Lukas Eigentler

Lukas.Eigentler --AT--

Assistant Professor at the University of Warwick, Warwick Mathematics Institute, The Zeeman Institute for Systems Biology & Infectious Disease Epidemiology Research (SBIDER)

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    Papers in peer-reviewed journals.

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  1. L. Eigentler, K. Reinhold: Maintenance and evolution of individual differences in a prey defence trait examined with a dynamic predator-prey model. Biorxiv preprint, DOI: 10.1101/2023.12.07.570589

    Abstract: Predator-prey systems often feature periodic population cycles. In an empirical system with a heritable prey defence trait, ecological oscillations were previously shown to cause evolution of prey defence on the timescale of the population cycles. In this paper, we develop an eco-evolutionary mathematical model comprising partial differential equations to investigate the evolutionary dynamics of prey defence during population cycles. We reveal that ecological population cycles induce evolutionary oscillations of the mean prey defence trait. In contrast to existing continuum modelling frameworks, our model allows for the evolution of individual variability. We show that eco-evolutionary oscillations select for increased individual variability close to the transition from stable to oscillatory states. We also reveal that evolution of prey defence requires both high efficiency and low cost of prey defence and highlight that more information on trade-offs between cost and efficiency is required.

  2. L. Eigentler, M. Sensi: Delayed loss of stability of periodic travelling waves: insights from the analysis of essential spectra ArXiv preprint, DOI: 10.48550/arXiv.2311.14717

    Abstract: Periodic travelling waves (PTW) are a common solution type of partial differential equations. Such models exhibit multistability of PTWs, typically visualised through the Busse balloon, and parameter changes typically lead to a cascade of wavelength changes through the Busse balloon. In the past, the stability boundaries of the Busse balloon have been used to predict such wavelength changes. Here, motivated by anecdotal evidence from previous work, we provide compelling evidence that the Busse balloon provides insufficient information to predict wavelength changes due to a delayed loss of stability phenomenon. Using two different reaction-advection-diffusion systems, we relate the delay that occurs between the crossing of a stability boundary in the Busse balloon and the occurrence of a wavelength change to features of the essential spectrum of the destabilised PTW. This leads to a predictive framework that can estimate the order of magnitude of such a time delay, which provides a novel "early warning sign" for pattern destabilization. We illustrate the implementation of the predictive framework to predict under what conditions a wavelength change of a PTW occurs.

  3. T. Rosazza, C.S. Earl, L. Eigentler, F.A. Davidson, N.R. Stanley-Wall: Reciprocal sharing of two classes of public goods facilitates Bacillus subtilis biofilm formation. Biorxiv preprint, DOI: 10.1101/2023.09.22.558988

    Abstract: Public goods support the development of microbial communities but are prone to exploitation. The extracellular matrix molecules of Bacillus subtilis are examples of public goods that are produced through the division of labour and support biofilm formation. Another class of public good that supports B. subtilis growth when nutrients are in a polymeric form are extracellular proteases. In this study, we investigated the role of extracellular proteases during B. subtilis biofilm formation. We established that the extracellular proteases support biofilm growth even when the nitrogen source is available as glutamic acid. Reviewing the contribution played by each protease identified that selected individual extracellular proteases have distinct functions, with AprE found to be crucial for development and WprA sufficient for biofilm formation. We found that changing the nutrient source affected biofilm structure, hydrophobicity, and architecture, likely due to the increased cost of protease dependency for growth. Co-culture of biofilm matrix and extracellular protease mutants rescued biofilm structure through public goods sharing, but the dependence on extracellular protease for growth influenced community coexistence. The intricate sharing among these two classes of public goods sheds light on microbial social dynamics during biofilm formation in different ecological contexts.

