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Sergei Petrovskii
School of Mathematics and Actuarial Science, University of Leicester, Leicester LE1 7RH, UK

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Journal article
Published: 28 April 2021 in Journal of The Royal Society Interface
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Spatial distribution of the human population is distinctly heterogeneous, e.g. showing significant difference in the population density between urban and rural areas. In the historical perspective, i.e. on the timescale of centuries, the emergence of densely populated areas at their present locations is widely believed to be linked to more favourable environmental and climatic conditions. In this paper, we challenge this point of view. We first identify a few areas at different parts of the world where the environmental conditions (quantified by the temperature, precipitation and elevation) show a relatively small variation in space on the scale of thousands of kilometres. We then examine the population distribution across those areas to show that, in spite of the approximate homogeneity of the environment, it exhibits a significant variation revealing a nearly periodic spatial pattern. Based on this apparent disagreement, we hypothesize that there may exist an inherent mechanism that may lead to pattern formation even in a uniform environment. We consider a mathematical model of the coupled demographic-economic dynamics and show that its spatially uniform, locally stable steady state can give rise to a periodic spatial pattern due to the Turing instability, the spatial scale of the emerging pattern being consistent with observations. Using numerical simulations, we show that, interestingly, the emergence of the Turing patterns may eventually lead to the system collapse.

ACS Style

Anna Zincenko; Sergei Petrovskii; Vitaly Volpert; Malay Banerjee. Turing instability in an economic–demographic dynamical system may lead to pattern formation on a geographical scale. Journal of The Royal Society Interface 2021, 18, 20210034 .

AMA Style

Anna Zincenko, Sergei Petrovskii, Vitaly Volpert, Malay Banerjee. Turing instability in an economic–demographic dynamical system may lead to pattern formation on a geographical scale. Journal of The Royal Society Interface. 2021; 18 (177):20210034.

Chicago/Turabian Style

Anna Zincenko; Sergei Petrovskii; Vitaly Volpert; Malay Banerjee. 2021. "Turing instability in an economic–demographic dynamical system may lead to pattern formation on a geographical scale." Journal of The Royal Society Interface 18, no. 177: 20210034.

Journal article
Published: 10 February 2021 in Mathematics
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Dynamics of human populations can be affected by various socio-economic factors through their influence on the natality and mortality rates, and on the migration intensity and directions. In this work we study an economic–demographic model which takes into account the dependence of the wealth production rate on the available resources. In the case of nonlocal consumption of resources, the homogeneous-in-space wealth–population distribution is replaced by a periodic-in-space distribution for which the total wealth increases. For the global consumption of resources, if the wealth redistribution is small enough, then the homogeneous distribution is replaced by a heterogeneous one with a single wealth accumulation center. Thus, economic and demographic characteristics of nonlocal and global economies can be quite different in comparison with the local economy.

ACS Style

Malay Banerjee; Sergei Petrovskii; Vitaly Volpert. Nonlocal Reaction–Diffusion Models of Heterogeneous Wealth Distribution. Mathematics 2021, 9, 351 .

AMA Style

Malay Banerjee, Sergei Petrovskii, Vitaly Volpert. Nonlocal Reaction–Diffusion Models of Heterogeneous Wealth Distribution. Mathematics. 2021; 9 (4):351.

Chicago/Turabian Style

Malay Banerjee; Sergei Petrovskii; Vitaly Volpert. 2021. "Nonlocal Reaction–Diffusion Models of Heterogeneous Wealth Distribution." Mathematics 9, no. 4: 351.

Article
Published: 01 February 2021 in Physical Review E
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With the growing number of discovered exoplanets, the Gaia concept finds its second wind. The Gaia concept defines that the biosphere of an inhabited planet regulates a planetary climate through feedback loops such that the planet remains habitable. Crunching the “Gaia” puzzle has been a focus of intense empirical research. Much less attention has been paid to the mathematical realization of this concept. In this paper, we consider the stability of a planetary climate system with the dynamic biosphere by linking a conceptual climate model to a generic population dynamics model with random parameters. We first show that the dynamics of the corresponding coupled system possesses multiple timescales and hence falls into the class of slow-fast dynamics. We then investigate the properties of a general dynamical system to which our model belongs and prove that the feedbacks from the biosphere dynamics cannot break the system's stability as long as the biodiversity is sufficiently high. That may explain why the climate is apparently stable over long time intervals. Interestingly, our coupled climate-biosphere system can lose its stability if biodiversity decreases; in this case, the evolution of the biosphere under the effect of random factors can lead to a global climate change.

