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Giannakou M, Waclaw B. Resonant noise amplification in a predator-prey model with quasi-discrete generations. Sci Rep 2024; 14:16783. [PMID: 39039177 PMCID: PMC11263699 DOI: 10.1038/s41598-024-67098-3] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/11/2024] [Accepted: 07/08/2024] [Indexed: 07/24/2024] Open
Abstract
Predator-prey models have been shown to exhibit resonance-like behaviour, in which random fluctuations in the number of organisms (demographic noise) are amplified when their frequency is close to the natural oscillatory frequency of the system. This behaviour has been traditionally studied in models with exponentially distributed replication and death times. Here we consider a biologically more realistic model, in which organisms replicate quasi-synchronously such that the distribution of replication times has a narrow maximum at some T > 0 corresponding to the mean doubling time. We show that when the frequency of replication f = 1 / T is tuned to the natural oscillatory frequency of the predator-prey model, the system exhibits oscillations that are much stronger than in the model with Poissonian (non-synchronous) replication and death. These oscillations lead to population instability and the extinction of one of the species much sooner than in the case of Poissonian replication. The effect can be explained by resonant amplification of coloured noise generated by quasi-synchronous replication events.
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Affiliation(s)
- M Giannakou
- School of Physics and Astronomy, University of Edinburgh, James Clerk Maxwell Building, Peter Guthrie Tait Road, Edinburgh, EH9 3FD, UK
- Institut für Physik, Johannes Gutenberg-Universität Mainz, Staudingerweg 9, 55128, Mainz, Germany
| | - B Waclaw
- School of Physics and Astronomy, University of Edinburgh, James Clerk Maxwell Building, Peter Guthrie Tait Road, Edinburgh, EH9 3FD, UK.
- Dioscuri Centre for Physics and Chemistry of Bacteria, Institute of Physical Chemistry PAS, Kasprzaka 44/52, 01-224, Warsaw, Poland.
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2
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Barraquand F. No sensitivity to functional forms in the Rosenzweig-MacArthur model with strong environmental stochasticity. J Theor Biol 2023; 572:111566. [PMID: 37422068 DOI: 10.1016/j.jtbi.2023.111566] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/03/2022] [Revised: 06/04/2023] [Accepted: 06/26/2023] [Indexed: 07/10/2023]
Abstract
The classic Rosenzweig-MacArthur predator-prey model has been shown to exhibit, like other coupled nonlinear ordinary differential equations (ODEs) from ecology, worrying sensitivity to model structure. This sensitivity manifests as markedly different community dynamics arising from saturating functional responses with nearly identical shapes but different mathematical expressions. Using a stochastic differential equation (SDE) version of the Rosenzweig-MacArthur model with the three functional responses considered by Fussmann & Blasius (2005), I show that such sensitivity seems to be solely a property of ODEs or stochastic systems with weak noise. SDEs with strong environmental noise have by contrast very similar fluctuations patterns, irrespective of the mathematical formula used. Although eigenvalues of linearized predator-prey models have been used as an argument for structural sensitivity, they can also be an argument against structural sensitivity. While the sign of the eigenvalues' real part is sensitive to model structure, its magnitude and the presence of imaginary parts are not, which suggests noise-driven oscillations for a broad range of carrying capacities. I then discuss multiple other ways to evaluate structural sensitivity in a stochastic setting, for predator-prey or other ecological systems.
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Affiliation(s)
- Frédéric Barraquand
- Institute of Mathematics of Bordeaux, CNRS & University of Bordeaux, Talence, France.
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3
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Greyson-Gaito CJ, Gellner G, McCann KS. Life-history speed, population disappearances and noise-induced ratchet effects. Proc Biol Sci 2023; 290:20222149. [PMID: 36987642 PMCID: PMC10050927 DOI: 10.1098/rspb.2022.2149] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/28/2022] [Accepted: 03/02/2023] [Indexed: 03/30/2023] Open
Abstract
Nature is replete with variation in the body sizes, reproductive output and generation times of species that produce life-history responses known to vary from small and fast to large and slow. Although researchers recognize that life-history speed likely dictates fundamental processes in consumer-resource interactions like productivity and stability, theoretical work remains incomplete in this critical area. Here, we examine the role of life-history speed on consumer-resource interactions by using a well-used mathematical approach that manipulates the speed of the consumer's growth rate in a consumer-resource interaction. Importantly, this approach holds the isocline geometry intact, allowing us to assess the impacts of altered life-history speed on stability (coefficient of variation, CV) without changing the underlying qualitative dynamics. Although slowing life history can be initially stabilizing, we find that in stochastic settings slowing ultimately drives highly destabilizing population disappearances, especially under reddened noise. Our results suggest that human-driven reddening of noise may decrease species stability because the autocorrelation of red noise enlarges the period and magnitude of perturbations, overwhelming a species' natural compensatory responses via a ratchet-like effect. This ratchet-like effect then pushes species' population dynamics far away from equilibria, which can lead to precipitous local extinction.
