Phytoplankton food quality control of planktonic food web processes

We developed a mechanistic model of nutrient, phytoplankton, zooplankton and fish interactions to test the effects of phytoplankton food quality for herbivorous zooplankton on planktonic food web processes. When phytoplankton food quality is high strong trophic cascades suppress phytoplankton biomass, the zooplankton can withstand intense zooplanktivory, and energy is efficiently transferred through the food web sustaining higher trophic level production. Low food quality results in trophic decoupling at the plant-animal interface, with phytoplankton biomass determined primarily by nutrient availability, zooplankton easily eliminated by fish predation, and poor energy transfer through the food web. At a given nutrient availability, food quality and zooplanktivory interact to determine zooplankton biomass which in turn determines algal biomass. High food quality resulted in intense zooplankton grazing which favored fast-growing phytoplankton taxa, whereas fish predation favored slow-growing phytoplankton. These results suggest algal food quality for herbivorous zooplankton can strongly influence the nature of aquatic food web dynamics, and can have profound effects on water quality and fisheries production. Handling editor: D. Hamilton     

Stochastic models for virus and immune system dynamics

New stochastic models are developed for the dynamics of a viral infection and an immune response during the early stages of infection. The stochastic models are derived based on the dynamics of deterministic models. The simplest deterministic model is a well-known system of ordinary differential equations which consists of three populations: uninfected cells, actively infected cells, and virus particles. This basic model is extended to include some factors of the immune response related to Human Immunodeficiency Virus-1 (HIV-1) infection. For the deterministic models, the basic reproduction number, R₀, is calculated and it is shown that if R₀<1, the disease-free equilibrium is locally asymptotically stable and is globally asymptotically stable in some special cases. The new stochastic models are systems of stochastic differential equations (SDEs) and continuous-time Markov chain (CTMC) models that account for the variability in cellular reproduction and death, the infection process, the immune system activation, and viral reproduction. Two viral release strategies are considered: budding and bursting. The CTMC model is used to estimate the probability of virus extinction during the early stages of infection. Numerical simulations are carried out using parameter values applicable to HIV-1 dynamics. The stochastic models provide new insights, distinct from the basic deterministic models. For the case R₀>1, the deterministic models predict the viral infection persists in the host. But for the stochastic models, there is a positive probability of viral extinction. It is shown that the probability of a successful invasion depends on the initial viral dose, whether the immune system is activated, and whether the release strategy is bursting or budding.     

Plankton cycles disguised by turbulent advection

Mathematical models used to represent plankton dynamics often display limit-cycle behavior in a range of realistic parameter values. However, experimental data do not show evidence of plankton oscillations besides externally driven seasonal blooms, casting doubts on the validity of the models themselves. In this work we show that spatial-temporal variability, coupled with advection by mesoscale turbulence, can disguise limit-cycle behavior to the point that it cannot be detected in fixed-point measurements of plankton abundance. The results presented here have more general implications as they indicate that the behavior of ecosystem models in the presence of advection can be very different from that occurring for homogeneous conditions. Care should thus be exercised in drawing general conclusions from the analysis of homogeneous ecosystem models.     

Itô versus Stratonovich calculus in random population growth

The context is the general stochastic differential equation (SDE) model dN/dt=N(g(N)+sigmaepsilon(t)) for population growth in a randomly fluctuating environment. Here, N=N(t) is the population size at time t, g(N) is the 'average' per capita growth rate (we work with a general almost arbitrary function g), and sigmaepsilon(t) is the effect of environmental fluctuations (sigma>0, epsilon(t) standard white noise). There are two main stochastic calculus used to interpret the SDE, It? calculus and Stratonovich calculus. They yield different solutions and even qualitatively different predictions (on extinction, for example). So, there is a controversy on which calculus one should use. We will resolve the controversy and show that the real issue is merely semantic. It is due to the informal interpretation of g(x) as being an (unspecified) 'average' per capita growth rate (when population size is x). The implicit assumption usually made in the literature is that the 'average' growth rate is the same for both calculi, when indeed this rate should be defined in terms of the observed process. We prove that, when using It? calculus, g(N) is indeed the arithmetic average growth rate R(a)(x) and, when using Stratonovich calculus, g(N) is indeed the geometric average growth rate R(g)(x). Writing the solutions of the SDE in terms of a well-defined average, R(a)(x) or R(g)(x), instead of an undefined 'average' g(x), we prove that the two calculi yield exactly the same solution. The apparent difference was due to the semantic confusion of taking the informal term 'average growth rate' as meaning the same average.     

