Phytoplankton blooms and fish recruitment rate: Effects of spatial distribution

We consider the spatio-temporal dynamics of a spatially-structured generalization of the phytoplankton-zooplankton-fish larvae model system proposed earlier (Biktashev et al., 2003, J. Plankton Res. 5, 21–33; James et al., 2003, Ecol. Model. 160, 77–90). In contrast to Pitchford and Brindley (2001, Bull. Math. Biol. 63, 527–546), who were concerned with small scale patchiness (i.e., 1–10m), on which the (stochastic) raptorial behaviour of individual larvae is important, we address here the much larger scale ‘patchy’ problems (i.e., 10–100 km), on which both larvae and plankton may be regarded as passive tracers of the fluid motion, dispersed and mixed by the turbulent diffusion processes. In particular, we study the dependence of the fish recruitment on carrying capacities of the plankton subsystem and on spatio-temporal evolution of that subsystem with respect to the larvae hatching site(s). It is shown that the main features found both in the nonstructured and age-structured spatially uniform models are observed in the spatially structured case, but that spatial effects can significantly modify the overall quantitative outcome. Spatial patterns in the metamorphosed fish distribution are a consequence of quasi-local interaction of larvae with plankton, in which the dispersion of larvae by large scale turbulent eddies plays little part due to the relatively short timescale of the larvae development. As a result, in a strong phyto/zooplankton subsystem, with fast reproduction rate and large carrying capacity of phytoplankton and high conversion ratio of zooplankton, recruitment success depends only on the localization and timing of the hatching with respect to the plankton patches. In a weak phyto/zooplankton system, with slow reproduction rate and small carrying capacity of phytoplankton and low conversion ratio of zooplankton, the larvae may significantly influence the evolution of the plankton patches, which may lead to nontrivial cooperative effects between different patches of larvae. However, in this case, recruitment is very low.     

Asymptotics of Conduction Velocity Restitution in Models of Electrical Excitation in the Heart

We extend a non-Tikhonov asymptotic embedding, proposed earlier, for calculation of conduction velocity restitution curves in ionic models of cardiac excitability. Conduction velocity restitution is the simplest non-trivial spatially extended problem in excitable media, and in the case of cardiac tissue it is an important tool for prediction of cardiac arrhythmias and fibrillation. An idealized conduction velocity restitution curve requires solving a non-linear eigenvalue problem with periodic boundary conditions, which in the cardiac case is very stiff and calls for the use of asymptotic methods. We compare asymptotics of restitution curves in four examples, two generic excitable media models, and two ionic cardiac models. The generic models include the classical FitzHugh–Nagumo model and its variation by Barkley. They are treated with standard singular perturbation techniques. The ionic models include a simplified “caricature” of Noble (J. Physiol. Lond. 160:317–352, 1962) model and Beeler and Reuter (J. Physiol. Lond. 268:177–210, 1977) model, which lead to non-Tikhonov problems where known asymptotic results do not apply. The Caricature Noble model is considered with particular care to demonstrate the well-posedness of the corresponding boundary-value problem. The developed method for calculation of conduction velocity restitution is then applied to the Beeler–Reuter model. We discuss new mathematical features appearing in cardiac ionic models and possible applications of the developed method.     

Biphasic constitutive laws for biological interface evolution

 A model of tissue differentiation at the bone–implant interface is proposed. The basic hypothesis of the model is that the mechanical environment determines the tissue differentiation. The stimulus chosen is related to the bone–implant micromotions. Equations governing the evolution of the interfacial tissue are proposed and combined with a finite element code to determine the evolution of the fibrous tissue around prostheses. The model is applied to the case of an idealized hip prosthesis. Received: 28 May 2002 / Accepted: 10 November 2002     

Phytoplankton depth profiles and their transitions near the critical sinking velocity

We consider a simple phytoplankton model introduced by Shigesada and Okubo which incorporates the sinking and self-shading effect of the phytoplankton. The amount of light the phytoplankton receives is assumed to be controlled by the density of the phytoplankton population above the given depth. We show the existence of non-homogeneous solutions for any water depth and study their profiles and stability. Depending on the sinking rate of the phytoplankton, light intensity and water depth, the plankton can concentrate either near the surface, at the bottom of the water column, or both, resulting in a “double-peak” profile. As the buoyancy passes a certain critical threshold, a sudden change in the phytoplankton profile occurs. We quantify this transition using asymptotic techniques. In all cases we show that the profile is locally stable. This generalizes the results of Shigesada and Okubo where infinite depth was considered.      