ACS Style

Sergey A. Vakulenko; Ivan Sudakov; Sergei V. Petrovskii; Dmitry Lukichev. Stability of a planetary climate system with the biosphere species competing for resources. Physical Review E 2021, 103, 022202 .

AMA Style

Sergey A. Vakulenko, Ivan Sudakov, Sergei V. Petrovskii, Dmitry Lukichev. Stability of a planetary climate system with the biosphere species competing for resources. Physical Review E. 2021; 103 (2):022202.

Chicago/Turabian Style

Sergey A. Vakulenko; Ivan Sudakov; Sergei V. Petrovskii; Dmitry Lukichev. 2021. "Stability of a planetary climate system with the biosphere species competing for resources." Physical Review E 103, no. 2: 022202.

Journal article
Published: 25 December 2020 in Insects
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Exploitation of heterogenous distributions of Deroceras reticulatum, in arable fields by targeting molluscicide applications toward areas with higher slug densities, relies on these patches displaying sufficient spatio-temporal stability. Regular sampling of slug activity/distribution was undertaken using 1 ha rectangular grids of 100 refuge traps established in 22 commercial arable field crops. Activity varied significantly between the three years of the study, and the degree of aggregation (Taylor’s Power Law) was higher in fields with higher mean trap catches. Hot spot analysis detected statistically significant spatial clusters in all fields, and in 162 of the 167 individual assessment visits. The five assessment visits in which no clusters were detected coincided with low slug activity (≤0.07 per trap). Generalized Linear Models showed significant spatial stability of patches in 11 fields, with non-significant fields also characterized by low slug activity (≤1.2 per trap). Mantel’s permutation tests revealed a high degree of correlation between location of individual patches between sampling dates. It was concluded that patches of higher slug density were spatio-temporally stable, but detection using surface refuge traps (which rely on slug activity on the soil surface) was less reliable when adverse environmental conditions resulted in slugs retreating into the upper soil horizons.

ACS Style

Emily Forbes; Matthew Back; Andrew Brooks; Natalia B. Petrovskaya; Sergei V. Petrovskii; Tom Pope; Keith F. A. Walters. Stability of Patches of Higher Population Density within the Heterogenous Distribution of the Gray Field Slug Deroceras reticulatum in Arable Fields in the UK. Insects 2020, 12, 9 .

AMA Style

Emily Forbes, Matthew Back, Andrew Brooks, Natalia B. Petrovskaya, Sergei V. Petrovskii, Tom Pope, Keith F. A. Walters. Stability of Patches of Higher Population Density within the Heterogenous Distribution of the Gray Field Slug Deroceras reticulatum in Arable Fields in the UK. Insects. 2020; 12 (1):9.

Chicago/Turabian Style

Emily Forbes; Matthew Back; Andrew Brooks; Natalia B. Petrovskaya; Sergei V. Petrovskii; Tom Pope; Keith F. A. Walters. 2020. "Stability of Patches of Higher Population Density within the Heterogenous Distribution of the Gray Field Slug Deroceras reticulatum in Arable Fields in the UK." Insects 12, no. 1: 9.

Journal article
Published: 19 December 2020 in Mathematics
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The fractional calculus is useful in describing the natural phenomena with memory effect. This paper addresses the fractional form of Ambartsumian equation with a delay parameter. It may be a challenge to obtain accurate approximate solution of such kinds of fractional delay equations. In the literature, several attempts have been conducted to analyze the fractional Ambartsumian equation. However, the previous approaches in the literature led to approximate power series solutions which converge in subdomains. Such difficulties are solved in this paper via the Homotopy Perturbation Method (HPM). The present approximations are expressed in terms of the Mittag-Leffler functions which converge in the whole domain of the studied model. The convergence issue is also addressed. Several comparisons with the previous published results are discussed. In particular, while the computed solution in the literature is physical in short domains, with our approach it is physical in the whole domain. The results reveal that the HPM is an effective tool to analyzing the fractional Ambartsumian equation.