Application of the lognormal equation to assess phytoplankton community structural changes induced by marine eutrophication

A methodological approach was developed for the quantification of the structural changes of phytoplankton communities induced by marine eutrophication. The lognormal equation assigning species abundance to doubling intervals (octaves) of individuals formed the basis of the proposed methodology and the field validation process was based on phytoplankton enumeration and classification data characteristic of eutrophic, mesotrophic and oligotrophic waters. Five octave sets with different sizes were tested for goodness of fit against field data and the set with the smallest size of doubling intervals was selected for further consideration. The application of the lognormal equation was evaluated statistically with field data and it was considered satisfactory at the 87% level. The changes in the shape of the lognormal equation induced by eutrophication were expressed by three characteristic parameters of the equation: the number of the modal octave, the number of species in the modal octave, and the shaping factor. Significant differences were observed for the three parameters among eutrophic, mesotrophic, and oligotrophic waters; the number of the modal octave was high in eutrophic and mesotrophic waters, the number of species in the modal octave has shown a trend of low values under mesotrophic conditions and the shaping factor has shown a considerable increase from eutrophic to oligotrophic waters. Handling editor: L. Naselli-Flores     

Models of dispersal in biological systems

In order to provide a general framework within which the dispersal of cells or organisms can be studied, we introduce two stochastic processes that model the major modes of dispersal that are observed in nature. In the first type of movement, which we call the position jump or kangaroo process, the process comprises a sequence of alternating pauses and jumps. The duration of a pause is governed by a waiting time distribution, and the direction and distance traveled during a jump is fixed by the kernel of an integral operator that governs the spatial redistribution. Under certain assumptions concerning the existence of limits as the mean step size goes to zero and the frequency of stepping goes to infinity the process is governed by a diffusion equation, but other partial differential equations may result under different assumptions. The second major type of movement leads to what we call a velocity jump process. In this case the motion consists of a sequence of runs separated by reorientations, during which a new velocity is chosen. We show that under certain assumptions this process leads to a damped wave equation called the telegrapher's equation. We derive explicit expressions for the mean squared displacement and other experimentally observable quantities. Several generalizations, including the incorporation of a resting time between movements, are also studied. The available data on the motion of cells and other organisms is reviewed, and it is shown how the analysis of such data within the framework provided here can be carried out.Supported in part by NIH Grant #GM 29123 and by NSF Grant #DMS-8301840Supported in part by NSF Grant #DMS-8301840Supported in part by the DFG Heisenberg Program     

Vertical distribution and composition of phytoplankton under the influence of an upper mixed layer

The vertical distribution of phytoplankton is of fundamental importance for the dynamics and structure of aquatic communities. Here, using an advection-reaction-diffusion model, we investigate the distribution and competition of phytoplankton species in a water column, in which inverse resource gradients of light and a nutrient can limit growth of the biomass. This problem poses a challenge for ecologists, as the location of a production layer is not fixed, but rather depends on many internal parameters and environmental factors. In particular, we study the influence of an upper mixed layer (UML) in this system and show that it leads to a variety of dynamic effects: (i) Our model predicts alternative density profiles with a maximum of biomass either within or below the UML, thereby the system may be bistable or the relaxation from an unstable state may require a long-lasting transition. (ii) Reduced mixing in the deep layer can induce oscillations of the biomass; we show that a UML can sustain these oscillations even if the diffusivity is less than the critical mixing for a sinking phytoplankton population. (iii) A UML can strongly modify the outcome of competition between different phytoplankton species, yielding bistability both in the spatial distribution and in the species composition. (iv) A light limited species can obtain a competitive advantage if the diffusivity in the deep layers is reduced below a critical value. This yields a subtle competitive exclusion effect, where the oscillatory states in the deep layers are displaced by steady solutions in the UML. Finally, we present a novel graphical approach for deducing the competition outcome and for the analysis of the role of a UML in aquatic systems.     