  4. K. Reinhold, L. Eigentler, David W. Kikuchi: Evolution of individual variation in a competitive trait: a theoretical analysis. J. Evol. Biol. voae036, DOI: 10.1093/jeb/voae036

    Abstract: When competitive traits are costly, negative frequency-dependence can maintain genetic variance. Most theoretical studies examining this problem assume binary polymorphisms, yet most trait variation in wild populations is continuous. We propose that continuous trait variation can result from continuous variation in resource quality and that, specifically, the shape of the resource distribution determines trait maintenance. We used an individual-based model to test which conditions favour the stable maintenance of variation and which cause temporal fluctuations in trait values. This approach, inspired by contrasting outcomes of previous studies regarding variance and fluctuations in trait values, clearly showed a decisive role played by the shape of resource distributions. Under extreme conditions, e.g. the absence of resource variation or with very scarce resources for weak competitors, traits evolved to a single non-competitive or highly competitive strategy, respectively. Most other distributions led to strong temporal fluctuations on trait values or the maintenance of stable, standing variation. Our results thus explain the contradicting outcomes of previous theoretical studies and at the same time provide hypotheses to explain the maintenance of genetic variation and individual differences. We suggest ways to empirically test the proposed effects of resource variation on trait maintenance.

  5. T. Rosazza, L. Eigentler, C.S. Earl, F.A. Davidson, N.R. Stanley-Wall: Bacillus subtilis extracellular protease production incurs a context-dependent cost. Mol. Microbiol. 120:2 (2023), 105--121, DOI: 10.1111/mmi.15110
    Editor's choice

    Abstract: Microbes encounter a wide range of polymeric nutrient sources in various environmental settings, which require processing to facilitate growth. Bacillus subtilis, a bacterium found in the rhizosphere and broader soil environment, is highly adaptable and resilient due to its ability to utilise diverse sources of carbon and nitrogen. Here, we explore the role of extracellular proteases in supporting growth and assess the cost associated with their production. We provide evidence of the essentiality of extracellular proteases when B. subtilis is provided with an abundant, but polymeric nutrient source and demonstrate the extracellular proteases as a shared public good that can operate over a distance. We show that B. subtilis is subjected to a public good dilemma, specifically in the context of growth sustained by the digestion of a polymeric food source. Furthermore, using mathematical simulations, we uncover that this selectively enforced dilemma is driven by the relative cost of producing the public good. Collectively, our findings reveal how bacteria can survive in environments that vary in terms of immediate nutrient accessibility and the consequent impact on the population composition. These findings enhance our fundamental understanding of how bacteria respond to diverse environments, which has importance to contexts ranging from survival in the soil to infection and pathogenesis scenarios.

  6. L. Eigentler, J.A. Sherratt: Long-range seed dispersal stabilises almost stationary patterns in a model for dryland vegetation. J. Math. Biol. 86:15 (2023), DOI: 10.1007/s00285-022-01852-x

    Abstract: Spatiotemporal patterns of vegetation are a ubiquitous feature of semi-arid ecosystems. On sloped terrain, vegetation patterns occur as stripes perpendicular to the contours. Field studies report contrasting long-term dynamics between different observation sites; some observe slow uphill migration of vegetation bands while some report stationary patterns. In this paper, we show that long-range seed dispersal provides a mechanism that enables the occurrence of both migrating and stationary patterns. We utilise a nonlocal PDE model in which seed dispersal is accounted for by a convolution term. The model represents vegetation patterns as periodic travelling waves and numerical continuation shows that both migrating and almost stationary patterns are stable if seed dispersal distances are sufficiently large. We use a perturbation theory approach to obtain analytical confirmation of the existence of almost stationary patterned solutions and provide a biological interpretation of the phenomenon.

  7. L. Eigentler, F.A. Davidson, N.R. Stanley-Wall: Mechanisms driving spatial distribution of residents in colony biofilms: an interdisciplinary perspective. Open Biol. 12:220194 (2022), DOI: 10.1098/rsob.220194 .

    Abstract: Biofilms are consortia of microorganisms that form collectives through the excretion of extracellular matrix compounds. The importance of biofilms in biological, industrial, and medical settings has long been recognised due to their emergent properties and impact on surrounding environments. In laboratory situations, one commonly used approach to study biofilm formation mechanisms is the colony biofilm assay, in which cell communities grow on solid-gas interfaces after the deposition of a population of founder cells. The residents of a colony biofilm can self-organise to form intricate spatial distributions. The assay is ideally suited to coupling with mathematical modelling due to the ability to extract a wide range of metrics. In this review, we highlight how interdisciplinary approaches have provided deep insights into mechanisms causing the emergence of these spatial distributions from well-mixed inocula.