ACS Style

Weam Alharbi; Sergei Petrovskii. Numerical Analysis for the Fractional Ambartsumian Equation via the Homotopy Herturbation Method. Mathematics 2020, 8, 2247 .

AMA Style

Weam Alharbi, Sergei Petrovskii. Numerical Analysis for the Fractional Ambartsumian Equation via the Homotopy Herturbation Method. Mathematics. 2020; 8 (12):2247.

Chicago/Turabian Style

Weam Alharbi; Sergei Petrovskii. 2020. "Numerical Analysis for the Fractional Ambartsumian Equation via the Homotopy Herturbation Method." Mathematics 8, no. 12: 2247.

Journal article
Published: 21 October 2020 in Scientific Reports
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We report the results of an experiment on radio-tracking of individual grey field slugs in an arable field and associated data modelling designed to investigate the effect of slug population density in their movement. Slugs were collected in a commercial winter wheat field in which a 5x6 trapping grid had been established with 2m distance between traps. The slugs were taken to the laboratory, radio-tagged using a recently developed procedure, and following a recovery period released into the same field. Seventeen tagged slugs were released singly (sparse release) on the same grid node on which they had been caught. Eleven tagged slugs were released as a group (dense release). Each of the slugs was radio-tracked for approximately 10 h during which their position was recorded ten times. The tracking data were analysed using the Correlated Random Walk framework. The analysis revealed that all components of slug movement (mean speed, turning angles and movement/resting times) were significantly different between the two treatments. On average, the slugs released as a group disperse more slowly than slugs released individually and their turning angle has a clear anticlockwise bias. The results clearly suggest that population density is a factor regulating slug movement.

ACS Style

John Ellis; Natalia Petrovskaya; Emily Forbes; Keith F. A. Walters; Sergei Petrovskii. Movement patterns of the grey field slug (Deroceras reticulatum) in an arable field. Scientific Reports 2020, 10, 1 -16.

AMA Style

John Ellis, Natalia Petrovskaya, Emily Forbes, Keith F. A. Walters, Sergei Petrovskii. Movement patterns of the grey field slug (Deroceras reticulatum) in an arable field. Scientific Reports. 2020; 10 (1):1-16.

Chicago/Turabian Style

John Ellis; Natalia Petrovskaya; Emily Forbes; Keith F. A. Walters; Sergei Petrovskii. 2020. "Movement patterns of the grey field slug (Deroceras reticulatum) in an arable field." Scientific Reports 10, no. 1: 1-16.

Journal article
Published: 05 August 2020 in Communications in Nonlinear Science and Numerical Simulation
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There is growing evidence that ecological interactions are often nonlocal. Correspondingly, increasing attention is paid to mathematical models with nonlocal terms as such models may provide a more realistic description of ecological dynamics. Here we consider a nonlocal prey-predator model where the movement of both species is described by the standard Fickian diffusion, and hence is local, but the intra-specific competition of prey is nonlocal and is described by a convolution-type term with the ‘top-hat’ (piecewise-constant) kernel. The prey growth rate also includes the strong Allee effect. The system is studied using a combination of analytical tools and extensive numerical simulations. We obtain that nonlocality makes possible the pattern formation due to the Turing instability, which is not possible in the corresponding local model. We also obtain that the nonlocality creates bistability: it depends on the initial conditions which of the two spatially heterogeneous distributions emerges. Finally, we show that the bifurcation structure of the system is less sensitive to the choice of parametrization than it is in the corresponding nonspatial case, suggesting that nonlocality may decrease the structural sensitivity of the system.

ACS Style

Swadesh Pal; Sergei Petrovskii; S. Ghorai; Malay Banerjee. Spatiotemporal pattern formation in 2D prey-predator system with nonlocal intraspecific competition. Communications in Nonlinear Science and Numerical Simulation 2020, 93, 105478 .

AMA Style

Swadesh Pal, Sergei Petrovskii, S. Ghorai, Malay Banerjee. Spatiotemporal pattern formation in 2D prey-predator system with nonlocal intraspecific competition. Communications in Nonlinear Science and Numerical Simulation. 2020; 93 ():105478.

