This contribution reviews the nonlinear stochastic properties of turbulent velocity and passive scalar intermittent fluctuations in Eulerian and Lagrangian turbulence. These properties are illustrated with original data sets of (i) velocity fluctuations collected in the field and in the laboratory, and (ii) temperature, salinity and in vivo fluorescence (a proxy of phytoplankton biomass, i.e. unicelled vegetals passively advected by turbulence) sampled from highly turbulent coastal waters. The strength of three of the most popular models describing intermittent fluctuations (the lognormal, log-Lévy and log-Poisson models) to fit the distribution of in vivo fluorescence has subsequently been critically assessed. A theoretical formulation for the stochastic properties of biologically active scalars is also provided and validated. Finally, the potential effect of the intermittent properties of turbulent velocity fluctuations on processes relevant to the life of plankton organisms are theoretically investigated. It is shown that the intermittent nature of microscale turbulence may result in (i) a decrease in the rate of nutrient fluxes towards non-motile phytoplankton cells (6-62 %), (ii) a decrease in the physical coagulation of phytoplankton cells (25-48 %) and in the subsequent phytoplankton aggregate volumes (22-41 %), and (iii) a decrease of the turbulence contribution to predator-prey encounter rates (25-50 %).

The spatio-temporal dynamics of a population present one of the most fascinating aspects and challenges for ecological modelling. In this article we review some simple mathematical models, based on one dimensional reaction-diffusion-advection equations, for the growth of a population on a heterogeneous habitat. Considering a number of models of increasing complexity we investigate the often contrary roles of advection and diffusion for the persistence of the population. When it is possible we demonstrate basic mathematical techniques and give the critical conditions providing the survival of a population in simple systems and in more complex resource-consumer models which describe the dynamics of phytoplankton in a water column.

This paper is devoted to the study of a predator-prey model in a patchy environment. The model represents the interactions between phytoplankton and zooplankton in the water column. Two patches are considered with respect to light availability: one patch is associated to the surface layer, the second patch describes the bottom layer. We show that this spatial heterogeneity may destabilize the predator-prey system, even in oligotrophic system where the nutrient is low enough to avoid ”paradox-enrichment” phenomenon. Indeed, in this case, an heterogeneity index can be used as a bifurcation parameter, leading to a Hopf bifurcation. Moreover, we assume that individuals can be dispersed in both patches via hydrodynamism processes, like in a mixed layer. The effect of mixing intensity is analysed as well as interactions between dispersion and enrichment. We also show that, in some cases, spatial heterogeneity has a stabilizing effect. These contrasted results are examined by considering the non linear interaction between heterogeneity, dispersal and enrichment and some mechanisms leading to stabilization/destabilization are exhibited.

We study the role of interactions between habitats in rotifer dynamics. For this purpose we use a modified version of the Consensus model. The Consensus model has been shown to be realistic enough to reproduce distinguishing features of the rotifer species dynamics. Being uncoupled, intrinsically bistable rotifer populations, which inhabit the regions under different environmental conditions, do not impact each other. We show that migration of the rotifers between the habitats leads to the transformation of bistability to oscillatory dynamics. The coupling-induced oscillations of the rotifer biomass are shown to be either regular or irregular depending on the value of the model parameters, which describe environmental conditions and the biomass exchange between the habitats. We show that the irregular oscillations are characterized by positive values of the Lyapunov exponent, which means that they are chaotic.

It is well established that resource variability generated by spatial patchiness and turbulence is an important influence on the growth and recruitment of planktonic fish larvae. Empirical data show fractal-like prey distributions, and simulations indicate that scale-invariant foraging strategies may be optimal. Here we show how larval growth and recruitment in a turbulent environment can be formulated as a hitting time problem for a jump-diffusion process. We present two theoretical results. Firstly, if jumps are of a fixed size and occur as a Poisson process (embedded within a drift-diffusion), recruitment is effectively described by a diffusion process alone. Secondly, in the absence of diffusion, and for “patchy” jumps (of negative binomial size with Pareto inter-arrivals), the encounter process becomes superdiffusive. To synthesise these results we conduct a strategic simulation study where “patchy” jumps are embedded in a drift-diffusion process. We conclude that increasingly Lévy-like predator foraging strategies can have a significantly positive effect on recruitment at the population level.

Parameterization of zooplankton functional response is crucial for constructing plankton models. Theoretical studies predict enhancing of system stability in case the response is of sigmoid type. Experiments on feeding in laboratories tell us in favor of non-sigmoid types for most herbivorous zooplankton species. However, recent field observations show that the overall functional response of zooplankton in the whole euphotic zone can exhibit a sigmoid behavior even when the response for the same species in laboratory mesocosms is non-sigmoid. Here we propose a simple model explaining the observed alterations of functional response. We divide the euphotic zone into a number of layers and take into account the food-dependent migration of zooplankton. In each layer, the functional response (local response) is suggested to be non-sigmoid. We show that the overall response of zooplankton exhibits different behavior compared to the patterns of the local response. In particular, the model predicts emergence of a sigmoid type as a result of zooplankton accumulation and feeding in layers with high phytoplankton density. We show the importance of light attenuation by phytoplankton on the alteration of functional response. The modelling results allow us to hypothesize that the sigmoid functional response in real communities should emerge more often than it was suggested earlier based only on experimental studies on zooplankton feeding.

Chattopadhyay et al. [Biosystems (2003), 68, pp. 5-17] proposed and analyzed an N – P model in the presence of viral infection on phytoplankton population. They studied the dynamics under the constant nutrient input. The present paper deals with the problem with seasonal variability on nutrient input. We use a general periodic function for nutrient input. We observe the dynamics of the system by considering (i) the infected phytoplankton consumes nutrient and (ii) the infected phytoplankton is not in a state to consume nutrient. Conditions for the persistence and extinction of populations are worked out. Our numerical experiments show that if the infected phytoplankton does not take nutrient then susceptible phytoplankton coexists with the infected ones. But if the infected phytoplankton consumes nutrient then there is a chance for extinction of susceptible phytoplankton for high rate of infection. We also observe that periodic nutrient input enforces the system to enter into chaotic region.