Phytoplankton-zooplankton System Research Articles

Numerical treatment for time fractional order phytoplankton-toxic phytoplankton-zooplankton system

<abstract><p>The study of time-fractional problems with derivatives in terms of Caputo is a recent area of study in biological models. In this article, fractional differential equations with phytoplankton-toxic phytoplankton-zooplankton (PTPZ) system were solved using the Laplace transform method (LTM), the Adomain decomposition method (ADM), and the differential transform method (DTM). This study demonstrates the good agreement between the results produced by using the specified computational techniques. The numerical results displayed as graphs demonstrate the accuracy of the computational methods. The approaches that have been established are thus quite relevant and suitable for solving nonlinear fractional models. Meanwhile, the impact of changing the fractional order of a time derivative and time $ t $ on populations of phytoplankton, toxic-phytoplankton, and zooplankton has been examined using graphical representations. Furthermore, the stability analysis of the LTM approach has been discussed.</p></abstract>

Self-organization of nested patterns in a coupled spatiotemporal phytoplankton-zooplankton system

This research theoretically investigates the emergence of nested structures in plankton patchiness. Considering the vertical migration of plankton populations, we develop a spatiotemporal dynamic model that describes the coupling of two phytoplankton-zooplankton systems. The conditions of pattern self-organization are derived, and importantly, the dispersion relation with two peaks is found to be the key for the occurrence of nested patterns. Based on the analysis of amplitude equations and simulations, four cases of nested patterns that are characterized by the nestedness of different sizes of hexagon and stripe patches are found. As the ratio of the wavenumbers corresponding to the critical Turing modes of patterns increases, the difference of patch sizes in the nested structures rises accordingly, leading to more small patches growing in the large ones. Moreover, the effect of advection on the self-organization of nested patterns is also discussed. The findings in this research may contribute to a new comprehension of the nestedness phenomenon occurring in the spatial distribution of plankton, as observed in field investigations.

Dynamical complexity driven by water temperature in a size-dependent phytoplankton–zooplankton model with environmental variability

In this paper, we propose a stochastic phytoplankton–zooplankton (PZ) system modeling the growth dynamic of phytoplankton, where the death rate of zooplankton and the growth rate of phytoplankton are considered to be dependent on water temperature, and random environmental perturbation is characterized by mean-reverting Ornstein–Uhlenbeck process. Mathematical theoretical work mainly studies the stability, persistence, and Hopf bifurcation of deterministic PZ model, as well as analyzes the stochastic extinction, persistence in mean, and ergodic stationary distribution of stochastic PZ model, which provides a theoretical basis for numerical simulations. Numerical simulation work mainly reveals that water temperature plays a crucial role in influencing the dynamic of size-dependent phytoplankton–zooplankton interactions, whether in deterministic or random environments. An important conclusion of this study is that the increase of water temperature is able to induce bubbling effect, however, the small phytoplankton cell size can weaken this bubbling effect, and the large phytoplankton cell size can strengthen this bubbling effect, which supports the experimental conclusion that body size plays a key role in mitigating the negative impact of warming on predator–prey interactions. Further, it is found that the plankton with large body size has the advantages of stable and sustainable survival in the cold environment, while the plankton with small body size has advantages of stable and sustainable survival in the warm environment. Therefore, it can be inferred that plankton will evolve into large body size in the cold environment, but they will evolve into small body size in the warm environment.

Dynamics of a stochastic phytoplankton–zooplankton system with defensive and offensive effects

In this paper, we propose a stochastic phytoplankton–zooplankton system considering phytoplankton defensive and zooplankton offensive effects. The aim of this paper is to study the effects of environmental fluctuations on plankton population dynamics. We prove the existence, uniqueness and stochastically ultimately boundedness of global positive solutions, and the extinction and persistence in the mean of plankton populations. When the system is persistent in the mean, there exists a unique stationary distribution. To further investigate the dynamics of the stochastic plankton system, we perform some numerical simulations and find that the white noise can directly affect the survival of plankton populations. The phytoplankton defense can strengthen the capability of phytoplankton protection that will benefit the plankton survival and weaken the impact of environmental fluctuations, but it has a negative effect on zooplankton population. Our findings reveal that zooplankton offense is beneficial to the survival of phytoplankton but may threaten the persistence of zooplankton population. An appropriate increase of phytoplankton defense or decrease of zooplankton offense can potentially change the survival state of the plankton system.