Chicago/Turabian Style

Swadesh Pal; Sergei Petrovskii; S. Ghorai; Malay Banerjee. 2020. "Spatiotemporal pattern formation in 2D prey-predator system with nonlocal intraspecific competition." Communications in Nonlinear Science and Numerical Simulation 93, no. : 105478.

Journal article
Published: 01 March 2020 in Physics of Life Reviews
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ACS Style

Andrew Morozov; Karen Abbott; Kim Cuddington; Tessa Francis; Gabriel Gellner; Alan Hastings; Ying-Cheng Lai; Sergei Petrovskii; Katherine Scranton; Mary Lou Zeeman. Long living transients: Enfant terrible of ecological theory? Physics of Life Reviews 2020, 32, 55 -58.

AMA Style

Andrew Morozov, Karen Abbott, Kim Cuddington, Tessa Francis, Gabriel Gellner, Alan Hastings, Ying-Cheng Lai, Sergei Petrovskii, Katherine Scranton, Mary Lou Zeeman. Long living transients: Enfant terrible of ecological theory? Physics of Life Reviews. 2020; 32 ():55-58.

Chicago/Turabian Style

Andrew Morozov; Karen Abbott; Kim Cuddington; Tessa Francis; Gabriel Gellner; Alan Hastings; Ying-Cheng Lai; Sergei Petrovskii; Katherine Scranton; Mary Lou Zeeman. 2020. "Long living transients: Enfant terrible of ecological theory?" Physics of Life Reviews 32, no. : 55-58.

Journal article
Published: 03 January 2020 in Mathematics
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Understanding of the dynamics of riots, protests, and social unrest more generally is important in order to ensure a stable, sustainable development of various social groups, as well as the society as a whole. Mathematical models of social dynamics have been increasingly recognized as a powerful research tool to facilitate the progress in this field. However, the question as to what should be an adequate mathematical framework to describe the corresponding social processes is largely open. In particular, a great majority of the previous studies dealt with non-spatial or spatially implicit systems, but the literature dealing with spatial systems remains meagre. Meanwhile, in many cases, the dynamics of social protests has a clear spatial aspect. In this paper, we attempt to close this gap partially by considering a spatial extension of a few recently developed models of social protests. We show that even a straightforward spatial extension immediately bring new dynamical behaviours, in particular predicting a new scenario of the protests’ termination.

ACS Style

Sergei Petrovskii; Weam Alharbi; Abdulqader Alhomairi; Andrew Morozov. Modelling Population Dynamics of Social Protests in Time and Space: The Reaction-Diffusion Approach. Mathematics 2020, 8, 78 .

AMA Style

Sergei Petrovskii, Weam Alharbi, Abdulqader Alhomairi, Andrew Morozov. Modelling Population Dynamics of Social Protests in Time and Space: The Reaction-Diffusion Approach. Mathematics. 2020; 8 (1):78.

Chicago/Turabian Style

Sergei Petrovskii; Weam Alharbi; Abdulqader Alhomairi; Andrew Morozov. 2020. "Modelling Population Dynamics of Social Protests in Time and Space: The Reaction-Diffusion Approach." Mathematics 8, no. 1: 78.

Review article
Published: 13 September 2019 in Physics of Life Reviews
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This paper discusses the recent progress in understanding the properties of transient dynamics in complex ecological systems. Predicting long-term trends as well as sudden changes and regime shifts in ecosystems dynamics is a major issue for ecology as such changes often result in population collapse and extinctions. Analysis of population dynamics has traditionally been focused on their long-term, asymptotic behavior whilst largely disregarding the effect of transients. However, there is a growing understanding that in ecosystems the asymptotic behavior is rarely seen. A big new challenge for theoretical and empirical ecology is to understand the implications of long transients. It is believed that the identification of the corresponding mechanisms along with the knowledge of scaling laws of the transient's lifetime should substantially improve the quality of long-term forecasting and crisis anticipation. Although transient dynamics have received considerable attention in physical literature, research into ecological transients is in its infancy and systematic studies are lacking. This text aims to partially bridge this gap and facilitate further progress in quantitative analysis of long transients in ecology. By revisiting and critically examining a broad variety of mathematical models used in ecological applications as well as empirical facts, we reveal several main mechanisms leading to the emergence of long transients and hence lays the basis for a unifying theory.

