Bloom dynamics in a seasonally forced phytoplankton–zooplankton model: Trigger mechanisms and timing effects

Investigation of a nutrient-plankton model with stochastic fluctuation and impulsive control

In this paper, we investigate a stochastic nutrient-plankton model with impulsive control of the nutrient concentration and zooplankton population. Analytically, we find that the population size is nonnegative for a sufficiently long time. We derive some sufficient conditions for the existence of stable periodic oscillations, which indicate that the plankton populations will behave periodically. The numerical results show that the plankton system experiences a transition from extinction to the coexistence of species due to the emergence of impulsive control. Additionally, we observe that the nutrient pulse has a stronger relationship with phytoplankton growth than the zooplankton pulse. Although the frequency of impulsive control and appropriate environmental fluctuations can promote the coexistence of plankton populations, an excessive intensity of noise can result in the collapse of the entire ecosystem. Our findings may provide some insights into the relationships among nutrients, phytoplankton and zooplankton in a stochastic environment.

Effects of solar irradiance noise on a complex marine trophic web

The analysis of experimental data of the solar irradiance, collected on the marine surface, clearly highlights the intrinsic stochasticity of such an environmental parameter. Given this result, effects of randomly fluctuating irradiance on the population dynamics of a marine ecosystem are studied on the basis of the stochastic 0-dimensional biogeochemical flux model. The noisy fluctuations of the irradiance are formally described as a multiplicative Ornstein-Uhlenbeck process, that is a self-correlated Gaussian noise. Nonmonotonic behaviours of the variance of the marine populations' biomass are found with respect to the intensity and the autocorrelation time of the noise source, manifesting a noise-induced transition of the ecosystem to an out-of-equilibrium steady state. Moreover, evidence of noise-induced effects on the organic carbon cycling processes underlying the food web dynamics are highlighted. The reported results clearly show the profound impact the stochastic environmental variables can have on both the populations and the biogeochemistry at the basis of a marine trophic network.

Phase tipping: how cyclic ecosystems respond to contemporary climate

We identify the phase of a cycle as a new critical factor for tipping points (critical transitions) in cyclic systems subject to time-varying external conditions. As an example, we consider how contemporary climate variability induces tipping from a predator–prey cycle to extinction in two paradigmatic predator–prey models with an Allee effect. Our analysis of these examples uncovers a counterintuitive behaviour, which we call phase tipping or P-tipping, where tipping to extinction occurs only from certain phases of the cycle. To explain this behaviour, we combine global dynamics with set theory and introduce the concept of partial basin instability for attracting limit cycles. This concept provides a general framework to analyse and identify easily testable criteria for the occurrence of phase tipping in externally forced systems, and can be extended to more complicated attractors.

State-dependent vulnerability of synchronization

A state-dependent vulnerability of synchronization is shown to exist in a complex network composed of numerically simulated electronic circuits. We demonstrate that disturbances to the local dynamics of network units can produce different outcomes to synchronization depending on the current state of its trajectory. We address such state dependence by systematically perturbing the synchronized system at states equally distributed along its trajectory. We find the states at which the perturbation desynchronizes the network to be complicatedly mixed with the ones that restore synchronization. Additionally, we characterize perturbation sets obtained for consecutive states by defining a safety index between them. Finally, we demonstrate that the observed vulnerability is due to the existence of an unstable chaotic set in the system's state space.

Effect of seasonal changing temperature on the growth of phytoplankton

An non-autonomous nutrient-phytoplankton interacting model incorporating the effect of time-varying temperature is established. The impacts of temperature on metabolism of phytoplankton such as nutrient uptake, death rate, and nutrient releasing from particulate nutrient are investigated. The ecological reproductive index is formulated to present a threshold criteria and to characterize the dynamics of phytoplankton. The positive invariance, dissipativity, and the existence and stability of boundary and positive periodic solution are established. The analyses rely on the comparison principle, the coincidence degree theory and Lyapunov direct method. The effect of seasonal temperature and daily temperature on phytoplankton biomass are simulated numerically. Numerical simulation shows that the phytoplankton biomass is very robust to the variation of water temperature. The dynamics of the model and model predictions agree with the experimental data. Our model and analysis provide a possible explanation of triggering mechanism of phytoplankton blooms.

An observer for an occluded reaction-diffusion system with spatially varying parameters

Spatially dependent parameters of a two-component chaotic reaction-diffusion partial differential equation (PDE) model describing ocean ecology are observed by sampling a single species. We estimate the model parameters and the other species in the system by autosynchronization, where quantities of interest are evolved according to misfit between model and observations, to only partially observed data. Our motivating example comes from oceanic ecology as viewed by remote sensing data, but where noisy occluded data are realized in the form of cloud cover. We demonstrate a method to learn a large-scale coupled synchronizing system that represents the spatio-temporal dynamics and apply a network approach to analyze manifold stability.