  8. L. Briganti Wiprachtiger, L. Eigentler: The effects of seasonality on competition for a limiting resource. SIAM Undergraduate Res. Online. 15 (2022), DOI: 10.1137/21S1458132.

    Abstract: Theoretical studies of PDE/ODE models describing ecosystem dynamics usually ignore seasonality in environmental conditions. In this paper I study a model of two generic consumer species that compete for a single limiting resource. I first consider constant resource input and then compare it to the case when resource input is dependent on time with a seasonal (periodic) pattern. The model with constant resource input is analysed analytically, by looking at the linear stability of every equilibrium. The model with seasonal resource input is analysed through numerical simulations. Results of the analysis show that seasonality has a significant effect on the outcome of the system, as when resource input is dependent on time, there could be stable coexistence, which is not possible under constant resource input. Moreover, metastable coexistence states exist for both resource input regimes if the average fitness difference between species is small. Finally, times until extinction become longer if resource input is not constant.

  9. L. Eigentler, N.R. Stanley-Wall, F.A. Davidson: A theoretical framework for multi-species range expansion in spatially heterogeneous landscapes. Oikos 2022.8 (2022), e09077, DOI: 10.1111/oik.09077. Preprint: 10.1101/2021.11.09.467881.

    Abstract: Range expansion is the spatial spread of a population into previously unoccupied regions. Understanding range expansion is important for the study and successful manipulation and management of ecosystems, with applications ranging from controlling bacterial biofilm formation in industrial and medical environments to large scale conservation programmes for species undergoing climate-change induced habitat disruption. During range expansion, species typically encounter competitors. Moreover, the environment into which expansion takes place is almost always heterogeneous when considered at the scale of the individual. Despite the ubiquitous nature of these features, the impact of competition and spatial landscape heterogeneities on range expansion remains understudied. In this paper we present a theoretical framework comprising two competing generic species undergoing range expansion and use it to investigate the impact of spatial landscape heterogeneities on range expansion with a particular focus on its effect on competition dynamics. We reveal that the area covered by range expansion during a fixed time interval is highly variable due to the fixed landscape heterogeneities. Moreover, we report significant variability in competitive outcome (relative abundance of a focal species) but determine that this is induced by low initial population densities, independent of landscape heterogeneities. We further show that both area covered by range expansion and competitive outcome can be accurately predicted by a Voronoi tessellation with respect to an appropriate metric, which only requires information on the spatial landscape and the response of each species to that landscape. Finally, we reveal that if species interact antagonistically during range expansion, the dominant mode of competition depends on the initial population density. Antagonistic actions determine competitive outcome if the initial population density is high, but competition for space is the dominant mode of competition if the initial population density is low.

  10. L. Eigentler, M. Kalamara, G. Ball, C.E. MacPhee, N.R. Stanley-Wall, F.A. Davidson: Founder cell configuration drives competitive outcome within colony biofilms. ISME J., 16.6 (2022), 1512--1522. DOI: 10.1038/s41396-022-01198-8. Preprint: 10.1101/2021.07.08.451560.

    Abstract: Bacteria can form dense communities called biofilms, where cells are embedded in a self-produced extracellular matrix. Exploiting competitive interactions between strains within the biofilm context can have potential applications in biological, medical, and industrial systems. By combining mathematical modelling with experimental assays, we reveal that spatial structure and competitive dynamics within biofilms are significantly affected by the location and density of the founder cells used to inoculate the biofilm. Using a species-independent theoretical framework describing colony biofilm formation, we show that the observed spatial structure and relative strain biomass in a mature biofilm comprising two isogenic strains can be mapped directly to the geographical distributions of founder cells. Moreover, we define a predictor of competitive outcome that accurately forecasts relative abundance of strains based solely on the founder cells’ potential for radial expansion. Consequently, we reveal that variability of competitive outcome in biofilms inoculated at low founder density is a natural consequence of the random positioning of founding cells in the inoculum. Extension of our study to non-isogenic strains that interact through local antagonisms, shows that even for strains with different competition strengths, a race for space remains the dominant mode of competition in low founder density biofilms. Our results, verified by experimental assays using Bacillus subtilis, highlight the importance of spatial dynamics on competitive interactions within biofilms and hence to related applications.