Stability Switches and Chaos in a Diffusive Toxic Phytoplankton–Zooplankton Model with Delay

To explore the effect of toxins produced by phytoplankton in algal blooms, a diffusive toxic phytoplankton–zooplankton system with delay is studied through detailed bifurcation analysis. The local stability of the positive equilibrium is studied by analyzing the characteristic equation. The critical values of delay at which Hopf bifurcations occur are obtained and ordered, and it is proved that the delay can induce stability switches, which means that the delay has a significant impact on both the formation and the termination of algal blooms. During the process of stability switching, double Hopf bifurcation arises. Taking the rate of toxin liberation and time delay as the bifurcation parameters, the normal form for double Hopf bifurcation is derived, from which the classification for dynamics and bifurcation sets near double Hopf bifurcation point is obtained. It is proved that the system has complex dynamics, such as periodic oscillations, quasi-periodic oscillations on two- or three-torus and even chaos. Numerical simulations are presented to support the theoretical results.

Fractional approach for mathematical model of phytoplankton–toxic phytoplankton–zooplankton system with Mittag-Leffler kernel

The solution for phytoplankton–toxic phytoplankton–zooplankton system with q-homotopy analysis transform method (q-HATM) is discussed. The projected system exemplifies three components (namely, zooplankton, toxic–phytoplankton as well as phytoplankton) and the corresponding nonlinear ordinary differential equations exemplify the zooplankton feeds on phytoplankton. The projected method is an amalgamation of q-homotopy analysis algorithm and Laplace transform and the derivative associated with the Atangana–Baleanu (AB) operator. The equilibrium points and stability have been discussed with the assistance of the Routh–Hurwitz rule in this work within the frame of generalized calculus. The fixed-point theorem is employed to present the existence and uniqueness of the attained result for the considered model, and we consider five different initial conditions for the projected system. Further, the physical nature of the achieved solution has been captured for fractional order, external force and diverse mass. The achieved consequences explicate that the proposed solution method is highly methodical, easy to implement and accurate to analyze the behavior of the nonlinear system relating to allied areas of science and technology.

Dynamics of a stochastic non-autonomous phytoplankton–zooplankton system involving toxin-producing phytoplankton and impulsive perturbations

This paper describes an analytical and numerical investigation of a stochastic non-autonomous phytoplankton–zooplankton system involving toxin-producing phytoplankton (TPP) and impulsive perturbations. White noise, impulsive perturbations, and TPP were incorporated into the system to stimulate natural aquatic ecological phenomena. The aim of this paper was to analyze how these factors affect the dynamics of the system. Mathematical derivations were utilized to investigate some key threshold conditions that ensure the existence and uniqueness of a global positive solution, population extinction, and persistence in the mean. In particular, we determined if there is a positive periodic solution for the system when the toxin liberation rate reaches a critical value. The numerical results indicated that both white noise and the impulsive control parameter can directly influence population extinction and persistence in the mean. Enhancing the toxin liberation rate of TPP increases the possibility of phytoplankton survival but reduces zooplankton biomass. These results improve our understanding of the dynamics of complex of aquatic ecosystems in a fluctuating environment.

Impact of nanoparticles on the dynamics of a Crowley–Martin type phytoplankton–zooplankton interaction model

The article discusses the effect of nanoparticles on a Crowley–Martin type phytoplankton–zooplankton interaction model. The mathematical model is proposed with the assumption that introducing nanoparticles reduces the growth rate of phytoplankton. The model system is analyzed with various dynamical tools such as stability and bifurcation along with boundedness. The stability conditions are derived with the Jacobian matrix and Routh–Hurwitz criterion. Hopf bifurcation is proved in with respect to parameter β, which is a contact rate of nanoparticles with phytoplankton. The direction of Hopf and stability of bifurcating periodic solutions are discussed by using the normal form theory. Numerical simulation is performed in support of analytical findings and to observe the effects of four key parameters: intrinsic growth rate (r), input rate of nanoparticles (A), contact rate of nanoparticles with phytoplankton (β), and the depletion rate of nanoparticles (e). The growth rate of phytoplankton and the input rate of nanoparticles have stabilizing effect on the system dynamics. On increasing the values of contact rate of nanoparticles, the system exhibits transition from limit cycle oscillation to stable dynamics. Hence nanoparticles stabilize the system if the system with phytoplankton and zooplankton is already oscillatory. Furthermore, we observe that the increase in the natural depletion rate of nanoparticles changes system stability from stable focus to limit cycle oscillation. Hence this study suggests that the introduction of nanoparticles (at a certain level) can be seen as a controlling factor for change in the phytoplankton–zooplankton system dynamics for a balanced ecosystem.