ACS Style

Andrew Morozov; Karen Abbott; Kim Cuddington; Tessa Francis; Gabriel Gellner; Alan Hastings; Ying-Cheng Lai; Sergei Petrovskii; Katherine Scranton; Mary Lou Zeeman. Long transients in ecology: Theory and applications. Physics of Life Reviews 2019, 32, 1 -40.

AMA Style

Andrew Morozov, Karen Abbott, Kim Cuddington, Tessa Francis, Gabriel Gellner, Alan Hastings, Ying-Cheng Lai, Sergei Petrovskii, Katherine Scranton, Mary Lou Zeeman. Long transients in ecology: Theory and applications. Physics of Life Reviews. 2019; 32 ():1-40.

Chicago/Turabian Style

Andrew Morozov; Karen Abbott; Kim Cuddington; Tessa Francis; Gabriel Gellner; Alan Hastings; Ying-Cheng Lai; Sergei Petrovskii; Katherine Scranton; Mary Lou Zeeman. 2019. "Long transients in ecology: Theory and applications." Physics of Life Reviews 32, no. : 1-40.

Article
Published: 05 September 2019 in Journal of Statistical Physics
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Reaction-telegraph equation (RTE) is a mathematical model that has often been used to describe natural phenomena, with specific applications ranging from physics to social sciences. In particular, in the context of ecology, it is believed to be a more realistic model to describe animal movement than the more traditional approach based on the reaction-diffusion equations. Indeed, the reaction-telegraph equation arises from more realistic microscopic assumptions about individual animal movement (the correlated random walk) and hence could be expected to be more relevant than the diffusion-type models that assume the simple, unbiased Brownian motion. However, the RTE has one significant drawback as its solutions are not positively defined. It is not clear at which stage of the RTE derivation the realism of the microscopic description is lost and/or whether the RTE can somehow be ‘improved’ to guarantee the solutions positivity. Here we show that the origin of the problem is twofold. Firstly, the RTE is not fully equivalent to the Cattaneo system from which it is obtained; the equivalence can only be achieved in a certain parameter range and only for the initial conditions containing a finite number of Fourier modes. Secondly, the Dirichlet type boundary conditions routinely used for reaction-diffusion equations appear to be meaningless if used for the RTE resulting in solutions with unrealistic properties. We conclude that, for the positivity to be regained, one has to use the Cattaneo system with boundary conditions of Robin type or Neumann type, and we show how relevant classes of solutions can be obtained.

ACS Style

Paulo F. C. Tilles; Sergei V. Petrovskii. On the Consistency of the Reaction-Telegraph Process Within Finite Domains. Journal of Statistical Physics 2019, 177, 569 -587.

AMA Style

Paulo F. C. Tilles, Sergei V. Petrovskii. On the Consistency of the Reaction-Telegraph Process Within Finite Domains. Journal of Statistical Physics. 2019; 177 (4):569-587.

Chicago/Turabian Style

Paulo F. C. Tilles; Sergei V. Petrovskii. 2019. "On the Consistency of the Reaction-Telegraph Process Within Finite Domains." Journal of Statistical Physics 177, no. 4: 569-587.

Journal article
Published: 18 December 2018 in Journal of Theoretical Biology
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Individual animal movement has been a focus of intense research and considerable controversy over the last two decades, however the understanding of wider ecological implications of various movement behaviours is lacking. In this paper, we consider this issue in the context of pattern formation. Using an individual-based modelling approach and computer simulations, we first show that density dependence (“auto-taxis”) of the individual movement in a population of random walkers typically results in the formation of a strongly heterogeneous population distribution consisting of clearly defined animals clusters or patches. We then show that, when the movement takes place in a large spatial domain, the properties of the clusters are significantly different in the populations of Brownian and non-Brownian walkers. Whilst clusters tend to be stable in the case of Brownian motion, in the population of Levy walkers clusters are dynamical so that the number of clusters fluctuates in the course of time. We also show that the population dynamics of non-Brownian walkers exhibits two different time scales: a short time scale of the relaxation of the initial condition and a long time scale when one type of dynamics is replaced by another. Finally, we show that the distribution of sample values in the populations of Brownian and non-Brownian walkers is significantly different.