Critical conditions for phytoplankton blooms

We motivate and analyse a reaction—advection—diffusion model for the dynamics of a phytoplankton species. The reproductive rate of the phytoplankton is determined by the local light intensity. The light intensity decreases with depth due to absorption by water and phytoplankton. Phytoplankton is transported by turbulent diffusion in a water column of given depth. Furthermore, it might be sinking or buoyant depending on its specific density. Dimensional analysis allows the reduction of the full problem to a problem with four dimensionless parameters that is fully explored. We prove that the critical parameter regime for which a stationary phytoplankton bloom ceases to exist, can be analysed by a reduced linearized equation with particular boundary conditions. This problem is mapped exactly to a Bessel function problem, which is evaluated both numerically and by asymptotic expansions. A final transformation from dimensionless parameters back to laboratory parameters results in a complete set of predictions for the conditions that allow phytoplankton bloom development. Our results show that the conditions for phytoplankton bloom development can be captured by a critical depth, a compensation depth, and zero, one or two critical values of the vertical turbulent diffusion coefficient. These experimentally testable predictions take the form of similarity laws: every plankton—water—light-system characterized by the same dimensionless parameters will show the same dynamics.     

Materials for plankton modelling: Vertical distribution of Atlantic zooplankton in summer

Current versions of plankton production models which allow variabilityin the vertical plane are based on very slender assumptionsabout vertical distributions of zooplankton. A late summer LHPRsection across the N.E. Atlantic and some spring and early summerLHPR hauls at 59°N indicate some of the details of verticaldistribution which seem to be important for model-formulation.It is suggested that a model which is designed to simulate verticalplankton structure in the North Atlantic should cause zooplanktonto be distributed during the production period so as to satisfythe following criteria: 60–70% of the biomass must beof carnivores, and 25–75% of all biomass should come tobe located within a discrete epiplankton layer within 40 d ofthe start of the seasonal production: this epiplankton shouldextend down to 3–4 times the depth of the mixed layer,with its lower limit near the base of the seasonal tbermocline;the epiplankton must extend deeper than the depth of negligiblechlorophyll, and 3–4 times deeper than the chorophyllmnTimnm; maximum concentrations of filter-feeders should belocated at aboout half the depth of the chlorophyll maximum,and the mmimnm of predators rather deeper; below the epiplankton,biomass should be rather constant down to 1,000 m, be about15–30 times less abundant than the epiplankton, and shouldcontain discrete layers of interzonal migrants. It is suggested,by reference to FLEX LHPR profiles, that there are importantdifferences between the oceanic plankton and the copepod-dominatedspring plankton in the North Sea, which will have importantconsequences for plankton modelling. ^*present address, Marine Ecology Laboratory, Bedford Instituteof Oceanography, Dartmouth, Canada     

The photophores of Meganyctiphanes norvegica (M. Sars) (Euphausiacea): mode of operation

The photophores of Meganyctiphanes were investigated with regard to the control of light production and with respect to their role in a hitherto unknown communication system using light flashes which became evident from observation of specialised signalling behaviour. To that purpose the light production was recorded during presentation of a range of stimuli delivered to the intact, tethered shrimp. Stimuli used were changes in ambient light, water turbulence, simulated predator approach and light flashes, as well as electric shocks and serotonin injections. Strong negative light gradients, exaggerating the natural sunset signal, reliably elicited light production, the peak of which lasted on average 2 min. In the late phase of this light production, low frequency water oscillations and turbulent flow (assumed intraspecific communication signals at close range) elicited transient increases in light production. Artificial light flashes presented to a group of shrimp evoked a signalling behaviour in which the animal points the light of its photophore beamers (positioned at the ventral side and normally directed downwards) for a fraction of a second at observers within the same depth level. The responses produced by the signalling behaviour indicate a fixed delay with respect to the triggering flash. Electric stimulation of the ventral nerve cord via implanted electrodes resulted in a strong light production with a latency of 160 ms. Injection of serotonin, resulting in haemolymph concentrations of 10^–5 M and higher, initiated increasingly strong and increasingly long-lasting continuous light production. Implications for the control of the photophores are discussed. Electronic Publication     