Dynamics of a stochastic phytoplankton–toxic phytoplankton–zooplankton system under regime switching

In this paper, a stochastic phytoplankton–toxic phytoplankton–zooplankton system with Beddington–DeAngelis functional response, where both the white noise and regime switching are taken into account, is studied analytically and numerically. The aim of this research is to study the combined effects of the white noise, regime switching, and toxin‐producing phytoplankton (TPP) on the dynamics of the system. Firstly, the existence and uniqueness of global positive solution of the system is investigated. Then some sufficient conditions for the extinction, persistence in the mean, and the existence of a unique ergodic stationary distribution of the system are derived. Significantly, some numerical simulations are carried to verify our analytical results and show that high intensity of white noise is harmful to the survival of plankton populations, but regime switching can balance the different survival states of plankton populations and decrease the risk of extinction. Additionally, it is found that an increase in the toxin liberation rate produced by TPP will increase the survival chance of phytoplankton, while it will reduce the biomass of zooplankton. All these results may provide some insightful understanding on the dynamics of phytoplankton–zooplankton systems in randomly disturbed aquatic environments.

Selective grazing of zooplankton on phytoplankton defines rapid algal succession and blooms in oceans

In a model plankton ecosystem containing phytoplankton and zooplankton, the later usually avoids toxic phytoplankton and prefers nontoxic species by modulating its grazing rate. The toxins released by toxic phytoplankton can lead to a decreased grazing rate of zooplankton on both phytoplankton, and the presence of nontoxic phytoplankton in the system also reduces the grazing pressure of zooplankton on the toxic phytoplankton. In this study, a set of functions were given that model the selective grazing of zooplankton on various phytoplankton. Preference coefficient and avoidance coefficient were introduced in the selective grazing functions to measure the level of zooplankton preference of nontoxic phytoplankton and avoidance of toxic phytoplankton, respectively. A dynamics model was established in a nontoxic phytoplankton-toxic phytoplankton-zooplankton system based on the grazing functions of zooplankton to study the existence and stability of an interior equilibrium and Hopf-bifurcation. Our results show that zooplankton selective grazing promotes both zooplankton and toxic phytoplankton biomass, and contributes to maintain the coexistence of all species. Low grazing selectivity, i.e., low values of preference coefficient and avoidance coefficient, decreases the zooplankton population and increases the opportunity for phytoplankton blooms. A moderate level of grazing selectivity helps keep the system in balance. With high grazing selectivity on phytoplankton, toxic phytoplankton becomes the dominant species, resulting in a rapid algal succession and eventual toxic phytoplankton blooms. When the preference coefficient and avoidance coefficient in the dynamics model crossed critical values respectively, the ecosystem enters into a Hopf-bifurcation around the interior equilibrium, inducing oscillations of these populations.

Persistence and Bifurcation Analysis of a Plankton Ecosystem with Cross-Diffusion and Double Delays

To investigate the joint effects of diffusion and multiple delays, a toxin producing phytoplankton–zooplankton system with cross-diffusion is studied and two-parameter bifurcation analysis is carried out. Firstly, for the delay-free system, the stability of equilibria, existence of solutions and priori bound are given. The globally asymptotic stability of equilibrium is investigated for the two situations with or without the diffusion terms. The linear stability analyses show that the cross-diffusion is a key mechanism for Turing instability to occur and Turing instability interval becomes larger with increasing cross diffusion coefficient. Secondly, the focus is on the effects of delays on the system including persistence, stability and Hopf bifurcation. The sufficient conditions for the diffusion system with delays to remain persistent are given. Under certain conditions, the boundary equilibrium is globally asymptotically stable. By analyzing the distribution of roots of the characteristic equation, using the crossing curve methods, the stable changes of positive equilibrium in two-delay plane are obtained. Explicit algorithm is derived for the properties of the bifurcating periodic solutions by normal form theory and the center manifold theorem for partial functional differential equations. Finally, numerical simulations confirm the correctness of theoretical analyses and find that double Hopf bifurcation may occur.

