Sergei Petrovskii

Spatiotemporal Complexity of Plankton and Fish Dynamics

Nonlinear dynamics and chaotic and complex systems constitute some of the most fascinating developments of late twentieth century mathematics and physics. The implications have changed our understanding of important phenomena in almost every field of science, including biology and ecology. This article investigates complexity and chaos in the spatiotemporal dynamics of aquatic ecosystems. The dynamics of these biological communities exhibit an interplay between processes acting on a scale from hundreds of meters to kilometers, controlled by biology, and processes acting on a scale from dozens to hundreds of kilometers, dominated by the heterogeneity of hydrophysical fields. We focus on how biological processes affect spatiotemporal pattern formation. Our results show that modeling by reaction-diffusion equations is an appropriate tool for investigating fundamental mechanisms of complex spatiotemporal plankton dynamics, fractal properties of planktivorous fish school movements, and their interrelationships.

Reaction–diffusion waves in biology

The theory of reaction-diffusion waves begins in the 1930s with the works in population dynamics, combustion theory and chemical kinetics. At the present time, it is a well developed area of research which includes qualitative properties of travelling waves for the scalar reaction-diffusion equation and for system of equations, complex nonlinear dynamics, numerous applications in physics, chemistry, biology, medicine. This paper reviews biological applications of reaction-diffusion waves.

Bifurcations and chaos in a predator-prey system with the Allee effect

It is known from many theoretical studies that ecological chaos may have numerous significant impacts on the population and community dynamics. Therefore, identification of the factors potentially enhancing or suppressing chaos is a challenging problem. In this paper, we show that chaos can be enhanced by the Allee effect. More specifically, we show by means of computer simulations that in a time–continuous predator–prey system with the Allee effect the temporal population oscillations can become chaotic even when the spatial distribution of the species remains regular. By contrast, in a similar system without the Allee effect, regular species distribution corresponds to periodic/quasi–periodic oscillations. We investigate the routes to chaos and show that in the spatially regular predator–prey system with the Allee effect, chaos appears as a result of series of period–doubling bifurcations. We also show that this system exhibits period–locking behaviour: a small variation of parameters can lead to alternating regular and chaotic dynamics.

Variation in individual walking behavior creates the impression of a Lévy flight

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

Exactly solvable models of biological invasion.

INTRODUCTION Why exactly solvable models are important Intra- and inter-species interactions and local population dynamics Basic mechanisms of species transport Biological invasion: main facts and constituting examples MODELS OF BIOLOGICAL INVASION Diffusion-reaction equations Integral-difference models Space-discrete models Stochastic models Concluding remarks BASIC METHODS AND RELEVANT EXAMPLES The Cole-Hopf transformation and the Burgers equation as a paradigm Further application of the Cole-Hopf transformation Method of piecewise linear approximation Exact solutions of a generalized Fisher equation More about ansatz SINGLE-SPECIES MODELS Impact of advection and migration Accelerating population waves The problem of critical aggregation DENSITY-DEPENDENT DIFFUSION The Aronson-Newman solution and its generalization Stratified diffusion and the Allee effect MODELS OF INTERACTING POPULATIONS Exact solution for a diffusive predator-prey system Migration waves in a resource-consumer system SOME ALTERNATIVE AND COMPLEMENTARY APPROACHES Wave speed and the eigenvalue problem Convergence of the initial conditions Convergence and the paradox of linearization Application of the comparison principle ECOLOGICAL EXAMPLES AND APPLICATIONS Invasion of Japanese beetle in the United States Mount St. Helens recolonization and the impact of predation Stratified diffusion and rapid plant invasion APPENDIX: BASIC BACKGROUND MATHEMATICS Ordinary differential equations and their solutions Phase plane and stability analysis Diffusion equation References Index

Comment on “Lévy Walks Evolve Through Interaction Between Movement and Environmental Complexity”

de Jager et al. (Reports, 24 June 2011, p. 1551) concluded that mussels Lévy walk. We confronted a larger model set with these data and found that mussels do not Lévy walk: Their movement is best described by a composite Brownian walk. This shows how model selection based on an impoverished set of candidate models can lead to incorrect inferences.