Numerical and exact solutions for continuum of alleles models

 Two results are presented for problems involving alleles with a continuous range of effects. The first result is a simple yet highly accurate numerical method that determines the equilibrium distribution of allelic effects, moments of this distribution, and the mutational load. The numerical method is explicitly applied to the mutation-selection balance problem of stabilising selection. The second result is an exact solution for the distribution of allelic effects under weak stabilising selection for a particular distribution of mutant effects. The exact solution is shown to yield a distribution of allelic effects that, depending on the mutation rate, interpolates between the ``House of Cards' approximation and the Gaussian approximation. The exact solution is also used to test the accuracy of the numerical method. Received: 7 November 2001 / Revised version: 5 September 2002 / Published online: 18 December 2002 Key words or phrases: Continuum of alleles – Numerical solution – Exact solution – Mutation selection balance – Stabilising selection     

The satellite-derived distribution of chlorophyll-a and its relation to ice cover, radiation and sea surface temperature in the Barents Sea

The response of oceanic phytoplankton to climate forcing in the Arctic Ocean has attracted increasing attention due to its special geographical position and potential susceptibility to global warming. Here, we examine the relationship between satellite-derived (sea-viewing wide field-of-view sensor, SeaWiFS) surface chlorophyll-a (CHL) distribution and climatic conditions in the Barents Sea (30–35°E, 70–80°N) for the period 1998–2002. We separately examined the regions north and south of the Polar Front (∼76°N). Although field data are rather limited, the satellite CHL distribution was generally consistent with cruise observations. The temporal and spatial distribution of CHL was strongly influenced by the light regime, mixed layer depth, wind speed and ice cover. Maximum CHL values were found in the marginal sea-ice zone (72–73°N) and not in the ice-free region further south (70–71°N). This indicates that melt-water is an important contributor to higher CHL production. The vernal phytoplankton bloom generally started in late March, reaching its peak in late April. A second, smaller CHL peak occurred regularly in late summer (September). Of the 5 years, 2002 had the highest CHL production in the southern region, likely due to earlier ice melting and stronger solar irradiance in spring and summer.     

Coexistence of different serotypes of dengue virus

 We formulate a non–linear system of differential equations that models the dynamics of dengue fever. This disease is produced by any of the four serotypes of dengue arbovirus. Each serotype produces permanent immunity to it, but only a certain degree of cross–immunity to heterologous serotypes. In our model we consider the relation between two serotypes. Our interest is to analyze the factors that allow the invasion and persistence of different serotypes in the human population. Analysis of the model reveals the existence of four equilibrium points, which belong to the region of biological interest. One of the equilibrium points corresponds to the disease–free state, the other three equilibria correspond to the two states where just one serotype is present, and the state where both serotypes coexist, respectively. We discuss conditions for the asymptotic stability of equilibria, supported by analytical and numerical methods. We find that coexistence of both serotypes is possible for a large range of parameters. Received: 7 July 1998 / Revised version: 12 July 2002 / Published online: 26 September 2002 Keywords or phrases: Dengue fever – Primary and secondary infections – Serotype – Coexistence – Threshold – Basic reproduction number – Persistence     

On the dynamics of discrete food chains: low- and high-frequency behavior and optimality of chaos

 In this paper we derive and analyze a discrete version of Rosenzweig's (Am. Nat. 1973) food-chain model. We provide substantial analytical and numerical evidence for the general dynamical patterns of food chains predicted by De Feo and Rinaldi (Am. Nat. 1997) remaining largely unaffected by this discretization. Our theoretical analysis gives rise to a classification of the parameter space into various regions describing distinct governing dynamical behaviors. Predator abundance has a local optimum at the edge of chaos. Received: 13 August 1999 / Revised version: 12 March 2002 / Published online: 17 October 2002 Mathematics Subject Classification (1991): 92D40 Keywords or phrases: Discrete food-chain – Discrete Hopf (Neimark-Sacker) bifurcation – Pulsewise birth processes – Mean yield maximization – Nicholson-Bailey model     

Alteration of Ecosystem Function by Zebra Mussels in Oneida Lake: Impacts on Submerged Macrophytes