ACS Style

John Ellis; Natalia Petrovskaya; Sergei Petrovskii. Effect of density-dependent individual movement on emerging spatial population distribution: Brownian motion vs Levy flights. Journal of Theoretical Biology 2018, 464, 159 -178.

AMA Style

John Ellis, Natalia Petrovskaya, Sergei Petrovskii. Effect of density-dependent individual movement on emerging spatial population distribution: Brownian motion vs Levy flights. Journal of Theoretical Biology. 2018; 464 ():159-178.

Chicago/Turabian Style

John Ellis; Natalia Petrovskaya; Sergei Petrovskii. 2018. "Effect of density-dependent individual movement on emerging spatial population distribution: Brownian motion vs Levy flights." Journal of Theoretical Biology 464, no. : 159-178.

Journal article
Published: 15 December 2018 in Journal of Theoretical Biology
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Spatial proliferation of invasive species often causes serious damage to agriculture, ecology and environment. Evaluation of the extent of the area potentially invadable by an alien species is an important problem. Landscape features that reduces dispersal space to narrow corridors can make some areas inaccessible to the invading species. On the other hand, the existence of stepping stones – small areas or ‘patches’ with better environmental conditions – is known to assist species spread. How an interplay between these factors can affect the invasion success remains unclear. In this paper, we address this question theoretically using a mechanistic model of population dynamics. Such models have been generally successful in predicting the rate and pattern of invasive spread; however, they usually consider the spread in an unbounded, uniform space hence ignoring the complex geometry of a real landscape. In contrast, here we consider a reaction-diffusion model in a domain of a complex shape combining corridors and stepping stones. We show that the invasion success depends on a subtle interplay between the stepping stone size, location and the strength of the Allee effect inside. In particular, for a stepping stone of a small size, there is only a narrow range of locations where it can unblock the otherwise impassable corridor.

ACS Style

Weam Alharbi; Sergei Petrovskii. Effect of complex landscape geometry on the invasive species spread: Invasion with stepping stones. Journal of Theoretical Biology 2018, 464, 85 -97.

AMA Style

Weam Alharbi, Sergei Petrovskii. Effect of complex landscape geometry on the invasive species spread: Invasion with stepping stones. Journal of Theoretical Biology. 2018; 464 ():85-97.

Chicago/Turabian Style

Weam Alharbi; Sergei Petrovskii. 2018. "Effect of complex landscape geometry on the invasive species spread: Invasion with stepping stones." Journal of Theoretical Biology 464, no. : 85-97.

Journal article
Published: 14 November 2018 in Computation
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Decreasing level of dissolved oxygen has recently been reported as a growing ecological problem in seas and oceans around the world. Concentration of oxygen is an important indicator of the marine ecosystem’s health as lack of oxygen (anoxia) can lead to mass mortality of marine fauna. The oxygen decrease is thought to be a result of global warming as warmer water can contain less oxygen. Actual reasons for the observed oxygen decay remain controversial though. Recently, it has been shown that it may as well result from a disruption of phytoplankton photosynthesis. In this paper, we further explore this idea by considering the model of coupled plankton-oxygen dynamics in two spatial dimensions. By means of extensive numerical simulations performed for different initial conditions and in a broad range of parameter values, we show that the system’s dynamics normally lead to the formation of a rich variety of patterns. We reveal how these patterns evolve when the system approaches the tipping point, i.e., the boundary of the safe parameter range beyond which the depletion of oxygen is the only possibility. In particular, we show that close to the tipping point the spatial distribution of the dissolved oxygen tends to become more regular; arguably, this can be considered as an early warning of the approaching catastrophe.

ACS Style

Yadigar Sekerci; Sergei Petrovskii. Pattern Formation in a Model Oxygen-Plankton System. Computation 2018, 6, 59 .

AMA Style

Yadigar Sekerci, Sergei Petrovskii. Pattern Formation in a Model Oxygen-Plankton System. Computation. 2018; 6 (4):59.

Chicago/Turabian Style

Yadigar Sekerci; Sergei Petrovskii. 2018. "Pattern Formation in a Model Oxygen-Plankton System." Computation 6, no. 4: 59.