Self-organised spatial patterns and chaos in a ratio-dependent predator–prey system

Mechanisms and scenarios of pattern formation in predator–prey systems have been a focus of many studies recently as they are thought to mimic the processes of ecological patterning in real-world ecosystems. Considerable work has been done with regards to both Turing and non-Turing patterns where the latter often appears to be chaotic. In particular, spatiotemporal chaos remains a controversial issue as it can have important implications for population dynamics. Most of the results, however, were obtained in terms of ‘traditional’ predator–prey models where the per capita predation rate depends on the prey density only. A relatively new family of ratio-dependent predator–prey models remains less studied and still poorly understood, especially when space is taken into account explicitly, in spite of their apparent ecological relevance. In this paper, we consider spatiotemporal pattern formation in a ratio-dependent predator–prey system. We show that the system can develop patterns both inside and outside of the Turing parameter domain. Contrary to widespread opinion, we show that the interaction between two different type of instability, such as the Turing–Hopf bifurcation, does not necessarily lead to the onset of chaos; on the contrary, the emerging patterns remain stationary and almost regular. Spatiotemporal chaos can only be observed for parameters well inside the Turing–Hopf domain. We then investigate the relative importance of these two instability types on the onset of chaos and show that, in a ratio-dependent predator–prey system, the Hopf bifurcation is indeed essential for the onset of chaos whilst the Turing instability is not.

Dispersal in a Statistically Structured Population: Fat Tails Revisited

Dispersal has long been recognized as a crucial factor affecting population dynamics. Several studies on long‐distance dispersal revealed a peculiarity now widely known as a problem of “fat tail”: instead of the rate of decay in the population density over large distances being described by a normal distribution, which is apparently predicted by the standard diffusion approach, field data often show much lower rates such as exponential or power law. The question as to what are the processes and mechanisms resulting in the fat tail is still largely open. In this note, by introducing the concept of a statistically structured population, we show that a fat‐tailed long‐distance dispersal is a consequence of the fundamental observation that individuals of the same species are not identical. Fat‐tailed dispersal thus appears to be an inherent property of any real population. We show that our theoretical predictions are in good agreement with available data.

Quantification of the spatial aspect of chaotic dynamics in biological and chemical systems

The need to study spatio-temporal chaos in a spatially extended dynamical system which exhibits not only irregular, initial-value sensitive temporal behavior but also the formation of irregular spatial patterns, has increasingly been recognized in biological science. While the temporal aspect of chaotic dynamics is usually characterized by the dominant Lyapunov exponent, the spatial aspect can be quantified by the correlation length. In this paper, using the diffusion-reaction model of population dynamics and considering the conditions of the system stability with respect to small heterogeneous perturbations, we derive an analytical formula for an ‘intrinsic length’ which appears to be in a very good agreement with the value of the correlation length of the system. Using this formula and numerical simulations, we analyze the dependence of the correlation length on the system parameters. We show that our findings may lead to a new understanding of some well-known experimental and field data as well as affect the choice of an adequate model of chaotic dynamics in biological and chemical systems.

Oscillations and waves in a virally infected plankton system: Part I: The lysogenic stage

A flow regulator for liquid to be administered parenterally to a patient comprises a first member having a flow passage therein for the liquid and a second member arranged to be adjustably telescoped in the passage. The first and second members are constructed such that they can be telescopingly positioned to form a flow rate controlling channel which restricts the flow rate as a function of the length of the channel. The length of the channel is adjustable by changing the relative position of the first and second members. An adjustment force applied to the outside of the regulator is transmitted to the second member within the flow regulator for changing the channel length and therefore the flow rate. The flow regulator can include a drip chamber positioned immediately above the second member for convenient adjustment of the flow rate. Sealing against leakage of air into the flow regulator is attainable because the flow rate controlling second member is located entirely within the first member.