing it as abrasive granules in their leaves. Increased plant silica levels reduce food quality for herbivores that feed on these grasses. Here we provide empirical evidence that a principal food species of an herbivorous rodent exhibits a delayed defensive response to grazing by increasing silica concentrations, and present theoretical modelling that predicts that such a response alone could lead to the population cycles observed in some herbivore populations. Experiments performed under greenhouse conditions revealed that the rate of deposition of silica defences in the grass Deschampsia caespitosa is a time-lagged, nonlinear function of grazing intensity and that, upon cessation of grazing, these defences take around one year to decay to within 5 % of control levels. Simple coupled grass-herbivore population models incorporating this functional response, and parameterised with empirical data, consistently predict population cycles for a wide range of realistic parameter values for a (Microtus) vole-grass system. Our results support the hypothesis that induced silica defences have the potential to strongly affect the population dynamics of their herbivores. Specifically, the feedback response we observed could be a driving mechanism behind the observed population cycles in graminivorous herbivores in cases where grazing levels in the field become sufficiently large and sustained to trigger an induced silica defence response.

Locating the transition from periodic oscillations to spatiotemporal chaos in the wake of invasion.

In systems with cyclic dynamics, invasions often generate periodic spatiotemporal oscillations, which undergo a subsequent transition to chaos. The periodic oscillations have the form of a wavetrain and occur in a band of constant width. In applications, a key question is whether one expects spatiotemporal data to be dominated by regular or irregular oscillations or to involve a significant proportion of both. This depends on the width of the wavetrain band. Here, we present mathematical theory that enables the direct calculation of this width. Our method synthesizes recent developments in stability theory and computation. It is developed for only 1 equation system, but because this is a normal form close to a Hopf bifurcation, the results can be applied directly to a wide range of models. We illustrate this by considering a classic example from ecology: wavetrains in the wake of the invasion of a prey population by predators.