Modelling phytoplankton allelopathy in a nutrient-plankton model with spatial heterogeneity

Stability switches and chaos induced by delay in a reaction-diffusion nutrient-plankton model

In this paper, we investigate a reaction-diffusion model incorporating dynamic variables for nutrient, phytoplankton, and zooplankton. Moreover, we account for the impact of time delay in the growth of phytoplankton following nutrient uptake. Our theoretical analysis reveals that the time delay can trigger the emergence of persistent oscillations in the model via a Hopf bifurcation. We also analytically track the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions. Our simulation results demonstrate stability switches occurring for the positive equilibrium with an increasing time lag. Furthermore, the model exhibits homogeneous periodic-2 and 3 solutions, as well as chaotic behaviour. These findings highlight that the presence of time delay in the phytoplankton growth can bring forth dynamical complexity to the nutrient-plankton system of aquatic habitats.

On the qualitative study of a discrete-time phytoplankton-zooplankton model under the effects of external toxicity in phytoplankton population

The current manuscript studies a discrete-time phytoplankton-zooplankton model with Holling type-II response. The original model is modified by considering the condition that the phytoplankton population is getting infected with an external toxic substance. To obtain the discrete counterpart from a continuous-time system, Euler's forward method is applied. Moreover, a consistent discrete-time phytoplankton-zooplankton model is obtained by using a nonstandard difference scheme. The boundedness character for every positive solution is discussed, and the local stability of obtained system about each of its fixed points is discussed. The existence of period-doubling bifurcation at a positive equilibrium point is discussed for the discrete system obtained by Euler's forward method. In addition, the comparison of the consistent discrete-time version with its inconsistent counterpart is provided. It is proved that the discrete-time system obtained by using a nonstandard scheme is dynamically consistent as there is no chance for the existence of period-doubling bifurcation in that system. In order to control the period-doubling bifurcation and Neimark-Sacker bifurcation, an improved hybrid control strategy is applied. Finally, we have provided some interesting numerical examples to explain our theoretical results.

Global dynamics of a diffusive phytoplankton-zooplankton model with toxic substances effect and delay

This paper examines a diffusive toxic-producing plankton system with delay. We first show the global attractivity of the positive equilibrium of the system without time-delay. We further consider the effect of delay on asymptotic behavior of the positive equilibrium: when the system undergoes Hopf bifurcation at some points of delay by the normal form and center manifold theory for partial functional differential equations. Global existence of periodic solutions is established by applying the global Hopf bifurcation theory.

Qualitative analysis of a discrete-time phytoplankton-zooplankton model with Holling type-II response and toxicity

The interaction among phytoplankton and zooplankton is one of the most important processes in ecology. Discrete-time mathematical models are commonly used for describing the dynamical properties of phytoplankton and zooplankton interaction with nonoverlapping generations. In such type of generations a new age group swaps the older group after regular intervals of time. Keeping in observation the dynamical reliability for continuous-time mathematical models, we convert a continuous-time phytoplankton-zooplankton model into its discrete-time counterpart by applying a dynamically consistent nonstandard difference scheme. Moreover, we discuss boundedness conditions for every solution and prove the existence of a unique positive equilibrium point. We discuss the local stability of obtained system about all its equilibrium points and show the existence of Neimark-Sacker bifurcation about unique positive equilibrium under some mathematical conditions. To control the Neimark-Sacker bifurcation, we apply a generalized hybrid control technique. For explanation of our theoretical results and to compare the dynamics of obtained discrete-time model with its continuous counterpart, we provide some motivating numerical examples. Moreover, from numerical study we can see that the obtained system and its continuous-time counterpart are stable for the same values of parameters, and they are unstable for the same parametric values. Hence the dynamical consistency of our obtained system can be seen from numerical study. Finally, we compare the modified hybrid method with old hybrid method at the end of the paper.

Chaos control in a multiple delayed phytoplankton-zooplankton model with group defense and predator's interference

Phytoplankton-zooplankton interaction is a topic of high interest among the interrelationships related to marine habitats. In the present manuscript, we attempt to study the dynamics of a three-dimensional system with three types of plankton: non-toxic phytoplankton, toxic producing phytoplankton, and zooplankton. We assume that both non-toxic and toxic phytoplankton are consumed by zooplankton via Beddington-DeAngelis and general Holling type-IV responses, respectively. We also incorporate gestation delay and toxic liberation delay in zooplankton's interactions with non-toxic and toxic phytoplankton correspondingly. First, we have studied the well-posedness of the system. Then, we analyze all the possible equilibrium points and their local and global asymptotic behavior. Furthermore, we assessed the conditions for the occurrence of Hopf-bifurcation and transcritical bifurcation. Using the normal form method and center manifold theorem, the conditions for stability and direction of Hopf-bifurcation are also studied. Various time-series, phase portraits, and bifurcation diagrams are plotted to confirm our theoretical findings. From the numerical simulation, we observe that a limited increase in inhibitory effect of toxic phytoplankton against zooplankton can support zooplankton's growth, and rising predator's interference can also boost zooplankton expansion in contrast to the nature of Holling type IV and Beddington-DeAngelis responses. Next, we notice that on variation of toxic liberation delay, the delayed system switches its stability multiple times and becomes chaotic. Furthermore, we draw the Poincaré section and evaluate the maximum Lyapunov exponent in order to verify the delayed system's chaotic nature. Results presented in this article might be helpful to interpret biological insights into phytoplankton-zooplankton interactions.