Editorial
Published: 13 September 2018 in Mathematics
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ACS Style

Sergei Petrovskii. Progress in Mathematical Ecology. Mathematics 2018, 6, 167 .

AMA Style

Sergei Petrovskii. Progress in Mathematical Ecology. Mathematics. 2018; 6 (9):167.

Chicago/Turabian Style

Sergei Petrovskii. 2018. "Progress in Mathematical Ecology." Mathematics 6, no. 9: 167.

Review article
Published: 06 September 2018 in Science
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The importance of transient dynamics in ecological systems and in the models that describe them has become increasingly recognized. However, previous work has typically treated each instance of these dynamics separately. We review both empirical examples and model systems, and outline a classification of transient dynamics based on ideas and concepts from dynamical systems theory. This classification provides ways to understand the likelihood of transients for particular systems, and to guide investigations to determine the timing of sudden switches in dynamics and other characteristics of transients. Implications for both management and underlying ecological theories emerge.

ACS Style

Alan Hastings; Karen C. Abbott; Kim Cuddington; Tessa Francis; Gabriel Gellner; Ying-Cheng Lai; Andrew Morozov; Sergei Petrovskii; Katherine Scranton; Mary Lou Zeeman. Transient phenomena in ecology. Science 2018, 361, eaat6412 .

AMA Style

Alan Hastings, Karen C. Abbott, Kim Cuddington, Tessa Francis, Gabriel Gellner, Ying-Cheng Lai, Andrew Morozov, Sergei Petrovskii, Katherine Scranton, Mary Lou Zeeman. Transient phenomena in ecology. Science. 2018; 361 (6406):eaat6412.

Chicago/Turabian Style

Alan Hastings; Karen C. Abbott; Kim Cuddington; Tessa Francis; Gabriel Gellner; Ying-Cheng Lai; Andrew Morozov; Sergei Petrovskii; Katherine Scranton; Mary Lou Zeeman. 2018. "Transient phenomena in ecology." Science 361, no. 6406: eaat6412.

Journal article
Published: 03 June 2018 in Geosciences
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We consider the effect of global warming on the coupled plankton-oxygen dynamics in the ocean. The net oxygen production by phytoplankton is known to depend on the water temperature and hence can be disrupted by warming. We address this issue theoretically by considering a mathematical model of the plankton-oxygen system. The model is generic and can account for a variety of biological factors. We first show that sustainable oxygen production by phytoplankton is only possible if the net production rate is above a certain critical value. This result appears to be robust to the details of model parametrization. We then show that, once the effect of zooplankton is taken into account (which consume oxygen and feed on phytoplankton), the plankton-oxygen system can only be stable if the net oxygen production rate is within a certain intermediate range (i.e., not too low and not too high). Correspondingly, we conclude that a sufficiently large increase in the water temperature is likely to push the system out of the safe range, which may result in ocean anoxia and even a global oxygen depletion. We then generalize the model by taking into account the effect of environmental stochasticity and show that, paradoxically, the probability of oxygen depletion may decrease with an increase in the rate of global warming.

ACS Style

Yadigar Sekerci; Sergei Petrovskii. Global Warming Can Lead to Depletion of Oxygen by Disrupting Phytoplankton Photosynthesis: A Mathematical Modelling Approach. Geosciences 2018, 8, 201 .

AMA Style

Yadigar Sekerci, Sergei Petrovskii. Global Warming Can Lead to Depletion of Oxygen by Disrupting Phytoplankton Photosynthesis: A Mathematical Modelling Approach. Geosciences. 2018; 8 (6):201.

Chicago/Turabian Style

Yadigar Sekerci; Sergei Petrovskii. 2018. "Global Warming Can Lead to Depletion of Oxygen by Disrupting Phytoplankton Photosynthesis: A Mathematical Modelling Approach." Geosciences 8, no. 6: 201.