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Affiliation(s)
| | - Gabriel Gellner
- Department of Integrative Biology, University of Guelph, Guelph, Ontario, Canada N1G 2W1
| | - Kevin S. McCann
- Department of Integrative Biology, University of Guelph, Guelph, Ontario, Canada N1G 2W1
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4
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Giacomini HC. Metabolic responses of predators to prey density. Front Ecol Evol 2022. [DOI: 10.3389/fevo.2022.980812] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/15/2022] Open
Abstract
The metabolic cost of foraging is the dark energy of ecological systems. It is much harder to observe and to measure than its beneficial counterpart, prey consumption, yet it is not inconsequential for the dynamics of prey and predator populations. Here I define the metabolic response as the change in energy expenditure of predators in response to changes in prey density. It is analogous and intrinsically linked to the functional response, which is the change in consumption rate with prey density, as they are both shaped by adjustments in foraging activity. These adjustments are adaptive, ubiquitous in nature, and are implicitly assumed by models of predator–prey dynamics that impose consumption saturation in functional responses. By ignoring the associated metabolic responses, these models violate the principle of energy conservation and likely underestimate the strength of predator–prey interactions. Using analytical and numerical approaches, I show that missing this component of interaction has broad consequences for dynamical stability and for the robustness of ecosystems to persistent environmental or anthropogenic stressors. Negative metabolic responses – those resulting from decreases in foraging activity when more prey is available, and arguably the most common – lead to lower local stability of food webs and a faster pace of change in population sizes, including higher excitability, higher frequency of oscillations, and quicker return times to equilibrium when stable. They can also buffer the effects of press perturbations, such as harvesting, on target populations and on their prey through top-down trophic cascades, but are expected to magnify bottom-up cascades, including the effects of nutrient enrichment or the effects of altering lower trophic levels that can be caused by environmental forcing and climate change. These results have implications for any resource management approach that relies on models of food web dynamics, which is the case of many applications of ecosystem-based fisheries management. Finally, besides having their own individual effects, metabolic responses have the potential to greatly alter, or even invert, functional response-stability relationships, and therefore can be critical to an integral understanding of predation and its influence on population dynamics and persistence.
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Esmaeili S, Hastings A, Abbott KC, Machta J, Nareddy VR. Noise-induced versus intrinsic oscillation in ecological systems. Ecol Lett 2022; 25:814-827. [PMID: 35007391 DOI: 10.1111/ele.13956] [Citation(s) in RCA: 4] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/26/2021] [Revised: 11/15/2021] [Accepted: 12/03/2021] [Indexed: 11/30/2022]
Abstract
Studies of oscillatory populations have a long history in ecology. A first-principles understanding of these dynamics can provide insights into causes of population regulation and help with selecting detailed predictive models. A particularly difficult challenge is determining the relative role of deterministic versus stochastic forces in producing oscillations. We employ statistical physics concepts, including measures of spatial synchrony, that incorporate patterns at all scales and are novel to ecology, to show that spatial patterns can, under broad and well-defined circumstances, elucidate drivers of population dynamics. We find that when neighbours are coupled (e.g. by dispersal), noisy intrinsic oscillations become distinguishable from noise-induced oscillations at a transition point related to synchronisation that is distinct from the deterministic bifurcation point. We derive this transition point and show that it diverges from the deterministic bifurcation point as stochasticity increases. The concept of universality suggests that the results are robust and widely applicable.
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Affiliation(s)
- Shadisadat Esmaeili
- Department of Environmental Science and Policy, University of California, Davis, California, USA
| | - Alan Hastings
- Department of Environmental Science and Policy, University of California, Davis, California, USA.,Santa Fe Institute, Santa Fe, New Mexico, USA
| | - Karen C Abbott
- Department of Biology, Case Western Reserve University, Cleveland, Ohio, USA
| | - Jonathan Machta
- Santa Fe Institute, Santa Fe, New Mexico, USA.,Physics Department, University of Massachusetts, Amherst, Massachusetts, USA
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6
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Boettiger C. From noise to knowledge: how randomness generates novel phenomena and reveals information. Ecol Lett 2018; 21:1255-1267. [PMID: 29790295 DOI: 10.1111/ele.13085] [Citation(s) in RCA: 31] [Impact Index Per Article: 5.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/06/2018] [Revised: 04/03/2018] [Accepted: 04/17/2018] [Indexed: 02/04/2023]
Abstract
Noise, as the term itself suggests, is most often seen a nuisance to ecological insight, a inconvenient reality that must be acknowledged, a haystack that must be stripped away to reveal the processes of interest underneath. Yet despite this well-earned reputation, noise is often interesting in its own right: noise can induce novel phenomena that could not be understood from some underlying deterministic model alone. Nor is all noise the same, and close examination of differences in frequency, colour or magnitude can reveal insights that would otherwise be inaccessible. Yet with each aspect of stochasticity leading to some new or unexpected behaviour, the time is right to move beyond the familiar refrain of "everything is important" (Bjørnstad & Grenfell ). Stochastic phenomena can suggest new ways of inferring process from pattern, and thus spark more dialog between theory and empirical perspectives that best advances the field as a whole. I highlight a few compelling examples, while observing that the study of stochastic phenomena are only beginning to make this translation into empirical inference. There are rich opportunities at this interface in the years ahead.