Dynamical analysis of a toxin-producing phytoplankton-zooplankton model with refuge

To study the impacts of toxin produced by phytoplankton and refuges provided for phytoplankton on phytoplankton-zooplankton interactions in lakes, we establish a simple phytoplankton-zooplankton system with Holling type II response function. The existence and stability of positive equilibria are discussed. Bifurcation analyses are given by using normal form theory which reveals reasonably the mechanisms and nonlinear dynamics of the effects of toxin and refuges, including Hopf bifurcation, Bogdanov-Takens bifurcation of co-dimension 2 and 3. Numerical simulations are carried out to intuitively support our analytical results and help to explain the observed biological behaviors. Our findings finally show that both phytoplankton refuge and toxin have a significant impact on the occurring and terminating of algal blooms in freshwater lakes.

Competition between a nonallelopathic phytoplankton and an allelopathic phytoplankton species under predation

We propose a model of two-species competition in the chemostat for a single growth-limiting, nonreproducing resource that extends that of Roy [38]. The response functions are specified to be Michaelis-Menten, and there is no predation in Roy's work. Our model generalizes Roy's model to general uptake functions. The competition is exploitative so that species compete by decreasing the common pool of resources. The model also allows allelopathic effects of one toxin-producing species, both on itself (autotoxicity) and on its nontoxic competitor (phytotoxicity). We show that a stable coexistence equilibrium exists as long as (a) there are allelopathic effects and (b) the input nutrient concentration is above a critical value. The model is reconsidered under instantaneous nutrient recycling. We further extend this work to include a zooplankton species as a fourth interacting component to study the impact of predation on the ecosystem. The zooplankton species is allowed to feed only on the two phytoplankton species which are its perfectly substitutable resources. Each of the models is analyzed for boundedness, equilibria, stability, and uniform persistence (or permanence). Each model structure fits very well with some harmful algal bloom observations where the phytoplankton assemblage can be envisioned in two compartments, toxin producing and non-toxic. The Prymnesium parvum literature, where the suppressing effects of allelochemicals are quite pronounced, is a classic example. This work advances knowledge in an area of research becoming ever more important, which is understanding the functioning of allelopathy in food webs.

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.

Dynamics of two phytoplankton populations under predation

The aim of this paper is to investigate the manner in which predation and single-nutrient competition affect the dynamics of a non-toxic and a toxic phytoplankton species in a homogeneous environment (such as a chemostat). We allow for the possibility that both species serve as prey for an herbivorous zooplankton species. We assume that the toxic phytoplankton species produces toxins that affect only its own growth (autotoxicity). The autotoxicity assumption is ecologically explained by the fact that the toxin-producing phytoplankton is not mature enough to produce toxins that will affect the growth of its nontoxic competitor. We show that, in the absence of phytotoxic interactions and nutrient recycling, our model exhibits uniform persistence. The removal rates are distinct and we use general response functions. Finally, numerical simulations are carried out to show consistency with theoretical analysis. Our model has similarities with other food-chain models. As such, our results may be relevant to a wider spectrum of population models, not just those focused on plankton. Some open problems are discussed at the end of this paper.

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.

Modelling the phytoplankton dynamics in a nutrient-rich solar saltern pond: predicting the impact of restoration and climate change

An ecological model for the solar saltern of Sfax (Tunisia) was established and validated by comparing simulation results to observed data relative to horizontal distributions of temperature, nutrients and phytoplankton biomass. Sensitivity analysis was performed in order to assess the influence of the main ecological model parameters. First applied at the saltern's pond A1, the model was calibrated with field data measured over 4 years of study (from 2000 to 2003), which allowed an evaluation of parameters such as maximum growth rate of phytoplankton, optimal growth temperature and constant of half saturation for P/N assimilation by phytoplankton. Simulation results showed that the model allowed us to predict realistic phytoplankton variations of the study area, though we were unable to accurately reproduce the nutrient variation. The model was then applied to simulations of the impact of changes in phytoplankton biomass through scenarios such as hypothetic climate changes and saltern restoration.

DIFFUSIVE INSTABILITY AND TRAVELING WAVES IN A MANGROVE ECOSYSTEM FOOD-CHAIN MODEL WITH DELAYED DETRITUS RECYCLING

In this paper, a food-chain model in a mangrove ecosystem with detritus recycling is analyzed. From the stability analysis of the delayed homogeneous system, an interval for the parameter representing detritus-detritivores interaction rate is obtained that imparts stability to the system around the coexistent state. Next, we have studied the model in a nonhomogeneous environment. The analysis revealed the existence of a subinterval of the above mentioned interval such that when the above interaction-rate lies within this interval, the system will undergo diffusion driven instability. Finally, we show the existence of travelling wave solutions for the said ecosystem. Numerical simulations are carried out to augment analytical results.