Journal article
Published: 28 May 2018 in Mathematical Modelling of Natural Phenomena
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Human population growth has been called the biggest issue the humanity faces in the 21st century, and although this statement is globally true, locally, many Western economies have been experiencing population decline. Europe is in fact homeland for population decline. By 2050 many large European economies are predicted to lose large parts of their population. In this work, we consider the dynamical system that corresponds to the model introduced by Volpert et al. [Nonlinear Anal. 159 (2017) 408–423]. With the help of this model, we illustrate scenarios that can lead, in the long-run, to sharp population decline and/or deterioration of the economy. We also illustrate that even when under certain conditions the population will go extinct, temporarily it might experience growth.

ACS Style

A. Zincenko; S. Petrovskii; V. Volpert. An economic-demographic dynamical system. Mathematical Modelling of Natural Phenomena 2018, 13, 27 .

AMA Style

A. Zincenko, S. Petrovskii, V. Volpert. An economic-demographic dynamical system. Mathematical Modelling of Natural Phenomena. 2018; 13 (3):27.

Chicago/Turabian Style

A. Zincenko; S. Petrovskii; V. Volpert. 2018. "An economic-demographic dynamical system." Mathematical Modelling of Natural Phenomena 13, no. 3: 27.

Journal article
Published: 17 April 2018 in Mathematics
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A telegraph equation is believed to be an appropriate model of population dynamics as it accounts for the directional persistence of individual animal movement. Being motivated by the problem of habitat fragmentation, which is known to be a major threat to biodiversity that causes species extinction worldwide, we consider the reaction–telegraph equation (i.e., telegraph equation combined with the population growth) on a bounded domain with the goal to establish the conditions of species survival. We first show analytically that, in the case of linear growth, the expression for the domain’s critical size coincides with the critical size of the corresponding reaction–diffusion model. We then consider two biologically relevant cases of nonlinear growth, i.e., the logistic growth and the growth with a strong Allee effect. Using extensive numerical simulations, we show that in both cases the critical domain size of the reaction–telegraph equation is larger than the critical domain size of the reaction–diffusion equation. Finally, we discuss possible modifications of the model in order to enhance the positivity of its solutions.

ACS Style

Weam Alharbi; Sergei Petrovskii. Critical Domain Problem for the Reaction–Telegraph Equation Model of Population Dynamics. Mathematics 2018, 6, 59 .

AMA Style

Weam Alharbi, Sergei Petrovskii. Critical Domain Problem for the Reaction–Telegraph Equation Model of Population Dynamics. Mathematics. 2018; 6 (4):59.

Chicago/Turabian Style

Weam Alharbi; Sergei Petrovskii. 2018. "Critical Domain Problem for the Reaction–Telegraph Equation Model of Population Dynamics." Mathematics 6, no. 4: 59.

Preprint
Published: 30 March 2018
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Many empirical and theoretical studies indicate that Brownian motion and diffusion models as its mean field counterpart provide appropriate modelling techniques for individual insect movement. However, this traditional approach has been challenged and conflicting evidence suggests that an alternative movement pattern such as Lévy walks can provide a better description. Lévy walks differ from Brownian motion since they allow for a higher frequency of large steps, resulting in a faster movement. Identification of the ‘correct’ movement model that would consistently provide the best fit for movement data is challenging and has become a highly controversial issue. In this paper, we show that this controversy may be superficial rather than real if the issue is considered in the context of trapping or, more generally, survival probabilities. In particular, we show that almost identical trap counts are reproduced for inherently different movement models (such as the Brownian motion and the Lévy walk) under certain conditions of equivalence. This apparently suggests that the whole ‘Levy or diffusion’ debate is rather senseless unless it is placed into a specific ecological context, e.g. pest monitoring programmes.

ACS Style

Danish A. Ahmed; Sergei V. Petrovskii; Paulo F.C. Tilles. The “Lévy or Diffusion” Controversy: How Important is the Movement Pattern in the Context of Trapping? 2018, 1 .

AMA Style

Danish A. Ahmed, Sergei V. Petrovskii, Paulo F.C. Tilles. The “Lévy or Diffusion” Controversy: How Important is the Movement Pattern in the Context of Trapping? . 2018; ():1.

Chicago/Turabian Style

Danish A. Ahmed; Sergei V. Petrovskii; Paulo F.C. Tilles. 2018. "The “Lévy or Diffusion” Controversy: How Important is the Movement Pattern in the Context of Trapping?" , no. : 1.