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Affiliation(s)
- Carl Boettiger
- Dept of Environmental Science, Policy, and Management, University of California Berkeley, Berkeley, CA, 94720-3114, USA
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7
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Hagen R, Heurich M, Storch I, Hanewinkel M, Kramer-Schadt S. Linking annual variations of roe deer bag records to large-scale winter conditions: spatio-temporal development in Europe between 1961 and 2013. EUR J WILDLIFE RES 2017. [DOI: 10.1007/s10344-017-1155-9] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/24/2022]
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8
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Barraquand F, Louca S, Abbott KC, Cobbold CA, Cordoleani F, DeAngelis DL, Elderd BD, Fox JW, Greenwood P, Hilker FM, Murray DL, Stieha CR, Taylor RA, Vitense K, Wolkowicz GS, Tyson RC. Moving forward in circles: challenges and opportunities in modelling population cycles. Ecol Lett 2017. [DOI: 10.1111/ele.12789] [Citation(s) in RCA: 89] [Impact Index Per Article: 12.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/11/2023]
Affiliation(s)
- Frédéric Barraquand
- Department of Arctic and Marine Biology University of Tromsø Tromsø Norway
- Integrative and Theoretical Ecology Chair, LabEx COTE University of Bordeaux Pessac France
| | - Stilianos Louca
- Institute of Applied Mathematics University of British Columbia Vancouver BC Canada
| | - Karen C. Abbott
- Department of Biology Case Western Reserve University Cleveland OH USA
| | | | - Flora Cordoleani
- Institute of Marine Science University of California Santa Cruz Santa Cruz CA USA
- Southwest Fisheries Science Center Santa Cruz CA USA
| | | | - Bret D. Elderd
- Department of Biological Sciences Lousiana State University Baton Rouge LA USA
| | - Jeremy W. Fox
- Department of Biological Sciences University of Calgary Calgary ABCanada
| | | | - Frank M. Hilker
- Institute of Environmental Systems Research, School of Mathematics/Computer Science Osnabrück University Osnabrück Germany
| | - Dennis L. Murray
- Integrative Wildlife Conservation Lab Trent University Peterborough ONCanada
| | - Christopher R. Stieha
- Department of Biology Case Western Reserve University Cleveland OH USA
- Department of Entomology Cornell University Ithaca NY USA
| | - Rachel A. Taylor
- Department of Integrative Biology University of South Florida Tampa FLUSA
| | - Kelsey Vitense
- Department of Fisheries, Wildlife, and Conservation Biology University of Minnesota Saint Paul MN USA
| | - Gail S.K. Wolkowicz
- Department of Mathematics and Statistics McMaster University Hamilton ON Canada
| | - Rebecca C. Tyson
- Department of Mathematics and Statistics University of British Columbia Okanagan Kelowna BC Canada
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9
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Singer A, Frank K. Viability of cyclic populations. Ecology 2016; 97:3143-3153. [PMID: 27870022 DOI: 10.1002/ecy.1564] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Received: 03/22/2016] [Revised: 06/13/2016] [Accepted: 08/05/2016] [Indexed: 11/05/2022]
Abstract
Theory on viability of small populations is well developed and has led to the standard methodology of population viability analysis (PVA) to assess vulnerability of single species. However, more complex situations involving community dynamics or environmental change violate theoretical assumptions. Synthesizing concepts from population, community, and conservation ecology, we develop a generic theory on the viability of cyclic populations. The interplay of periodic population decline and demography causes varying risk patterns that aggregate during cycles and modify the temporal structure of viability. This variability is visualized and quantitatively assessed. For two standard viability metrics that summarize immediate extinction risk and the general long-term conditions of populations, we mathematically describe the impact of population cycles. Finally, we suggest and demonstrate PVA for cyclic populations that respond to, e.g., seasonality, interannual variation, or trophic interactions. Our theoretical and methodological advancement opens a route to viability analysis in food webs and trophic meta-communities and equips biodiversity conservation with a long-missing tool.
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Affiliation(s)
- Alexander Singer
- Department of Ecological Modelling, Helmholtz-Centre for Environmental Research-UFZ, Permoserstrasse 15, D-04318, Leipzig, Germany.,German Centre for Integrative Biodiversity Research (iDiv) Halle-Jena-Leipzig, Deutscher Platz 5e, D-04103, Leipzig, Germany
| | - Karin Frank
- Department of Ecological Modelling, Helmholtz-Centre for Environmental Research-UFZ, Permoserstrasse 15, D-04318, Leipzig, Germany.,German Centre for Integrative Biodiversity Research (iDiv) Halle-Jena-Leipzig, Deutscher Platz 5e, D-04103, Leipzig, Germany.,Institute for Environmental Systems Research, University of Osnabrück, Barbarastrasse 12, D-49076, Osnabrück, Germany
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10
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Gellner G, McCann KS, Hastings A. The duality of stability: towards a stochastic theory of species interactions. THEOR ECOL-NETH 2016. [DOI: 10.1007/s12080-016-0303-2] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/21/2022]
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11
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Júlvez J. A straightforward method to compute average stochastic oscillations from data samples. BMC Bioinformatics 2015; 16:333. [PMID: 26482438 PMCID: PMC4615616 DOI: 10.1186/s12859-015-0765-z] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/04/2015] [Accepted: 10/07/2015] [Indexed: 11/10/2022] Open
Abstract
BACKGROUND Many biological systems exhibit sustained stochastic oscillations in their steady state. Assessing these oscillations is usually a challenging task due to the potential variability of the amplitude and frequency of the oscillations over time. As a result of this variability, when several stochastic replications are averaged, the oscillations are flattened and can be overlooked. This can easily lead to the erroneous conclusion that the system reaches a constant steady state. RESULTS This paper proposes a straightforward method to detect and asses stochastic oscillations. The basis of the method is in the use of polar coordinates for systems with two species, and cylindrical coordinates for systems with more than two species. By slightly modifying these coordinate systems, it is possible to compute the total angular distance run by the system and the average Euclidean distance to a reference point. This allows us to compute confidence intervals, both for the average angular speed and for the distance to a reference point, from a set of replications. CONCLUSIONS The use of polar (or cylindrical) coordinates provides a new perspective of the system dynamics. The mean trajectory that can be obtained by averaging the usual cartesian coordinates of the samples informs about the trajectory of the center of mass of the replications. In contrast to such a mean cartesian trajectory, the mean polar trajectory can be used to compute the average circular motion of those replications, and therefore, can yield evidence about sustained steady state oscillations. Both, the coordinate transformation and the computation of confidence intervals, can be carried out efficiently. This results in an efficient method to evaluate stochastic oscillations.
