Dynamics induced by delay in a nutrient–phytoplankton model with diffusion

In this paper, a nutrient–phytoplankton model described by a couple of reaction-diffusion equations with delay is studied analytically and numerically. The aim of this research is to provide an understanding of the impact of delay on the nutrient–phytoplankton dynamics. Significantly, the delay can not only induce instability of a positive equilibrium, but also promote the formation of patchiness (an irregular pattern) via Hopf bifurcation. However, if the delay does not exist, the positive equilibrium is always globally asymptotically stable when it exists. In addition, the numerical analysis indicates that the input rate and the loss rate of nutrient also play an important role in the growth of phytoplankton, which supports that eutrophic conditions may be a significant reason inducing phytoplankton blooms. Numerical results are consistent with the analytical results.     

Boundary forcing-induced oscillations in a phytoplankton–zooplankton patch

We consider one- and two-dimensional minimal models of a phytoplankton–zooplankton dynamical system subject to boundary forced oscillations in the prey. As a consequence, waves propagate in the space dimensions in both species. The difference between the peaks and the troughs are at times of orders of magnitude.     

Strength of phytoplankton–nutrient relationship: evidence from 13 biomanipulated ponds

Phytoplankton biomass–nutrient relationship is widely used by lake managers to assess the eutrophication impact and to set the nutrient targets. Submerged vegetation and large zooplankton grazing have long been identified as factors weakening the relationship by decoupling phytoplankton from nutrients. Proving this decoupling unambiguously is difficult because, in natural systems, many factors act together, blurring each other’s effect. In this article, we present the results of continuous monitoring of 13 ponds where the effects of submerged vegetation and zooplankton grazing were enhanced by biomanipulation (fish removal). The monitoring allowed these effects to be assessed and compared with the pre-biomanipulation situations when phytoplankton biomass was mainly nutrient driven. The comparison showed a strong weakening effect of submerged vegetation and large zooplankton grazing on the chlorophyll a–total phosphorus relationship suggesting that a considerable degree of ecological quality of ponds affected by eutrophication can be restored even when nutrient-loading reduction is not feasible.     

The Population and Evolutionary Dynamics of Phage and Bacteria with CRISPR–Mediated Immunity

Clustered Regularly Interspaced Short Palindromic Repeats (CRISPR), together with associated genes (cas), form the CRISPR–cas adaptive immune system, which can provide resistance to viruses and plasmids in bacteria and archaea. Here, we use mathematical models, population dynamic experiments, and DNA sequence analyses to investigate the host–phage interactions in a model CRISPR–cas system, Streptococcus thermophilus DGCC7710 and its virulent phage 2972. At the molecular level, the bacteriophage-immune mutant bacteria (BIMs) and CRISPR–escape mutant phage (CEMs) obtained in this study are consistent with those anticipated from an iterative model of this adaptive immune system: resistance by the addition of novel spacers and phage evasion of resistance by mutation in matching sequences or flanking motifs. While CRISPR BIMs were readily isolated and CEMs generated at high rates (frequencies in excess of 10⁻⁶), our population studies indicate that there is more to the dynamics of phage–host interactions and the establishment of a BIM–CEM arms race than predicted from existing assumptions about phage infection and CRISPR–cas immunity. Among the unanticipated observations are: (i) the invasion of phage into populations of BIMs resistant by the acquisition of one (but not two) spacers, (ii) the survival of sensitive bacteria despite the presence of high densities of phage, and (iii) the maintenance of phage-limited communities due to the failure of even two-spacer BIMs to become established in populations with wild-type bacteria and phage. We attribute (i) to incomplete resistance of single-spacer BIMs. Based on the results of additional modeling and experiments, we postulate that (ii) and (iii) can be attributed to the phage infection-associated production of enzymes or other compounds that induce phenotypic phage resistance in sensitive bacteria and kill resistant BIMs. We present evidence in support of these hypotheses and discuss the implications of these results for the ecology and (co)evolution of bacteria and phage.     

Dynamics of the ‘echo’ effect in a phytoplankton system with nitrogen fixation

Consideration of nitrogen fixation adds a positive nonlinear feedback to plankton ecosystem models. We investigate the consequences of this feedback for secondary phytoplankton blooms and the response of phytoplankton dynamics to physical forcing. The dynamics of phytoplankton, Trichodesmium (the nitrogen fixer), and nutrients is modeled with a system of three differential equations. The model includes two types of nonlinear interactions: the competition of phytoplankton and Trichodesmium for light, and the positive feedback resulting from Trichodesmium recycling. A typical simulation of the model in time, with forcing by a varying mixed-layer depth, reveals a clear successional sequence including a secondary or ‘echo’ bloom of the phytoplankton. We explain this sequence of events through the stability analysis of three different steady states of the model. Our analysis shows the existence of a critical biological parameter, the ratio of normalized growth rates, determining the occurrence of ‘echo’ blooms and the specific sequence of events following a physical perturbation. The interplay of positive and negative feedbacks appears essential to the timing and the type of events following such a perturbation.     