Numerical investigation of fractional model of phytoplankton–toxic phytoplankton–zooplankton system with convergence analysis

In this paper, a fractional order model of the phytoplankton–toxic phytoplankton–zooplankton system with Caputo fractional derivative is investigated via three computational methods, namely, residual power series method (RPSM), homotopy perturbation Sumudu transform method (HPSTM) and the homotopy analysis Sumudu transform method (HASTM). This model is constituted by three components: phytoplankton, toxic phytoplankton and zooplankton. Phytoplankton species are self-feeding members of the plankton community and play a very significant role in ecosystems. A wide range of sea creatures get food through phytoplankton. This paper focuses on the implementation of the three above-mentioned computational methods for a nonlinear time-fractional phytoplankton–toxic phytoplankton–zooplankton (PTPZ) model with a perception to study the dynamics of a model. This study shows that the solutions obtained by employing the suggested computational methods are in good agreement with each other. The computational procedures reveal that the HASTM solution generates a more general solution as compared to RPSM and HPSTM and incorporates their results as a special case. The numerical results presented in the form of graphs authenticate the accuracy of computational schemes. Hence, the implemented methods are very appropriate and relevant to handle nonlinear fractional models. In addition, the effect of variation of fractional order of a time derivative and time [Formula: see text] on populations of phytoplankton, toxic–phytoplankton and zooplankton has also been studied through graphical presentations. Moreover, the uniqueness and convergence analyses of HASTM solution have also been discussed in view of the Banach fixed-point theory.

Dynamics of Delayed Phytoplankton–Zooplankton System with Disease Spread Among Zooplankton

In this paper, we study the dynamics of phytoplankton–zooplankton system with delay, where delay means that releasing toxin for phytoplankton is not instantaneous. First, we prove the positivity and boundedness of solutions, discuss the Hopf bifurcation caused by delay. Furthermore, we study the property of Hopf bifurcation by center manifold and normal form. Then, we study the global existence of bifurcated periodic solution. Finally by simulation, we show the influence of delay, disease spread and recovery from infected to susceptible on the dynamics of phytoplankton–zooplankton system.

Global Hopf Bifurcation in a Phytoplankton–Zooplankton System with Delay and Diffusion

In this paper, the dynamics of a phytoplankton–zooplankton system with delay and diffusion are investigated. The positivity and persistence are studied by using the comparison theorem and upper and lower solutions method. The stability of steady states and the existence of local Hopf bifurcation are obtained by analyzing the distribution of eigenvalues. And the global existence of positive periodic solutions is established by using the global Hopf bifurcation result given by Wu [1996]. Finally, some numerical simulations are carried out to illustrate the analytical results.

Global bifurcation in a toxin producing phytoplankton–zooplankton system with prey-taxis

In this paper, we first study the stability of the positive constant equilibrium and Turing instability induced by prey-taxis for phytoplankton–zooplankton system with prey-taxis. Then by applying Crandall–Rabinowitz bifurcation theory, the local existence of the nonconstant positive steady state bifurcating from the positive constant equilibrium E∗ is proven. In one dimension case, when ϕ(Z)=Z, we show that the local solution of the bifurcating branches can be extended to a global one. Of course, a priori estimates play an important role in the proof.

Bifurcation and control of a planktonic ecological system with double delays by delayed feedback control

In this paper, a delayed feedback controller with the delay-dependent coefficient is introduced into a multiple delay phytoplankton-zooplankton system. For uncontrolled system, choosing delays as the bifurcation parameters, we prove that Hopf bifurcation can occur when the delays change and cross some values. Then, the delays are still chosen as the bifurcation parameters to research the dynamic behaviors of the controlled system. Under this control mechanism, the onset of Hopf bifurcation can be delayed by selecting the appropriate control parameters and the stability domain can be extended as feedback gain (the decay rate) decreases (increases), and the influence of the decay rate cannot be ignored. Furthermore, using the crossing curve methods, the stable changes of equilibrium in two delay plane can be obtained. Some numerical simulations are given to verify the correctness and validity of the delayed feedback controller in the bifurcation control.

A nonautonomous model for the effects of refuge and additional food on the dynamics of phytoplankton-zooplankton system

In this paper, a mathematical model for the interacting dynamics of phytoplankton-zooplankton is proposed. The phytoplankton have the ability to take refuge and release toxins to avoid over predation by zooplankton. The zooplankton are provided some additional food to persist in the system. The phytoplankton are assumed to be affected directly by external toxic substances whereas zooplankton are affected indirectly by feeding on the affected phytoplankton. We incorporate seasonal variations in the model, assuming the level of nutrients, refuge and the rate of toxins released by phytoplankton as functions of time. Our results show that when high toxicity and refuge cause extinction of zooplankton, providing additional food supports the survival of zooplankton population and controls the phytoplankton population. Prey refuge and additional food have stabilizing effects on the system; higher values of the former results in extinction of zooplankton whereas phytoplankton disappear for larger values of the latter. Seasonality in nutrients level and toxins released by phytoplankton generate higher periodic solutions while time-dependent refuge of phytoplankton causes the occurrence of a period-three solution. The possibility of finding additional food for zooplankton may push back the ecosystem to a simple stable state from a complex dynamics.