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Affiliation(s)
- Jorge Júlvez
- Cambridge Systems Biology Centre, University of Cambridge, Tennis Court RoadCB2 1GA, Cambridge, United Kingdom. .,Department of Computer Science and Systems Engineering, University of Zaragoza, María de Luna 1, Zaragoza, 50018, Spain.
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12
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13
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Shih HY, Goldenfeld N. Path-integral calculation for the emergence of rapid evolution from demographic stochasticity. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:050702. [PMID: 25493725 DOI: 10.1103/physreve.90.050702] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/24/2014] [Indexed: 06/04/2023]
Abstract
Genetic variation in a population can sometimes arise so fast as to modify ecosystem dynamics. Such phenomena have been observed in natural predator-prey systems and characterized in the laboratory as showing unusual phase relationships in population dynamics, including a π phase shift between predator and prey (evolutionary cycles) and even undetectable prey oscillations compared to those of the predator (cryptic cycles). Here we present a generic individual-level stochastic model of interacting populations that includes a subpopulation of low nutritional value to the predator. Using a master equation formalism and by mapping to a coherent state path integral solved by a system-size expansion, we show that evolutionary and cryptic quasicycles can emerge generically from the combination of intrinsic demographic fluctuations and clonal mutations alone, without additional biological mechanisms.
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Affiliation(s)
- Hong-Yan Shih
- Loomis Laboratory of Physics, Department of Physics, Center for the Physics of Living Cells and Institute for Genomic Biology, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, Illinois 61801-3080, USA
| | - Nigel Goldenfeld
- Loomis Laboratory of Physics, Department of Physics, Center for the Physics of Living Cells and Institute for Genomic Biology, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, Illinois 61801-3080, USA
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14
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Distinguishing intrinsic limit cycles from forced oscillations in ecological time series. THEOR ECOL-NETH 2014. [DOI: 10.1007/s12080-014-0225-9] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/25/2022]
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15
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Louca S, Lampo M, Doebeli M. Assessing host extinction risk following exposure to Batrachochytrium dendrobatidis. Proc Biol Sci 2014; 281:20132783. [PMID: 24807247 DOI: 10.1098/rspb.2013.2783] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
Wildlife diseases are increasingly recognized as a major threat to biodiversity. Chytridiomycosis is an emerging infectious disease of amphibians caused by the fungus Batrachochytrium dendrobatidis (Bd). Using a mathematical model and simulations, we study its effects on a generic riparian host population with a tadpole and adult life stage. An analytical expression for the basic reproduction quotient, Qo, of the pathogen is derived. By sampling the entire relevant parameter space, we perform a statistical assessment of the importance of all considered parameters in determining the risk of host extinction, upon exposure to Bd. We find that Qo not only gives a condition for the initial invasion of the fungus, but is in fact the best predictor for host extinction. We also show that the role of tadpoles, which in some species tolerate infections, is ambivalent. While tolerant tadpoles may provide a reservoir for the fungus, thus facilitating its persistence or even amplifying its outbreaks, they can also act as a rescue buffer for a stressed host population. Our results have important implications for amphibian conservation efforts.
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Affiliation(s)
- Stilianos Louca
- Institute of Applied Mathematics, University of British Columbia, , 121-1984 Mathematics Road, Vancouver, British Columbia, Canada , V6T 1Z2, Centro de Ecología, Instituto Venezolano de Investigaciones Científicas, , Apartado 21827, Caracas 1020-A, Venezuela, Department of Zoology, University of British Columbia, , 6270 University Boulevard, Vancouver, British Columbia, Canada , V6T 1Z4, Department of Mathematics, University of British Columbia, , 6270 University Boulevard, Vancouver, British Columbia, Canada , V6T 1Z4
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16
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Ashcroft P, Galla T. Pattern formation in individual-based systems with time-varying parameters. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 88:062104. [PMID: 24483383 DOI: 10.1103/physreve.88.062104] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/28/2013] [Indexed: 06/03/2023]
Abstract
We study the patterns generated in finite-time sweeps across symmetry-breaking bifurcations in individual-based models. Similar to the well-known Kibble-Zurek scenario of defect formation, large-scale patterns are generated when model parameters are varied slowly, whereas fast sweeps produce a large number of small domains. The symmetry breaking is triggered by intrinsic noise, originating from the discrete dynamics at the microlevel. Based on a linear-noise approximation, we calculate the characteristic length scale of these patterns. We demonstrate the applicability of this approach in a simple model of opinion dynamics, a model in evolutionary game theory with a time-dependent fitness structure, and a model of cell differentiation. Our theoretical estimates are confirmed in simulations. In further numerical work, we observe a similar phenomenon when the symmetry-breaking bifurcation is triggered by population growth.