The distribution and nutrient status of phytoplankton in the Southern Ocean between 20° and 70° W

Summary The distribution of phytoplankton along transects amounting to about 10,000 nautical miles in the sector of the Southern Ocean between 20° and 70°W was determined during the austral summer of 1978/79. Chlorophyll a concentration was monitored by the continuous measurement of in vivo fluorescence (IVF). Surface samples were collected for the determination of temperature, salinity, chlorophyll a concentration, carbon fixation rate and species of the phytoplankton. Phytoplankton distribution was found to be extremely patchy both locally and regionally. High phytoplankton concentrations were often associated with either hydrographic features, such as upwelling or the presence of sea-ice, or with bathymetric features, such as shelf breaks, submarine mountain ranges or islands. Enrichment experiments, in which the effects of various nutrient additions on the rate of ¹⁴C fixation by the natural phytoplankton were compared, and bioassay experiments, in which the growth of Thalassiosira pseudonana (Hustedt) Hasle and Heimdal in enriched water samples was measured, were carried out using water samples collected at various stations throughout the study area. Although these techniques were effective in demonstrating nutrient limitation elsewhere, the results suggest that availability of nitrate, phosphate, silicate, trace metals or vitamins exerts no primary control over phytoplankton abundance south of the Polar Front.     

Molecular Bases of Cell–Cell Junctions Stability and Dynamics

Epithelial cell–cell junctions are formed by apical adherens junctions (AJs), which are composed of cadherin adhesion molecules interacting in a dynamic way with the cortical actin cytoskeleton. Regulation of cell–cell junction stability and dynamics is crucial to maintain tissue integrity and allow tissue remodeling throughout development. Actin filament turnover and organization are tightly controlled together with myosin-II activity to produce mechanical forces that drive the assembly, maintenance, and remodeling of AJs. In this review, we will discuss these three distinct stages in the lifespan of cell–cell junctions, using several developmental contexts, which illustrate how mechanical forces are generated and transmitted at junctions, and how they impact on the integrity and the remodeling of cell–cell junctions.Cell–cell junction formation and remodeling occur repeatedly throughout development. Epithelial cells are linked by apical adherens junctions (AJs) that rely on the cadherin-catenin-actin module. Cadherins, of which epithelial E-cadherin (E-cad) is the most studied, are Ca²⁺-dependent transmembrane adhesion proteins forming homophilic and heterophilic bonds in trans between adjacent cells. Cadherins and the actin cytoskeleton are mutually interdependent (Jaffe et al. 1990; Matsuzaki et al. 1990; Hirano et al. 1992; Oyama et al. 1994; Angres et al. 1996; Orsulic and Peifer 1996; Adams et al. 1998; Zhang et al. 2005; Pilot et al. 2006). This has long been attributed to direct physical interaction of E-cad with β-catenin (β-cat) and of α-catenin (α-cat) with actin filaments (for reviews, see Gumbiner 2005; Leckband and Prakasam 2006; Pokutta and Weis 2007). Recently, biochemical and protein dynamics analyses have shown that such a link may not exist and that instead, a constant shuttling of α-cat between cadherin/β-cat complexes and actin may be key to explain the dynamic aspect of cell–cell adhesion (Drees et al. 2005; Yamada et al. 2005). Regardless of the exact nature of this link, several studies show that AJs are indeed physically attached to actin and that cadherins transmit cortical forces exerted by junctional acto-myosin networks (Costa et al. 1998; Sako et al. 1998; Pettitt et al. 2003; Dawes-Hoang et al. 2005; Cavey et al. 2008; Martin et al. 2008; Rauzi et al. 2008). In addition, physical association depends in part on α-cat (Cavey et al. 2008) and additional intermediates have been proposed to represent alternative missing links (Abe and Takeichi 2008) (reviewed in Gates and Peifer 2005; Weis and Nelson 2006). Although further work is needed to address the molecular nature of cadherin/actin dynamic interactions, association with actin is crucial all throughout the lifespan of AJs. In this article, we will review our current understanding of the molecular mechanisms at work during three different developmental stages of AJs biology: assembly, stabilization, and remodeling, with special emphasis on the mechanical forces controlling AJs integrity and development.     

