This contribution reviews the nonlinearstochastic properties of turbulent velocity and passive scalarintermittent fluctuations in Eulerian and Lagrangian turbulence.These properties are illustrated with original data sets of (i)velocity fluctuations collected in the field and in thelaboratory, and (ii) temperature, salinity and in vivofluorescence (a proxy of phytoplankton biomass, i.e. unicelledvegetals passively advected by turbulence) sampled from highlyturbulent coastal waters. The strength of three of the mostpopular models describing intermittent fluctuations (thelognormal, log-Lévy and log-Poisson models) to fit thedistribution of in vivo fluorescence has subsequently beencritically assessed. A theoretical formulation for the stochasticproperties of biologically active scalars is also provided andvalidated. Finally, the potential effect of the intermittentproperties of turbulent velocity fluctuations on processesrelevant to the life of plankton organisms are theoreticallyinvestigated. It is shown that the intermittent nature ofmicroscale turbulence may result in (i) a decrease in the rate ofnutrient 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 topredator-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 thesurface layer, the second patch describes the bottom layer. We show that this spatial heterogeneitymay destabilize the predator-prey system, even in oligotrophic system where the nutrient is lowenough to avoid ”paradox-enrichment” phenomenon. Indeed, in this case, an heterogeneity indexcan be used as a bifurcation parameter, leading to a Hopf bifurcation. Moreover, we assume that individualscan be dispersed in both patches via hydrodynamism processes, like in a mixed layer. Theeffect 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 contrastedresults are examined by considering the non linear interaction between heterogeneity, dispersal andenrichment 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 importantinfluence on the growth and recruitment of planktonic fish larvae. Empirical data show fractal-like prey distributions, and simulationsindicate that scale-invariant foraging strategies may be optimal. Here we show how larval growth and recruitment in a turbulent environment canbe formulated as a hitting time problem for a jump-diffusion process. We present two theoretical results. Firstly, if jumps are of a fixed sizeand 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 processbecomes superdiffusive. To synthesise these results we conduct a strategic simulation study where “patchy” jumps are embedded in adrift-diffusion process. We conclude that increasingly Lévy-like predator foraging strategies can have a significantly positive effect onrecruitment at the population level.

Parameterization of zooplankton functional response is crucial for constructing plankton models. Theoreticalstudies predict enhancing of system stability in case the response is of sigmoid type. Experiments on feeding in laboratories tell us infavor of non-sigmoid types for most herbivorous zooplankton species. However, recent field observations show that the overall functionalresponse of zooplankton in the whole euphotic zone can exhibit a sigmoid behavior even when the response for the same species in laboratorymesocosms is non-sigmoid. Here we propose a simple model explaining the observed alterations of functional response. We divide the euphoticzone into a number of layers and take into account the food-dependent migration of zooplankton. In each layer, the functional response (localresponse) is suggested to be non-sigmoid. We show that the overall response of zooplankton exhibits different behavior compared to thepatterns of the local response. In particular, the model predicts emergence of a sigmoid type as a result of zooplankton accumulation andfeeding in layers with high phytoplankton density. We show the importance of light attenuation by phytoplankton on the alteration offunctional response. The modelling results allow us to hypothesize that the sigmoid functional response in real communities should emergemore 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 presenceof viral infection on phytoplankton population. They studied thedynamics under the constant nutrient input. The present paper dealswith the problem with seasonal variability on nutrient input. We usea general periodic function for nutrient input. We observe thedynamics of the system by considering (i) the infected phytoplanktonconsumes nutrient and (ii) the infected phytoplankton is not in astate to consume nutrient. Conditions for the persistence andextinction of populations are worked out. Our numerical experimentsshow that if the infected phytoplankton does not take nutrient thensusceptible phytoplankton coexists with the infected ones. But ifthe infected phytoplankton consumes nutrient then there is a chancefor extinction of susceptible phytoplankton for high rate ofinfection. We also observe that periodic nutrient input enforces thesystem to enter into chaotic region.