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Affiliation(s)
- Peter Ashcroft
- Theoretical Physics, School of Physics and Astronomy, The University of Manchester, Manchester M13 9PL, United Kingdom
| | - Tobias Galla
- Theoretical Physics, School of Physics and Astronomy, The University of Manchester, Manchester M13 9PL, United Kingdom
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17
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Barraquand F. Functional responses and predator–prey models: a critique of ratio dependence. THEOR ECOL-NETH 2013. [DOI: 10.1007/s12080-013-0201-9] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/26/2022]
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18
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Brett T, Galla T. Stochastic processes with distributed delays: chemical Langevin equation and linear-noise approximation. PHYSICAL REVIEW LETTERS 2013; 110:250601. [PMID: 23829723 DOI: 10.1103/physrevlett.110.250601] [Citation(s) in RCA: 27] [Impact Index Per Article: 2.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/26/2013] [Indexed: 06/02/2023]
Abstract
We develop a systematic approach to the linear-noise approximation for stochastic reaction systems with distributed delays. Unlike most existing work our formalism does not rely on a master equation; instead it is based upon a dynamical generating functional describing the probability measure over all possible paths of the dynamics. We derive general expressions for the chemical Langevin equation for a broad class of non-Markovian systems with distributed delay. Exemplars of a model of gene regulation with delayed autoinhibition and a model of epidemic spread with delayed recovery provide evidence of the applicability of our results.
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Affiliation(s)
- Tobias Brett
- Theoretical Physics, School of Physics and Astronomy, The University of Manchester, Manchester M13 9PL, United Kingdom.
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19
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Simonis JL. Demographic stochasticity reduces the synchronizing effect of dispersal in predator-prey metapopulations. Ecology 2012; 93:1517-24. [PMID: 22919898 DOI: 10.1890/11-0460.1] [Citation(s) in RCA: 16] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
Dispersal may affect predator-prey metapopulations by rescuing local sink populations from extinction or by synchronizing population dynamics across the metapopulation, increasing the risk of regional extinction. Dispersal is likely influenced by demographic stochasticity, however, particularly because dispersal rates are often very low in metapopulations. Yet the effects of demographic stochasticity on predator-prey metapopulations are not well known. To that end, I constructed three models of a two-patch predator-prey system. The models constitute a hierarchy of complexity, allowing direct comparisons. Two models included demographic stochasticity (pure jump process [PJP] and stochastic differential equations [SDE]), and the third was deterministic (ordinary differential equations [ODE]). One stochastic model (PJP) treated population sizes as discrete, while the other (SDE) allowed population sizes to change continuously. Both stochastic models only produced synchronized predator-prey dynamics when dispersal was high for both trophic levels. Frequent dispersal by only predators or prey in the PJP and SDE spatially decoupled the trophic interaction, reducing synchrony of the non-dispersive species. Conversely, the ODE generated synchronized predator-prey dynamics across all dispersal rates, except when initial conditions produced anti-phase transients. These results indicate that demographic stochasticity strongly reduces the synchronizing effect of dispersal, which is ironic because demographic stochasticity is often invoked post hoc as a driver of extinctions in synchronized metapopulations.
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Affiliation(s)
- Joseph L Simonis
- Department of Ecology and Evolutionary Biology, Cornell University, Ithaca, New York 14853, USA.
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20
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Black AJ, McKane AJ. Stochastic formulation of ecological models and their applications. Trends Ecol Evol 2012; 27:337-45. [PMID: 22406194 DOI: 10.1016/j.tree.2012.01.014] [Citation(s) in RCA: 125] [Impact Index Per Article: 10.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/30/2011] [Revised: 01/25/2012] [Accepted: 01/27/2012] [Indexed: 10/28/2022]
Abstract
The increasing use of computer simulation by theoretical ecologists started a move away from models formulated at the population level towards individual-based models. However, many of the models studied at the individual level are not analysed mathematically and remain defined in terms of a computer algorithm. This is not surprising, given that they are intrinsically stochastic and require tools and techniques for their study that may be unfamiliar to ecologists. Here, we argue that the construction of ecological models at the individual level and their subsequent analysis is, in many cases, straightforward and leads to important insights. We discuss recent work that highlights the importance of stochastic effects for parameter ranges and systems where it was previously thought that such effects would be negligible.