Dynamics analysis of a predator–prey system with harvesting prey and disease in prey species

In this paper, a predator–prey system with harvesting prey and disease in prey species is given. In the absence of time delay, the existence and stability of all equilibria are investigated. In the presence of time delay, some sufficient conditions of the local stability of the positive equilibrium and the existence of Hopf bifurcation are obtained by analysing the corresponding characteristic equation, and the properties of Hopf bifurcation are given by using the normal form theory and centre manifold theorem. Furthermore, an optimal harvesting policy is investigated by applying the Pontryagin's Maximum Principle. Numerical simulations are performed to support our analytic results.     

Dynamics of plant–pollinator–robber systems

Plant–pollinator–robber systems are considered, where the plants and pollinators are mutualists, the plants and nectar robbers are in a parasitic relation, and the pollinators and nectar robbers consume a common limiting resource without interfering competition. My aim is to show a mechanism by which pollination–mutualism could persist when there exist nectar robbers. Through the dynamics of a plant–pollinator–robber model, it is shown that (i) when the plants alone (i.e., without pollination–mutualism) cannot provide sufficient resources for the robbers’ survival but pollination–mutualism can persist in the plant–pollinator system, the pollination–mutualism may lead to invasion of the robbers, while the pollinators will not be driven into extinction by the robbers’ invasion. (ii) When the plants alone cannot support the robbers’ survival but persistence of pollination–mutualism in the plant–pollinator system is density-dependent, the pollinators and robbers could coexist if the robbers’ efficiency in translating the plant–robber interactions into fitness is intermediate and the initial densities of the three species are in an appropriate region. (iii) When the plants alone can support the robbers’ survival, the pollinators will not be driven into extinction by the robbers if their efficiency in translating the plant–pollinator interactions into fitness is relatively larger than that of the robbers. The analysis leads to an explanation for the persistence of pollination–mutualism in the presence of nectar robbers in real situations.     

Modelling of phytoplankton allelopathy with Monod–Haldane-type functional response—A mathematical study

In this paper, a three-tier model of phytoplankton, zooplankton and nutrient is considered and stability of different equilibrium points is analyzed along with Hopf-bifurcation around coexisting equilibrium point. Here, we have assumed toxication process as the guiding factor for bloom formation as well as its termination and this process is incorporated into our model by choosing the zooplankton grazing function as a Monod–Haldane function due to the phytoplankton toxicity. Extensive numerical simulations have been performed to validate the analytical findings and these simulation work reveal the chaotic oscillation exhibited by the model system for certain choice of the parameter values.     

Heteroclinic cycling and extinction in May–Leonard models with demographic stochasticity

Journal of Mathematical Biology - Reactions involving three or more reactants, called higher-molecular reactions, play an important role in mathematical modelling in systems and synthetic biology....     

Production characteristics of bacteria and phytoplankton in the Sea of Okhotsk and Bering Sea during spring–summer

Production parameters for bacterioplankton were assessed during the spring–summer period in the western parts of the Sea of Okhotsk and the Bering Sea, as well as in northwestern Pacific Ocean. The lowest values of bacterial production were observed in early June during the spring phytoplankton bloom (0.08 mg C day^–1 m^–3), while the maximum values (up to 55 mg C day^–1 m^–3) occurred in late July?early August, 1.5 to 2 months after the bloom. The concentration of dissolved organic matter, the substrate for bacterioplankton, was assessed using satellite data. The ratio between bacterial and primary production in the surface samples varied from 0.5% at the peak of phytoplankton bloom to 180% at the peak of bacterioplankton development.     

Erratum to: How do UV radiation, temperature, and zooplankton influence the dynamics of alpine phytoplankton communities?

Plankton in mountain lakes are confronted with generally higher levels of incident ultraviolet radiation (UVR), lower temperatures, and shorter growing seasons than their lower elevation counterparts. The direct inhibitory effects of high UVR and low temperatures on montane phytoplankton are widely recognized. Yet little is known about the indirect effects of these two abiotic factors on phytoplankton, and more specifically whether they alter zooplankton grazing rates which may in turn influence phytoplankton. Here, we report the results of field microcosm experiments that examine the impact of temperature and UVR on phytoplankton growth rates and zooplankton grazing rates (by adult female calanoid copepods). We also examine consequent changes in the absolute and relative abundance of the four dominant phytoplankton species present in the source lake (Asterionella formosa, Dinobryon sp., Discostella stelligera, and Fragilaria crotonensis). All four species exhibited higher growth rates at higher temperatures and three of the four species (all except Dinobryon) exhibited lower growth rates in the presence of UVR versus when shielded from UVR. The in situ grazing rates of zooplankton had significant effects on all species except Asterionella. Lower temperatures significantly reduced grazing rates on Fragilaria and Discostella, but not Dinobryon. While UVR had no effect on zooplankton grazing on any of the four species, there was a significant interaction effect of temperature and UVR on zooplankton grazing on Dinobryon. Discostella and Dinobryon increased in abundance relative to the other species in the presence of UVR. Colder temperatures, the presence of zooplankton, and UVR all had consistently negative effects on rates of increase in overall phytoplankton biomass. These results demonstrate the importance of indirect as well as direct effects of climate forcing by UVR and temperature on phytoplankton community composition in mountain lakes, and suggest that warmer climates and higher UVR levels may favor certain species over others.     