Cycling expression and cooperative operator interaction in the trp operon of Escherichia coli

Oscillatory behaviour in the tryptophan operon of an Escherichia coli mutant strain lacking the enzyme-inhibition regulatory mechanism has been observed by Bliss et al. but not confirmed by others. This behaviour could be important from the standpoint of synthetic biology, whose goals include the engineering of intracellular genetic oscillators. This work is devoted to investigating, from a mathematical modelling point of view, the possibility that the trp operon of the E. coli inhibition-free strain expresses cyclically. For that we extend a previously introduced model for the regulatory pathway of the tryptophan operon in Escherichia coli to account for the observed multiplicity and cooperativity of repressor binding sites. Thereafter we investigate the model dynamics using deterministic numeric solutions, stochastic simulations, and analytic studies. Our results suggest that a quasi-periodic behaviour could be observed in the trp operon expression level of single bacteria.     

The evolution of tumor metastases during clonal expansion

Cancer is a leading cause of morbidity and mortality in many countries. Solid tumors generally initiate at one particular site called the primary tumor, but eventually disseminate and form new colonies in other organs. The development of such metastases greatly diminishes the potential for a cure of patients and is thought to represent the final stage of the multi-stage progression of human cancer. The concept of early metastatic dissemination, however, postulates that cancer cell spread might arise early during the development of a tumor. It is important to know whether metastases are present at diagnosis since this determines treatment strategies and outcome. In this paper, we design a stochastic mathematical model of the evolution of tumor metastases in an expanding cancer cell population. We calculate the probability of metastasis at a given time during tumor evolution, the expected number of metastatic sites, and the total number of cancer cells as well as metastasized cells. Furthermore, we investigate the effect of drug administration and tumor resection on these quantities and predict the survival time of cancer patients. The model presented in this paper allows us to determine the probability and number of metastases at diagnosis and to identify the optimum treatment strategy to maximally prolong survival of cancer patients.     

Propagation of Fluctuations in Biochemical Systems,I: Linear SSC Networks

We investigate the propagation of random fluctuations through biochemical networks in which the number of molecules of each species is large enough so that the concentrations are well modeled by differential equations. We study the effect of network topology on the emergent properties of the reaction system by characterizing the behavior of variance as fluctuations propagate down chains and studying the effect of side chains and feedback loops. We also investigate the asymptotic behavior of the system as one reaction becomes fast relative to the others.     

一类具有随机周期移民扰动的非线性人口发展方程随机周期解的存在唯一性

本文给出了一类具有随机周期移民扰动的非线性m增生人口发展方程随机周期解的存在性和唯一性结论。     

Noise-Induced Coherence and Network Oscillations in a Reduced Bursting Model

The dynamics of the Hindmarsh-Rose (HR) model of bursting thalamic neurons is reduced to a system of two linear differential equations that retains the subthreshold resonance properties of the HR model. Introducing a reset mechanism after a threshold crossing, we turn this system into a resonant integrate-and-fire (RIF) model. Using Monte-Carlo simulations and mathematical analysis, we examine the effects of noise and the subthreshold dynamic properties of the RIF model on the occurrence of coherence resonance (CR). Synchronized burst firing occurs in a network of such model neurons with excitatory pulse-coupling. The coherence level of the network oscillations shows a stochastic resonance-like dependence on the noise level. Stochastic analysis of the equations shows that the slow recovery from the spike-induced inhibition is crucial in determining the frequencies of the CR and the subthreshold resonance in the original HR model. In this particular type of CR, the oscillation frequency strongly depends on the intrinsic time scales but changes little with the noise intensity. We give analytical quantities to describe this CR mechanism and illustrate its influence on the emerging network oscillations. We discuss the profound physiological roles this kind of CR may have in information processing in neurons possessing a subthreshold resonant frequency and in generating synchronized network oscillations with a frequency that is determined by intrinsic properties of the neurons. PACS 05.45.-a, 05.40.Ca, 87.18.Sn, 87.19     