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Affiliation(s)
- Andrew J Black
- School of Mathematical Sciences, The University of Adelaide, Adelaide, SA 5005, Australia
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21
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Bladon AJ, Galla T. Learning dynamics in public goods games. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:041132. [PMID: 22181112 DOI: 10.1103/physreve.84.041132] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/06/2011] [Indexed: 05/31/2023]
Abstract
We extend recent analyses of stochastic effects in game dynamical learning to cases of multiplayer games and to games defined on networked structures. By means of an expansion in the noise strength we consider the weak-noise limit and present an analytical computation of spectral properties of fluctuations in multiplayer public goods games. This extends existing work on two-player games. In particular we show that coherent cycles may emerge driven by noise in the adaptation dynamics. These phenomena are not too dissimilar from cyclic strategy switching observed in experiments of behavioral game theory.
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Affiliation(s)
- Alex J Bladon
- Theoretical Physics, School of Physics and Astronomy, University of Manchester, Manchester M13 9PL, United Kingdom.
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Biancalani T, Galla T, McKane AJ. Stochastic waves in a Brusselator model with nonlocal interaction. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:026201. [PMID: 21929075 DOI: 10.1103/physreve.84.026201] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/04/2011] [Indexed: 05/31/2023]
Abstract
We show that intrinsic noise can induce spatiotemporal phenomena such as Turing patterns and traveling waves in a Brusselator model with nonlocal interaction terms. In order to predict and to characterize these stochastic waves we analyze the nonlocal model using a system-size expansion. The resulting theory is used to calculate the power spectra of the stochastic waves analytically and the outcome is tested successfully against simulations. We discuss the possibility that nonlocal models in other areas, such as epidemic spread or social dynamics, may contain similar stochastically induced patterns.
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Affiliation(s)
- Tommaso Biancalani
- Theoretical Physics Division, School of Physics and Astronomy, University of Manchester, Manchester M13 9PL, United Kingdom.
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23
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Butler T, Goldenfeld N. Fluctuation-driven Turing patterns. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2011; 84:011112. [PMID: 21867118 DOI: 10.1103/physreve.84.011112] [Citation(s) in RCA: 61] [Impact Index Per Article: 4.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/04/2010] [Indexed: 05/21/2023]
Abstract
Models of diffusion-driven pattern formation that rely on the Turing mechanism are utilized in many areas of science. However, many such models suffer from the defect of requiring fine tuning of parameters or an unrealistic separation of scales in the diffusivities of the constituents of the system in order to predict the formation of spatial patterns. In the context of a very generic model of ecological pattern formation, we show that the inclusion of intrinsic noise in Turing models leads to the formation of "quasipatterns" that form in generic regions of parameter space and are experimentally distinguishable from standard Turing patterns. The existence of quasipatterns removes the need for unphysical fine tuning or separation of scales in the application of Turing models to real systems.
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Affiliation(s)
- Thomas Butler
- Department of Physics and Institute for Genomic Biology, University of Illinois at Urbana Champaign, 1110 West Green Street, Urbana, IL 61801, USA
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24
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Antonovics J, Edwards M. Spatio-temporal dynamics of bumblebee nest parasites (Bombus subgenus Psythirus ssp.) and their hosts (Bombus spp.). J Anim Ecol 2011; 80:999-1011. [PMID: 21644977 DOI: 10.1111/j.1365-2656.2011.01846.x] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Abstract
1. A 39-year bumblebee data base was used to study the codistribution of six cuckoo bumblebees in the subgenus Psythirus of Bombus (hereafter called Psythirus) and their free-living bumblebee hosts in the British Isles. 2. A model of nest parasitism predicted host threshold densities and stable deterministic dynamics, with fluctuations only emerging as a result of environmental or demographic stochasticity. 3. Standardized transects indicated that variation in total number of records could be largely attributed to variation in observer effort; analyses were therefore carried out using relative abundance. 4. Spatially, parasite-free zones were evident in areas of low host abundance, but the host threshold for parasite presence differed among species and locations. 5. Temporally, the relative numbers of the parasite and host species remained relatively constant, except that the nest parasite P. campestris declined significantly since 1990. 6. There were consistent negative effects of the parasitic species on the numbers of hosts in the following year, and this pattern was seen over large geographic areas. 7. The spatio-temporal patterns confirmed a high degree of host specificity, except that P. campestris may be parasitizing not only B. pascuorum but also other species in the subgenus Thoracobombus.
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Affiliation(s)
- Janis Antonovics
- Biology Department, University of Virginia, Charlottesville, VA 22904, USA.
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25
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Wallace E, Benayoun M, van Drongelen W, Cowan JD. Emergent oscillations in networks of stochastic spiking neurons. PLoS One 2011; 6:e14804. [PMID: 21573105 PMCID: PMC3089610 DOI: 10.1371/journal.pone.0014804] [Citation(s) in RCA: 73] [Impact Index Per Article: 5.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/08/2011] [Accepted: 03/21/2011] [Indexed: 12/03/2022] Open
Abstract
Networks of neurons produce diverse patterns of oscillations, arising from the network's global properties, the propensity of individual neurons to oscillate, or a mixture of the two. Here we describe noisy limit cycles and quasi-cycles, two related mechanisms underlying emergent oscillations in neuronal networks whose individual components, stochastic spiking neurons, do not themselves oscillate. Both mechanisms are shown to produce gamma band oscillations at the population level while individual neurons fire at a rate much lower than the population frequency. Spike trains in a network undergoing noisy limit cycles display a preferred period which is not found in the case of quasi-cycles, due to the even faster decay of phase information in quasi-cycles. These oscillations persist in sparsely connected networks, and variation of the network's connectivity results in variation of the oscillation frequency. A network of such neurons behaves as a stochastic perturbation of the deterministic Wilson-Cowan equations, and the network undergoes noisy limit cycles or quasi-cycles depending on whether these have limit cycles or a weakly stable focus. These mechanisms provide a new perspective on the emergence of rhythmic firing in neural networks, showing the coexistence of population-level oscillations with very irregular individual spike trains in a simple and general framework.