Transmission Dynamics of an Influenza Model with Vaccination and Antiviral Treatment

Vaccination and antiviral treatment are two important prevention and control measures for the spread of influenza. However, the benefit of antiviral use can be compromised if drug-resistant strains arise. In this paper, we develop a mathematical model to explore the impact of vaccination and antiviral treatment on the transmission dynamics of influenza. The model includes both drug-sensitive and resistant strains. Analytical results of the model show that the quantities ℛ _SC and ℛ _RC , which represent the control reproduction numbers of the sensitive and resistant strains, respectively, provide threshold conditions that determine the competitive outcomes of the two strains. These threshold conditions can be used to gain important insights into the effect of vaccination and treatment on the prevention and control of influenza. Numerical simulations are also conducted to confirm and extend the analytic results. The findings imply that higher levels of treatment may lead to an increase of epidemic size, and the extent to which this occurs depends on other factors such as the rates of vaccination and resistance development. This suggests that antiviral treatment should be implemented appropriately.     

Interactions between phytoplankton,zooplankton and fish in a shallow,hypertrophic lake: a study of phytoplankton collapses in Lake Søbygård,Denmark

Since 1983 severe phytoplankton collapses have occurred 1–4 times every summer in the shallow and hypertrophic Lake Søbygård, which is recovering after a ten-fold decrease of the external phosphorus loading in 1982. In July 1985, for example, chlorophyll a changed from 650 µg l^–1 to about 12 µg 1^–1 within 3–5 days. Simultaneously, oxygen concentration dropped from 20–25 mg O₂l^–1 to less than 1 mg O₂l^–1, and pH decreased from 10.7 to 8.9. Less than 10 days later the phytoplankton biomass had fully recovered. During all phytoplankton collapses the density of filter-feeding zooplankton increased markedly, and a clear-water period followed. Due to marked changes in age structure of the fish stock, different zooplankton species were responsible for the density increase in different years, and consequently different collapse patterns and frequencies were observed.The sudden increase in density of filter-feeding zooplankton from a generally low summer level to extremely high levels during algae collapses, which occurred three times from July 1984 to June 1986, could neither be explained by changes in regulation from below (food) nor from above (predation). The density increase was found after a period with high N/P ratios in phytoplankton or nitrate depletion in the lake. During that period phytoplankton biomass, primary production and thus pH decreased, the latter from 10.8–11.0 to 10.5. We hypothesize that direct or indirect effects of high pH are important in controlling the filter-feeding zooplankton in this hypertrophic lake. Secondarily, this situation affects the trophic interactions in the lake water and the net internal loading of nutrients. Consequently, not only a high content of planktivorous fish but also a high pH may promote uncoupling of the grazing food-web in highly eutrophic shallow lakes, and thereby enhance eutrophication.A tentative model is presented for the occurrence of collapses, and their pattern in hypertrophic lakes with various fish densities.     

Dynamics of a Prey–Predator System with Herd Behaviour in Both and Strong Allee Effect in Prey

Biophysics - This paper mainly deals with the prey?predator dynamics where both the prey and predator exhibit herd behavior. Positivity, boundedness, some extinction criteria, stability of...     

Dynamics of additional food provided predator–prey system with mutually interfering predators

Use of additional/alternative food source to predators is one of the widely recognised practices in the field of biological control. Both theoretical and experimental works point out that quality and quantity of additional food play a vital role in the controllability of the pest. Theoretical studies carried out previously in this direction indicate that incorporating mutual interference between predators can stabilise the system. Experimental evidence also point out that mutual interference between predators can affect the outcome of the biological control programs. In this article dynamics of additional food provided predator–prey system in the presence of mutual interference between predators has been studied. The mutual interference between predators is modelled using Beddington–DeAngelis type functional response. The system analysis highlights the role of mutual interference on the success of biological control programs when predators are provided with additional food. The model results indicate the possibility of stable coexistence of predators with low prey population levels. This is in contrast to classical predator–prey models wherein this stable co-existence at low prey population levels is not possible. This study classifies the characteristics of biological control agents and additional food (of suitable quality and quantity), permitting the eco-managers to enhance the success rate of biological control programs.