Bifurcation analysis of phytoplankton-fish model through parametric control by fish mortality rate and food transfer efficiency

An Algae-zooplankton fish model is studied in this article. First the proposed model is evaluated for positive invariance and boundedness. Then,the Routh-Hurwitz parameters and the Lyapunov function are used to determine the presence of a positive interior steady state and the criteria for plankton model stability (both local and global). Taylor’s sequence is also used to discuss Hopf bifurcation and the stability of bifurcated periodic solutions. The model’s bifurcation analysis reveals that Hopf-bifurcation can occur when mortality rate and food transfer efficiency are used as bifurcation parameters. Finally, we use numerical simulation to validate the analytical results

End-to-end models of marine ecosystems: exploring the consequences of climate change and fishing using a minimal framework

Marine ecosystems are vital to human society: as a source of food, for economic growth and for their potential to mitigate climate change. With marine ecosystems threatened by climate change and overfishing, there is a need for sustainable fisheries management, which has been the basis for an ecosystem-based approach to management. This has led to considerable interest in end-to-end ecosystem models, where the physical effects of the environment and the population dynamics of all marine organisms are coupled together into one framework. In this thesis, I studied an end-to-end model which coupled together a box-component model representing phytoplankton and zooplankton, with a size-structured fish community model. I investigated the potential artefacts in model results, caused by numerical methods or by model architecture. I found that care needs to be taken with the choice of numerical method used to simulate size-structured models, as the choice of numerical resolution can yield numerically stable results but can also affect large-scale behaviours of the system, such as the slope and mathematical stability of the size-spectra solutions. With regards to model architecture, coupling together two submodels which differ in structure and resolution can lead to large-scale behaviours of the system which appear plausible and consistent with empirical data, but which impose serious discrepancies in the underlying life-histories of the fish. By distinguishing model artefacts from ecosystem-effects, the interactions and feedbacks between the higher and lower trophic level organisms can be investigated. I studied the potential impact of climate change upon the marine ecosystem, and in particular, upon the seasonal dynamics of phytoplankton. I found that under a warming climate, the spring phytoplankton bloom occurs earlier and for a longer duration, and the model predicts the loss of the autumn phytoplankton bloom. These changes were not solely due to the direct effect of temperature, but also due to the indirect effect of the interactions of the fish population with zooplankton. The effect of fishing upon the marine ecosystem was also explored with this end-to-end model, with two potential fishing strategies applied to the system. Regardless of the choice of fishing strategy, intensive exploitation of fish stocks can lead to a significant shift in the dynamics of phytoplankton. The phytoplankton's dynamics change from stable annual patterns to unpredictable periodic behaviours. This thesis has developed an end-to-end model which uses a minimal framework to study the interactions of organisms at different trophic levels, and highlights the importance of these interactions and the associated feedbacks under different scenarios. It combines important theoretical insights into the consequences of model architecture and model-derived artefacts upon ecosystem-scale behaviours, at the same time as highlighting the potential for end-to-end models as a practical and flexible management tool

Nonlinear dynamics of plankton ecosystem with impulsive control and environmental fluctuations