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Affiliation(s)
- Edward Wallace
- Department of Mathematics, University of Chicago, Chicago, Illinois, United States of America.
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26
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Scott M, Poulin FJ, Tang H. Approximating intrinsic noise in continuous multispecies models. Proc Math Phys Eng Sci 2010. [DOI: 10.1098/rspa.2010.0275] [Citation(s) in RCA: 20] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
In small-scale chemical reaction networks, the local density of molecules is changed by discrete jumps owing to reactive collisions, and through transport. A systematic perturbation scheme is developed to analytically characterize these non-equilibrium intrinsic fluctuations in a multispecies spatially varying system. The method is illustrated on a variety of model systems. In all cases, the continuous approximation method is corroborated with extensive stochastic simulation. As an example of our technique applied to a spatially varying steady state, we demonstrate that a model for embryonic patterning mediated by regulatory mRNA is surprisingly robust to intrinsic fluctuations.
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Affiliation(s)
- Matthew Scott
- Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
| | - Francis J. Poulin
- Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
| | - Herbert Tang
- Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
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27
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Bladon AJ, Galla T, McKane AJ. Evolutionary dynamics, intrinsic noise, and cycles of cooperation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 81:066122. [PMID: 20866493 DOI: 10.1103/physreve.81.066122] [Citation(s) in RCA: 19] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/27/2010] [Indexed: 05/29/2023]
Abstract
We use analytical techniques based on an expansion in the inverse system size to study the stochastic evolutionary dynamics of finite populations of players interacting in a repeated prisoner's dilemma game. We show that a mechanism of amplification of demographic noise can give rise to coherent oscillations in parameter regimes where deterministic descriptions converge to fixed points with complex eigenvalues. These quasicycles between cooperation and defection have previously been observed in computer simulations; here we provide a systematic and comprehensive analytical characterization of their properties. We are able to predict their power spectra as a function of the mutation rate and other model parameters and to compare the relative magnitude of the cycles induced by different types of underlying microscopic dynamics. We also extend our analysis to the iterated prisoner's dilemma game with a win-stay lose-shift strategy, appropriate in situations where players are subject to errors of the trembling-hand type.
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Affiliation(s)
- Alex J Bladon
- Theoretical Physics, School of Physics and Astronomy, University of Manchester, Manchester M13 9PL, United Kingdom.
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28
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Murase Y, Shimada T, Ito N, Rikvold PA. Effects of demographic stochasticity on biological community assembly on evolutionary time scales. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 81:041908. [PMID: 20481754 DOI: 10.1103/physreve.81.041908] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/10/2009] [Revised: 01/25/2010] [Indexed: 05/29/2023]
Abstract
We study the effects of demographic stochasticity on the long-term dynamics of biological coevolution models of community assembly. The noise is induced in order to check the validity of deterministic population dynamics. While mutualistic communities show little dependence on the stochastic population fluctuations, predator-prey models show strong dependence on the stochasticity, indicating the relevance of the finiteness of the populations. For a predator-prey model, the noise causes drastic decreases in diversity and total population size. The communities that emerge under influence of the noise consist of species strongly coupled with each other and have stronger linear stability around the fixed-point populations than the corresponding noiseless model. The dynamics on evolutionary time scales for the predator-prey model are also altered by the noise. Approximate 1/f fluctuations are observed with noise, while 1/f2 fluctuations are found for the model without demographic noise.
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Affiliation(s)
- Yohsuke Murase
- Department of Applied Physics, School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
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29
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Svanbäck R, Pineda-Krch M, Doebeli M. Fluctuating population dynamics promotes the evolution of phenotypic plasticity. Am Nat 2009; 174:176-89. [PMID: 19519278 DOI: 10.1086/600112] [Citation(s) in RCA: 68] [Impact Index Per Article: 4.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/03/2022]
Abstract
Theoretical and empirical studies are showing evidence in support of evolutionary branching and sympatric speciation due to frequency-dependent competition. However, phenotypic diversification due to underlying genetic diversification is only one possible evolutionary response to disruptive selection. Another potentially general response is phenotypic diversification in the form of phenotypic plasticity. It has been suggested that genetic variation is favored in stable environments, whereas phenotypic plasticity is favored in unstable and fluctuating environments. We investigate the "competition" between the processes of evolutionary branching and the evolution of phenotypic plasticity in a predator-prey model that allows both processes to occur. In this model, environmental fluctuations can be caused by complicated population dynamics. We found that the evolution of phenotypic plasticity was generally more likely than evolutionary branching when the ecological dynamics exhibited pronounced predator-prey cycles, whereas the opposite was true when the ecological dynamics was more stable. At intermediate levels of density cycling, trimorphisms with two specialist branches and a phenotypically plastic generalist branch sometimes occurred. Our theoretical results suggest that ecological dynamics and evolutionary dynamics can often be tightly linked and that an explicit consideration of population dynamics may be essential to explain the evolutionary dynamics of diversification in natural populations.