  11. L. Eigentler: Species coexistence in resource-limited patterned ecosystems is facilitated by the interplay of spatial self-organisation and intraspecific competition. Oikos, 130.4 (2021), 609--623. DOI: 10.1111/oik.07880. Post-peer-review, pre-copyedit version: 10.1101/2020.01.13.903179

    Abstract: The exploration of mechanisms that enable species coexistence under competition for a sole limiting resource is widespread across ecology. Two examples of such facilitative processes are intraspecific competition and spatial self-organisation. These processes determine the outcome of competitive dynamics in many resource-limited patterned ecosystems, classical examples of which include dryland vegetation patterns, intertidal mussel beds and Sub-alpine ribbon forests. Previous theoretical investigations have explained coexistence within patterned ecosystems by making strong assumptions on the differences between species (e.g. contrasting dispersal behaviours or different functional responses to resource availability). In this paper, I show that the interplay between the detrimental effects of intraspecific competition and the facilitative nature of self-organisation forms a coexistence mechanism that does not rely on species-specific assumptions and captures coexistence across a wide range of the environmental stress gradient. I use a theoretical model that captures the interactions of two generic consumer species with an explicitly modelled resource to show that coexistence relies on a balance between species’ colonisation abilities and their local competitiveness, provided intraspecific competition is sufficiently strong. Crucially, the requirements on species’ self-limitation for coexistence to occur differ on opposite ends of the resource input spectrum. For low resource levels, coexistence is facilitated by strong intraspecific dynamics of the species superior in its colonisation abilities, but for larger volumes of resource input, strong intraspecific competition of the locally superior species enables coexistence. Results presented in this paper also highlight the importance of hysteresis in understanding tipping points, in particular extinction events. Finally, the theoretical framework provides insights into spatial species distributions within single patches, supporting verbal hypotheses on co-existence of herbaceous and woody species in dryland vegetation patterns and suggesting potential empirical tests in the context of other patterned ecosystems.

  12. L. Eigentler, J.A. Sherratt: An integrodifference model for vegetation patterns in semi-arid environments with seasonality. J. Math. Biol., 81.3 (2020), 875--904. DOI: 10.1007/s00285-020-01530-w Post-peer-review, pre-copyedit version: arXiv:1911.10964

    Abstract: 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.

  13. L. Eigentler: Intraspecific competition in models for vegetation patterns: decrease in resilience to aridity and facilitation of species coexistence. Ecol. Complexity, 42 (2020), 100835. DOI: 10.1016/j.ecocom.2020.100835 Post-peer-review, pre-copyedit version: arXiv:2002.05677

    Abstract: Patterned vegetation is a characteristic feature of many dryland ecosystems. While plant densities on the ecosystem-wide scale are typically low, a spatial self-organisation principle leads to the occurrence of alternating patches of high biomass and patches of bare soil. Nevertheless, intraspecific competition dynamics other than competition for water over long spatial scales are commonly ignored in mathematical models for vegetation patterns. In this paper, I address the impact of local intraspecific competition on a modelling framework for banded vegetation patterns. Firstly, I show that in the context of a single-species model, neglecting local intraspecific competition leads to an overestimation of a patterned ecosystem’s resilience to increases in aridity. Secondly, in the context of a multispecies model, I argue that local intraspecific competition is a key element in the successful capture of species coexistence in model solutions representing a vegetation pattern. For both models, a detailed bifurcation analysis is presented to analyse the onset, existence and stability of patterns. Besides the strengths of local intraspecific competition, also the difference between two species has a significant impact on the bifurcation structure, providing crucial insights into the complex ecosystem dynamics. Predictions on future ecosystem dynamics presented in this paper, especially on pattern onset and pattern stability, can aid the development of conservation programs.