It is well known that the density of plankton populations always increases and decreases or keeps invariant for a long time, and the variation of plankton density is an important factor influencing the real aquatic environments, why do these situations occur? It is an interesting topic which has become the common interest for many researchers. As the basis of the food webs in oceans, lakes, and reservoirs, plankton plays a significant role in the material circulation and energy flow for real aquatic ecosystems that have a great effect on the economic and social values. Planktonic blooms can occur in some environments, however, and the direct or indirect adverse effects of planktonic blooms on real aquatic ecosystems, such as water quality, water landscape, aquaculture development, are sometimes catastrophic, and thus planktonic blooms have become a challenging and intractable problem worldwide in recent years. Therefore, to understand these effects so that some necessary measures can be taken, it is important and meaningful to investigate the dynamic growth mechanism of plankton and reveal the dynamics mechanisms of formation and disappearance of planktonic blooms. To this end, based on the background of the ecological environments in the subtropical lakes and reservoirs, this dissertation research takes mainly the planktonic algae as the research objective to model the mechanisms of plankton growth and evolution. In this dissertation, some theories related to population dynamics, impulsive control dynamics, stochastic dynamics, as well as the methods of dynamic modeling, dynamic analysis and experimental simulation, are applied to reveal the effects of some key biological factors on the dynamics mechanisms of the spatial-temporal distribution of plankton and the termination of planktonic blooms, and to predict the dynamics evolutionary processes of plankton growth. The main results are as follows: Firstly, to discuss the prevention and control strategies on planktonic blooms, an impulsive reaction-diffusion hybrid system was developed. On the one hand, the dynamic analysis showed that impulsive control can significantly influence the dynamics of the system, including the ultimate boundedness, extinction, permanence, and the existence and uniqueness of positive periodic solution of the system. On the other hand, some experimental simulations were preformed to reveal that impulsive control can lead to the extinction and permanence of population directly. More precisely, the prey and intermediate predator populations can coexist at any time and location of their inhabited domain, while the top predator population undergoes extinction when the impulsive control parameter exceeds some a critical value, which can provide some key arguments to control population survival by means of some reaction-diffusion impulsive hybrid systems in the real life. Additionally, a heterogeneous environment can affect the spatial distribution of plankton and change the temporal-spatial oscillation of plankton distribution. All results are expected to be helpful in the study of dynamic complex of ecosystems. Secondly, a stochastic phytoplankton-zooplankton system with toxic phytoplankton was proposed and the effects of environmental stochasticity and toxin-producing phytoplankton (TPP) on the dynamics mechanisms of the termination of planktonic blooms were discussed. The research illustrated that white noise can aggravate the stochastic oscillation of plankton density and a high-level intensity of white noise can accelerate the extinction of plankton and may be advantageous for the disappearance of harmful phytoplankton, which imply that the white noise can help control the biomass of plankton and provide a guide for the termination of planktonic blooms. Additionally, some experimental simulations were carried out to reveal that the increasing toxin liberation rate released by TPP can increase the survival chance of phytoplankton population and reduce the biomass of zooplankton population, but the combined effects of those two toxin liberation rates on the changes in plankton are stronger than that of controlling any one of the two TPP. All results suggest that both white noise and TPP can play an important role in controlling planktonic blooms. Thirdly, we established a stochastic phytoplankton-toxic producing phytoplankton-zooplankton system under regime switching and investigated how the white noise, regime switching and TPP affect the dynamics mechanisms of planktonic blooms. The dynamical analysis indicated that both white noise and toxins released by TPP are disadvantageous to the development of plankton and may increase the risk of plankton extinction. Also, a series of experimental simulations were carried out to verify the correctness of the dynamical analysis and further reveal the effects of the white noise, regime switching and TPP on the dynamics mechanisms of the termination of planktonic blooms. On the one hand, the numerical study revealed that the system can switch from one state to another due to regime shift, and further indicated that the regime switching can balance the different survival states of plankton density and decrease the risk of plankton extinction when the density of white noise are particularly weak. On the other hand, an increase in the toxin liberation rate can increase the survival chance of phytoplankton but reduce the biomass of zooplankton, which implies that the presence of toxic phytoplankton may have a positive effect on the termination of planktonic blooms. These results may provide some insightful understanding on the dynamics of phytoplankton-zooplankton systems in randomly disturbed aquatic environments. Finally, a stochastic non-autonomous phytoplankton-zooplankton system involving TPP and impulsive perturbations was studied, where the white noise, impulsive perturbations and TPP are incorporated into the system to simulate the natural aquatic ecological phenomena. The dynamical analysis revealed some key threshold conditions that ensure the existence and uniqueness of a global positive solution, plankton 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. Some experimental simulations also revealed that both white noise and impulsive control parameter can directly influence the plankton extinction and persistence in the mean. Significantly, enhancing the toxin liberation rate released by TPP increases the possibility of phytoplankton survival but reduces the zooplankton biomass. All these results can improve our understanding of the dynamics of complex of aquatic ecosystems in a fluctuating environment

Dynamical Complexity of a Spatial Phytoplankton-Zooplankton Model with an Alternative Prey and Refuge Effect

The spatiotemporal dynamics of a phytoplankton-zooplankton model with an alternative prey and refuge effect is investigated mathematically and numerically. The stability of the equilibrium point and the traveling wave solution of the phytoplankton-zooplankton model are described based on theoretical mathematical work, which provides the basis of the numerical simulation. The numerical analysis shows that refuges have a strong effect on the spatiotemporal dynamics of the model according to the pattern formation. These results may help us to understand prey-predator interactions in water ecosystems. They are also relevant to research into phytoplankton-zooplankton ecosystems

On the Dynamical Behavior of Toxic-Phytoplankton-Zooplankton Model with Delay

A toxin producing phytoplankton-zooplankton model with inhibitory exponential substrate and time delay has been formulated and analyzed. Since the liberation of toxic substances by phytoplankton species is not an instantaneous process but is mediated by some time lag required for maturity of the species and the zooplankton mortality due to the toxic phytoplankton bloom occurs after some time laps of the bloom of toxic phytoplankton, we induced a discrete time delay to both of the consume response function and distribution of toxic substance term. Furthermore, based on the fact that the predation rate decreases at large toxic-phytoplankton density, the system is modelled via a Tissiet type functional response. We study the dynamical behaviour and investigate the conditions to guarantee the coexistence of two species. Analytical methods and numerical simulations are used to obtain information about the qualitative behaviour of the models

Discrete-time prey-predator model with θ-logistic growth for prey incorporating square root functional response

This article presents the dynamics of a discrete-time prey-predator system with square root functional response incorporating θ-logistic growth. This type of functional response is used to study the dynamics of the prey--predator system where the prey population exhibits herd behavior, i.e., the interaction between prey and predator occurs along the boundary of the population. The existence and stability of fixed points and Neimark-Sacker Bifurcation (NSB) are analyzed. The phase portraits, bifurcation diagrams and Lyapunov exponents are presented and analyzed for different parameters of the model. Numerical simulations show that the discrete model exhibits rich dynamics as the effect of θ-logistic growth

On a three-tier ecological food chain model with deterministic and random harvesting: A mathematical study

In the present study, we consider a nutrient-autotroph-herbivore ecosystem model where the herbivore species is assumed to have a commercial value. We use a Holling type-II harvest function to model density dependent herbivore harvesting. Stability criteria of the resulting model is investigated both from analytical and numerical viewpoints. The investigation revealed the existence of a number of threshold values of the harvest rate that have a remarkable influence on the system dynamics. Next we incorporate a noise term in the parameter representing harvest rate to model the phenomenon of poaching as random harvesting. The stochastic model is analyzed for exponential mean square stability and the resulting criteria in terms of harvest related parameters obtained. These parameter thresholds could be utilized to develop effective harvesting strategies and wildlife management policies which take into account the overall survival of the ecological populations