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Affiliation(s)
- Richard Svanbäck
- Department of Zoology, University of British Columbia, Vancouver, British Columbia V6T 1Z4, Canada.
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30
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Butler T, Goldenfeld N. Robust ecological pattern formation induced by demographic noise. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 80:030902. [PMID: 19905053 DOI: 10.1103/physreve.80.030902] [Citation(s) in RCA: 41] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/30/2009] [Indexed: 05/21/2023]
Abstract
We demonstrate that demographic noise can induce persistent spatial pattern formation and temporal oscillations in the Levin-Segel predator-prey model for plankton-herbivore population dynamics. Although the model exhibits a Turing instability in mean-field theory, demographic noise greatly enlarges the region of parameter space where pattern formation occurs. To distinguish between patterns generated by fluctuations and those present at the mean-field level in real ecosystems, we calculate the power spectrum in the noise-driven case and predict the presence of fat tails not present in the mean-field case. These results may account for the prevalence of large-scale ecological patterns, beyond that expected from traditional nonstochastic approaches.
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Affiliation(s)
- Thomas Butler
- Department of Physics and Institute for Genomic Biology, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, Illinois 61801, USA
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31
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Galla T. Intrinsic fluctuations in stochastic delay systems: theoretical description and application to a simple model of gene regulation. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 80:021909. [PMID: 19792153 DOI: 10.1103/physreve.80.021909] [Citation(s) in RCA: 21] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/21/2009] [Indexed: 05/28/2023]
Abstract
The effects of intrinsic noise on stochastic delay systems is studied within an expansion in the inverse system size. We show that the stochastic nature of the underlying dynamics may induce oscillatory behavior in parameter ranges where the deterministic system does not sustain cycles, and compute the power spectra of these stochastic oscillations analytically, in good agreement with simulations. The theory is developed in the context of a simple one-dimensional toy model, but is applicable more generally. Gene regulatory systems in particular often contain only a small number of molecules, leading to significant fluctuations in messenger RNA (mRNA) and protein concentrations. As an application we therefore study a minimalistic model of the expression levels of hes1 mRNA and Hes1 protein, representing the simple motif of an autoinhibitory feedback loop and motivated by its relevance to somite segmentation.
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Affiliation(s)
- Tobias Galla
- Theoretical Physics, School of Physics and Astronomy, The University of Manchester, Manchester M13 9PL, United Kingdom.
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32
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Tomé T, de Oliveira MJ. Role of noise in population dynamics cycles. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 79:061128. [PMID: 19658494 DOI: 10.1103/physreve.79.061128] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/12/2008] [Revised: 05/22/2009] [Indexed: 05/08/2023]
Abstract
Noise is an intrinsic feature of population dynamics and plays a crucial role in oscillations called phase-forgetting quasicycles by converting damped into sustained oscillations. This function of noise becomes evident when considering Langevin equations whose deterministic part yields only damped oscillations. We formulate here a consistent and systematic approach to population dynamics, leading to a Fokker-Planck equation and the associate Langevin equations in accordance with this conceptual framework, founded on stochastic lattice-gas models that describe spatially structured predator-prey systems. Langevin equations in the population densities and predator-prey pair density are derived in two stages. First, a birth-and-death stochastic process in the space of prey and predator numbers and predator-prey pair number is obtained by a contraction method that reduces the degrees of freedom. Second, a van Kampen expansion in the inverse of system size is then performed to get the Fokker-Planck equation. We also study the time correlation function, the asymptotic behavior of which is used to characterize the transition from the cyclic coexistence of species to the ordinary coexistence.
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Affiliation(s)
- Tânia Tomé
- Instituto de Física, Universidade de São Paulo, Caixa Postal 66318, 05314-970 São Paulo, SP, Brazil
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33
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Boland RP, Galla T, McKane AJ. Limit cycles, complex Floquet multipliers, and intrinsic noise. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2009; 79:051131. [PMID: 19518440 DOI: 10.1103/physreve.79.051131] [Citation(s) in RCA: 21] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/30/2009] [Indexed: 05/27/2023]
Abstract
We study the effects of intrinsic noise on chemical reaction systems, which in the deterministic limit approach a limit cycle in an oscillatory manner. Previous studies of systems with an oscillatory approach to a fixed point have shown that the noise can transform the oscillatory decay into sustained coherent oscillations with a large amplitude. We show that a similar effect occurs when the stable attractors are limit cycles. We compute the correlation functions and spectral properties of the fluctuations in suitably comoving Frenet frames for several model systems including driven and coupled Brusselators, and the Willamowski-Rössler system. Analytical results are confirmed convincingly in numerical simulations. The effect is quite general, and occurs whenever the Floquet multipliers governing the stability of the limit cycle are complex, with the amplitude of the oscillations increasing as the instability boundary is approached.
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Affiliation(s)
- Richard P Boland
- Theoretical Physics, School of Physics and Astronomy, University of Manchester, Manchester M13 9PL, United Kingdom.
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