Seasonal nutrient and plankton dynamics in a physical-biological model of Crater Lake

A coupled 1D physical-biological model of Crater Lake is presented. The model simulates the seasonal evolution of two functional phytoplankton groups, total chlorophyll, and zooplankton in good quantitative agreement with observations from a 10-year monitoring study. During the stratified period in summer and early fall the model displays a marked vertical structure: the phytoplankton biomass of the functional group 1, which represents diatoms and dinoflagellates, has its highest concentration in the upper 40 m; the phytoplankton biomass of group 2, which represents chlorophyta, chrysophyta, cryptomonads and cyanobacteria, has its highest concentrations between 50 and 80 m, and phytoplankton chlorophyll has its maximum at 120 m depth. A similar vertical structure is a reoccurring feature in the available data. In the model the key process allowing a vertical separation between biomass and chlorophyll is photoacclimation. Vertical light attenuation (i.e., water clarity) and the physiological ability of phytoplankton to increase their cellular chlorophyll-to-biomass ratio are ultimately determining the location of the chlorophyll maximum. The location of the particle maxima on the other hand is determined by the balance between growth and losses and occurs where growth and losses equal. The vertical particle flux simulated by our model agrees well with flux measurements from a sediment trap. This motivated us to revisit a previously published study by Dymond et al. (1996). Dymond et al. used a box model to estimate the vertical particle flux and found a discrepancy by a factor 2.5–10 between their model-derived flux and measured fluxes from a sediment trap. Their box model neglected the exchange flux of dissolved and suspended organic matter, which, as our model and available data suggests is significant for the vertical exchange of nitrogen. Adjustment of Dymond et al.’s assumptions to account for dissolved and suspended nitrogen yields a flux estimate that is consistent with sediment trap measurements and our model.     

Modeling the simple epidemic with deterministic differential equations and random initial conditions

In a simple epidemic the only transition in the population is from susceptible to infected and the total population size is fixed for all time. This paper investigates the effect of random initial conditions on the deterministic model for the simple epidemic. By assuming a Beta distribution on the initial proportion of susceptibles, we define a distribution that describes the proportion of susceptibles in a population at any time during an epidemic. The mean and variance for this distribution are derived as hypergeometric functions, and the behavior of these functions is investigated. Lastly, we define a distribution to describe the time until a given proportion of the population remains susceptible. A method for finding the quantiles of this distribution is developed and used to make confidence statements regarding the time until a given proportion of the population is susceptible.     

Scale-free extinction dynamics in spatially structured host-parasitoid systems

Much of the work on extinction events has focused on external perturbations of ecosystems, such as climatic change, or anthropogenic factors. Extinction, however, can also be driven by endogenous factors, such as the ecological interactions between species in an ecosystem. Here we show that endogenously driven extinction events can have a scale-free distribution in simple spatially structured host-parasitoid systems. Due to the properties of this distribution there may be many such simple ecosystems that, although not strictly permanent, persist for arbitrarily long periods of time. We identify a critical phase transition in the parameter space of the host-parasitoid systems, and explain how this is related to the scale-free nature of the extinction process. Based on these results, we conjecture that scale-free extinction processes and critical phase transitions of the type we have found may be a characteristic feature of many spatially structured, multi-species ecosystems in nature. The necessary ingredient appears to be competition between species where the locally inferior type disperses faster in space. If this condition is satisfied then the eventual outcome depends subtly on the strength of local superiority of one species versus the dispersal rate of the other.     

河口最大浑浊带浮游植物生态动力过程研究进展

通过对近十几年来河口最大浑浊带浮游植物生态动力过程研究的报道进行综述,阐明该方面研究的最新进展。研究结果表明,河口最大浑浊带的湍流混合过程增大了浮游植物细胞光合作用的机会;重力环流致使浮游植物及其光合作用所需的物质有较长时间的停留;再悬浮过程使微型底栖藻类对最大浑浊带水体中叶绿素产生明显贡献;锋面强烈的辐合聚集作用则可使浮游植物在锋面附近出现高值现象。最后对河口最大浑浊带浮游植物生态动力过程的继续研究提出了几点看法。