Dynamical study of harmful algal bloom in Sundarban mangrove wetland with spatial interaction and competing effects

Sundarbans is the largest mangrove wetland ecosystem of the world with rich biodiversity suffering from deteriorated water quality due to excessive fertilization that leads to an uncontrolled increase in phytoplankton. Such eutrophication also changes the community structure and increases the harmful algal blooms (HABs). In this work, we propose an interacting population model for phytoplankton–zooplankton system in which the density of zooplankton is influenced by non-toxic phytoplankton (NTP) and toxin-producing phytoplankton (TPP) followed by Holling type II and Monod–Haldane (MH)-type functional response. The growth of zooplankton species is assumed to reduce due to toxic chemicals released by TPP population. The mathematical model of the proposed system includes the competition terms between TPP and NTP. System dynamics is studied in both cases, i.e., system with diffusion and without diffusion. For the non-diffusive system, we have investigated the condition for boundedness along with the existing criteria of all feasible equilibrium point. Stability analysis of the model system is carried out in detail for each equilibrium point. Forward and backward bifurcation diagrams are obtained for the temporal system in order to understand the behavior of different parameters that control the system dynamics. Theoretically, stability criteria and Turing instability of diffusive system are derived. In this study, we have taken a case of Sundarban mangrove wetland which is suffering from algal blooms due to the presence of toxic Dinoflagellates and Cyanophyceae. Our numerical investigation shows that the lower value of intraspecific interference of zooplankton promotes the complex spatiotemporal dynamics for the population of non-toxic, toxic phytoplankton and zooplankton. The higher value of inter-specific competition coefficient of NTP leads to reduction in zooplankton density that may cause bad health of the wetland system. This investigation renders the importance of diffusion in algal blooms by the occurrence of different Turing patterns and the role of time delay in destabilization of stationary points through the creation of limit cycles. We observed that the increasing value of diffusion coefficient of zooplankton and time allows the algal blooms to settle down from spot-strip mixture to spot patterns.

Dynamical Study of an Eco-Epidemiological Delay Model for Plankton System with Toxicity.

In this paper, we analyze the complexity of an eco-epidemiological model for phytoplankton–zooplankton system in presence of toxicity and time delay. Holling type II function response is incorporated to address the predation rate as well as toxic substance distribution in zooplankton. It is also presumed that infected phytoplankton does recover from the viral infection. In the absence of time delay, stability and Hopf-bifurcation conditions are investigated to explore the system dynamics around all the possible equilibrium points. Further, in the presence of time delay, conditions for local stability are derived around the interior equilibria and the properties of the periodic solution are obtained by applying normal form theory and central manifold arguments. Computational simulation is performed to illustrate our theoretical findings. It is explored that system dynamics is very sensitive corresponding to carrying capacity and toxin liberation rate and able to generate chaos. Further, it is observed that time delay in the viral infection process can destabilize the phytoplankton density whereas zooplankton density remains in its old state. Incorporation of time delay also gives the scenario of double Hopf-bifurcation. Some control parameters are discussed to stabilize system dynamics. The effect of time delay on (i) growth rate of susceptible phytoplankton shows the extinction and double Hopf-bifurcation in the zooplankton population, (ii) a sufficiently large value of carrying capacity stabilizes the chaotic dynamics or makes the whole system chaotic with further increment.

Dynamics induced by environmental stochasticity in a phytoplankton-zooplankton system with toxic phytoplankton.

Environmental stochasticity and toxin-producing phytoplankton (TPP) are the key factors that affect the aquatic ecosystems. To investigate the effects of environmental stochasticity and TPP on the dynamics of plankton populations, a stochastic phytoplankton-zooplankton system with two TPP is studied theoretically and numerically in this paper. Theoretically, we first prove that the system possesses a unique and global positive solution with positive initial values, and then derive some sufficient conditions guaranteeing the extinction and persistence in the mean of the system. Significantly, it is shown that the system has a stationary distribution when toxin liberation rate reaches some a critical value. Additionally, numerical analysis shows that the white noise can affect the survival of plankton populations directly. Furthermore, it has been observed that the increasing one toxin liberation rate can increase the survival chance of phytoplankton and reduce the biomass of zooplankton, but the combined effects of two liberation rates on the changes in plankton populations are stronger than that of controlling any one of the two TPP.