Dynamic analysis of a fractional order system

In this paper, we investigate the dynamical behavior of a fractional order phytoplankton–zooplankton system. In this paper, stability analysis of the phytoplankton–zooplankton model (PZM) is studied by using the fractional Routh–Hurwitz stability conditions. We have studied the local stability of the equilibrium points of PZM. We applied an efficient numerical method based on converting the fractional derivative to integer derivative to solve the PZM.

Stability, bifurcation and chaos analysis of vector-borne disease model with application to Rift Valley fever

This paper investigates a RVF epidemic model by qualitative analysis and numerical simulations. Qualitative analysis have been used to explore the stability dynamics of the equilibrium points while visualization techniques such as bifurcation diagrams, Poincaré maps, maxima return maps and largest Lyapunov exponents are numerically computed to confirm further complexity of these dynamics induced by the seasonal forcing on the mosquitoes oviposition rates. The obtained results show that ordinary differential equation models with external forcing can have rich dynamic behaviour, ranging from bifurcation to strange attractors which may explain the observed fluctuations found in RVF empiric outbreak data, as well as the non deterministic nature of RVF inter-epidemic activities. Furthermore, the coexistence of the endemic equilibrium is subjected to existence of certain number of infected Aedes mosquitoes, suggesting that Aedes have potential to initiate RVF epidemics through transovarial transmission and to sustain low levels of the disease during post epidemic periods. Therefore we argue that locations that may serve as RVF virus reservoirs should be eliminated or kept under control to prevent multi-periodic outbreaks and consequent chains of infections. The epidemiological significance of this study is: (1) low levels of birth rate (in both Aedes and Culex) can trigger unpredictable outbreaks; (2) Aedes mosquitoes are more likely capable of inducing unpredictable behaviour compared to the Culex; (3) higher oviposition rates on mosquitoes do not in general imply manifestation of irregular behaviour on the dynamics of the disease. Finally, our model with external seasonal forcing on vector oviposition rates is able to mimic the linear increase in livestock seroprevalence during inter-epidemic period showing a constant exposure and presence of active transmission foci. This suggests that RVF outbreaks partly build upon RVF inter-epidemic activities. Therefore, active RVF surveillance in livestock is recommended.

A fractional-order toxin producing phytoplankton and zooplankton system

In this work, we investigate the dynamical behavior of a fractional-order toxin producing on a phytoplankton–zooplankton (TPPZ) system with nutrient cycling. We propose a mathematical system to model this situation. All the feasible equilibria of the system are obtained and the conditions for the existence of the equilibriums are determined. Local stability analysis of the TPPZ is studied by using the fractional Routh–Hurwitz stability conditions. Numerical simulations are carried out for a hypothetical set of parameter values to substantiate our analytical findings.

Thermal stratification influences maturation and timing of spawning in a local Clupea harengus population

Maturation and timing of spawning in relation to temperature were studied in a local Atlantic herring Clupea harengus population inhabiting a small semi-enclosed ecosystem (7 km(2)) separated from the larger outer fjord system by narrow sills on the west coast of Norway. Ambient temperatures varied annually up to 4° C during both the pre-spawning and spawning periods from February to April, but without affecting the spawning time. Instead, the timing of spawning was found to be related to thermal stratification in response to spring warming, which occurred about the same time every year regardless of initial temperatures.

Phytoplankton-zooplankton dynamics in periodic environments taking into account eutrophication

In this paper, we derive and analyze a mathematical model for the interactions between phytoplankton and zooplankton in a periodic environment, in which the growth rate and the intrinsic carrying-capacity of phytoplankton are changing with respect to time and nutrient concentration. A threshold value: "Predator's average growth rate" is introduced and it is proved that the phytoplankton-zooplankton ecosystem is permanent (both populations survive cronically) and possesses a periodic solution if and only if the value is positive. We use TP (Total Phosphorus) concentration to mark the degree of eutrophication. Based on experimental data, we fit the growth rate function and the environmental carrying capacity function with temperature and nutrient concentration as independent variables. Using measured data of temperature on water bodies we fit a periodic temperature function of time, and this leads the growth rate and intrinsic carrying-capacity of phytoplankton to be periodic functions of time. Thus we establish a periodic system with TP concentration as parameter. The simulation results reveal a high diversity of population levels of the ecosystem that are mainly sensitive to TP concentration and the death-rate of zooplankton. It illustrates that the eruption of algal bloom is mainly resulted from the increasing of nutrient concentration while zooplankton only plays a role to alleviate the scale of algal bloom, which might be used to explain the mechanism of algal bloom occurrence in many natural waters. What is more, our results provide a better understanding of the traditional manipulation method.

Effect of demographic noise in a phytoplankton-zooplankton model of bloom dynamics

An extension of the Truscott-Brindley model [Bull. Math. Biol. 56, 981 (1994)] is derived to account for the effect of demographic fluctuations. In the presence of seasonal forcing and sufficiently shallow water conditions, the fluctuations induced by the discreteness of the zooplankton component appear sufficient to cause switching between the bloom and no-bloom cycles predicted at the mean-field level by the model. The destabilization persists in the thermodynamic limit of a water basin infinitely extended in the horizontal direction.