Dreissenid mussels (the zebra mussel Dreissena polymorpha and the quagga mussel D. bugensis) are ecosystem engineers that modify the physical environment by increasing light penetration. Such a change is likely to affect the distribution and diversity of submerged macrophytes. Filter-feeding by these mussels has been associated with increased water clarity in many North American and European lakes. In this study, we report the increase in water clarity of Oneida Lake, New York, USA, for 1975–2002 and argue that the increase was caused by zebra mussel invasion rather than declines in nutrients. Over the study period, although mean total phosphorus decreased significantly, the main increase in water clarity occurred after the zebra mussel invasion in 1991. The average depth receiving 1% surface light increased from 6.7 m to 7.8 m after the invasion of zebra mussels, representing a 23% areal expansion. The maximum depth of macrophyte colonization, as measured by diver and hydroacoustic surveys, increased from 3.0 m before the invasion of zebra mussels to 5.1 m after their establishment. In addition, macrophyte species richness increased, the frequency of occurrence increased for most species, and the composition of the macrophyte community changed from low-light–tolerant species to those tolerating a wide range of light conditions. Comparisons with observations reported in the literature indicate that increased light penetration alone could explain these changes in macrophyte distribution and diversity. Such changes will increase the importance of benthic primary production over pelagic production in the food web, thereby representing an overall alteration of ecosystem function, a process we refer to as “benthification”.     

The onset of oscillatory dynamics in models of multiple disease strains

 We examine a generalised SIR model for the infection dynamics of four competing disease strains. This model contains four previously-studied models as special cases. The different strains interact indirectly by the mechanism of cross-immunity; individuals in the host population may become immune to infection by a particular strain even if they have only been infected with different but closely related strains. Several different models of cross-immunity are compared in the limit where the death rate is much smaller than the rate of recovery from infection. In this limit an asymptotic analysis of the dynamics of the models is possible, and we are able to compute the location and nature of the Takens–Bogdanov bifurcation associated with the presence of oscillatory dynamics observed by previous authors. Received: 5 December 2001 / Revised version: 5 May 2002 / Published online: 17 October 2002 Keywords or phrases: Infection – Pathogen – Epidemiology – Multiple strains – Cross-immunity – Oscillations – Dynamics – Bifurcations     

Ecological Values of Shallow-Water Habitats: Implications for the Restoration of Disturbed Ecosystems

A presumed value of shallow-habitat enhanced pelagic productivity derives from the principle that in nutrient-rich aquatic systems phytoplankton growth rate is controlled by light availability, which varies inversely with habitat depth. We measured a set of biological indicators across the gradient of habitat depth within the Sacramento–San Joaquin River Delta (California) to test the hypothesis that plankton biomass, production, and pelagic energy flow also vary systematically with habitat depth. Results showed that phytoplankton biomass and production were only weakly related to phytoplankton growth rates whereas other processes (transport, consumption) were important controls. Distribution of the invasive clam Corbicula fluminea was patchy, and heavily colonized habitats all supported low phytoplankton biomass and production and functioned as food sinks. Surplus primary production in shallow, uncolonized habitats provided potential subsidies to neighboring recipient habitats. Zooplankton in deeper habitats, where grazing exceeded phytoplankton production, were likely supported by significant fluxes of phytoplankton biomass from connected donor habitats. Our results provide three important lessons for ecosystem science: (a) in the absence of process measurements, derived indices provide valuable information to improve our mechanistic understanding of ecosystem function and to benefit adaptive management strategies; (b) the benefits of some ecosystem functions are displaced by water movements, so the value of individual habitat types can only be revealed through a regional perspective that includes connectedness among habitats; and (c) invasive species can act as overriding controls of habitat function, adding to the uncertainty of management outcomes.     

Mean annual characteristics of zooplankton in the Sea of Okhotsk,Bering Sea and Northwestern Pacific (Annual and seasonal biomass values and predominance)

Averaged data on fractions, groups and species of zooplankton in the epipelagic of the Sea of Okhotsk, Bering Sea and Northwestern Pacific are based on the results of plankton studies in 1984–2006 carried out during TINRO-Center expeditions. The material is presented at the following levels: (1) general mean annual quantitative characteristics for the Sea of Okhotsk, Bering Sea, and Northwestern Pacific; (2) the same characteristics for more fractional subdivisions of regions, such as the biotopes of outer and inner shelves and deepwater areas; (3) specification of data for certain time spans (1984–1990; 1991–1996; 1997–1998; 1999–2006); (4) seasonal dynamics of the basic components; and (5) the predominance of species in each biotope. The obtained quantitative data allowed determining a series of peculiarities in the main constituents of plankton communities on both seasonal and long-term scales. These averaged parameters can serve as some criterion for the evaluation of probable changes that could take place in the plankton of the studied areas.     