  14. L. Eigentler, J.A. Sherratt: Effects of precipitation intermittency on vegetation patterns in semi-arid landscapes. Physica D , 405 (2020), 132396. DOI: 10.1016/j.physd.2020.132396 Post-peer-review, pre-copyedit version: arXiv:1911.10878

    Abstract: Patterns of vegetation are a characteristic feature of many semi-arid regions. The limiting resource in these ecosystems is water, which is added to the system through short and intense rainfall events that cause a pulse of biological processes such as plant growth and seed dispersal. We propose an impulsive model based on the Klausmeier reaction-advection-diffusion system, analytically investigate the effects of rainfall intermittency on the onset of patterns, and augment our results by numerical simulations of model extensions. Our investigation focuses on the parameter region in which a transition between uniform and patterned vegetation occurs. Results show that decay-type processes associated with a low frequency of precipitation pulses inhibit the onset of patterns and that under intermittent rainfall regimes, a spatially uniform solution is sustained at lower total precipitation volumes than under continuous rainfall, if plant species are unable to efficiently use low soil moisture levels. Unlike in the classical setting of a reaction-diffusion model, patterns are not caused by a diffusion-driven instability but by a combination of sufficiently long periods of droughts between precipitation pulses and water diffusion. Our results further indicate that the introduction of pulse-type seed dispersal weakens the effects of changes to width and shape of the plant dispersal kernel on the onset of patterns.

  15. L. Eigentler, J.A. Sherratt: Spatial self-organisation enables species coexistence in a model for savanna ecosystems. J. Theor. Biol., 487 (2020), 110122. DOI: 10.1016/j.jtbi.2019.110122. Post-peer-review, pre-copyedit version: arXiv:1911.10801

    Abstract: The savanna biome is characterised by a continuous vegetation cover, comprised of herbaceous and woody plants. The coexistence of species in arid savannas, where water availability is the main limiting resource for plant growth, provides an apparent contradiction to the classical principle of competitive exclusion. Previous theoretical work using nonspatial models has focussed on the development of an understanding of coexistence mechanisms through the consideration of resource niche separation and ecosystem disturbances. In this paper, we propose that a spatial self-organisation principle, caused by a positive feedback between local vegetation growth and water redistribution, is sufficient for species coexistence in savanna ecosystems. We propose a spatiotemporal ecohydrological model of partial differential equations, based on the Klausmeier reaction-advection-diffusion model for vegetation patterns, to investigate the effects of spatial interactions on species coexistence on sloped terrain. Our results suggest that species coexistence is a possible model outcome, if a balance is kept between the species' average fitness (a measure of a species' competitive abilities in a spatially uniform setting) and their colonisation abilities. Spatial heterogeneities in resource availability are utilised by the superior coloniser (grasses), before it is outcompeted by the species of higher average fitness (trees). A stability analysis of the spatially nonuniform coexistence solutions further suggests that grasses act as ecosystem engineers and facilitate the formation of a continuous tree cover for precipitation levels unable to support a uniform tree density in the absence of a grass species.

  16. L. Eigentler, J.A. Sherratt: Metastability as a coexistence mechanism in a model for dryland vegetation patterns. Bull. Math. Biol., 81.7 (2019), 2290--2322. DOI: 10.1007/s11538-019-00606-z, Post-peer-review, pre-copyedit version: arXiv:1911.11022

    Abstract: 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 $10^3$ 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.

  17. L. Eigentler, J.A. Sherratt: Analysis of a model for banded vegetation patterns in semi-arid environments with nonlocal dispersal. J. Math. Biol., 77.3 (2018), 739--763, DOI: 10.1007/s00285-018-1233-y, Post-peer-review, pre-copyedit version: arXiv:1911.11037

    Abstract: 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.

Other publications.

  1. L. Eigentler: The Reinhart-Heinrich Doctoral Thesis Award 2020. European Communications in Mathematical and Theoretical Biology , 25(2022), 4--9. Link

A copy of my PhD thesis can be found here.