The estimation of a mackerel (Scomber scombrus L.) spawning stock size by plankton survey

The International Council for the Exploration of the Sea (ICES)Mackerel Working Group recognises two mackerel stocks in Europeanwaters, the North Sea stock and the Western stock. From Marchto July 1977 an Anglo-French plankton survey was carried outin the shelf waters of the Bay of Biscay, Celtic Sea and westof Ireland to estimate the size of the Western stock. Sampleswere collected with a 76 cm Lowestoft pattern high-speed planktonsampler fished to a maximum depth of 100 m. The number of eggsm^–2 at each of five development stages were raised todaily production estimates with development rate coefficientsmeasured with live material collected in March. The mean numberof stage I eggs m^–2 d^–1 was raised to a total dailyproduction estimate for each month by the area within the minimumcontour level, and a production curve was drawn for the spawningseason. The total egg production was estimated at 1.98 x 10¹⁵eggs, + 30%, – 20%. With a mean fecundity of 361 000 anda sex ratio of 0.64 males per female this is equivalent to 8995 million fish.     

Distribution of dissolved organic carbon and nitrogen in a coral reef

Dissolved organic matter (DOM) concentrations in a fringing coral reef were measured for both carbon and nitrogen with the analytical technique of high-temperature catalytic oxidation. Because of high precision of the analytical system, not only the concentrations of dissolved organic carbon and nitrogen (DOC and DON, respectively) but the C:N ratio was also determined from the distribution of DOC and DON concentrations. The observed concentrations of DOC and DON ranged 57–76 and 3.8–5.6 μmol l⁻¹, respectively. The C:N ratios of the DOM that was produced on the reef flat were very similar between seagrass- and coral-dominated areas; the C:N ratio was 10 on average. The C:N ratio of DOM was significantly higher than that of particulate organic matter (POM) that was produced on the reef flat. Production rates of DOC were measured on the reef flat during stagnant periods and accounted for 3–7% of the net primary production, depending on the sampling site. The production rate of DON was estimated to be 10–30% of the net uptake of dissolved inorganic N in the reef community. Considering that the DOM and POM concentrations were not correlated with each other, a major source of the reef-derived DOM may be the benthic community and not POM such as phytoplankton. It was concluded that a widely distributed benthic community in the coral reef released C-rich DOM to the overlying seawater, conserving N in the community.     

Implications of resting eggs of zooplankton for the paradox of enrichment

In this study, we numerically investigated to what extent introducing resting-egg dynamics would stabilize simple Daphnia–algae consumer–resource models. In the models, the density of viable resting eggs was explicitly expressed, and we assumed that zooplankton produced resting eggs seasonally or in response to food deficiency and that resting eggs hatched seasonally. The models predicted that, although the paradox of enrichment was not completely resolved (i.e., the system was destabilized by eutrophication), we found the following conditions under which the stabilizing effects of resting eggs would be significantly large: (1) resting eggs are produced seasonally (rather than in response to food deficiency), (2) the annual average allocation ratio to resting eggs is large, and (3) the annual average hatching rate of resting eggs is low. The results suggest that resting-egg dynamics can significantly reduce the paradox of enrichment within the biologically meaningful parameter space and contribute to the stability of plankton community dynamics.     

Ecological balance between ciliate plankton and its prey candidates,pico- and nanoplankton,in the East China Sea

Standing stocks of ciliate plankton and its prey candidates, both picoplankton and nanoplankton, were investigated in spring in the East China Sea. The former was 1.36 × 10⁵–1.54 × 10⁸ μm³ l⁻¹ in biovolume, and the latter was 5.33 × 10⁶–1.11 × 10⁸ μm³ l⁻¹. The biovolume ratio of ciliate plankton to prey candidates ranged from 1.31 × 10⁻² to 2.00 × 10⁰; it was larger in abundant prey conditions and smaller in sparse preys. Making some plausible assumptions about physiological activity on both organisms, every ratio meet the quantitative restriction that prey production should be equal to or larger than ciliate consumption. However, prey candidates would be so sparsely distributed that ciliate plankton could not capture sufficient prey organisms in its random filter-feeding manner. Even though planktonic ciliates must have some extraordinary mechanisms to capture preys efficiently, this quantitative imbalance might be one of the reasons for decreasing ciliate/prey ratio in sparse prey conditions. Handling editor: K. Martens