Dynamics of populations in fish farm: Analysis of stability and direction of Hopf-bifurcating periodic oscillation

The paper deals with the dynamical behavior of fish and mussel population in a fish farm where external food is supplied. The ecosystem of the fish farm is represented by a set of nonlinear differential equations involving nutrient (food), fish and mussel. We have listed some results already obtained. We have analyzed for the direction of Hopf-bifurcation, stability of the Hopf-bifurcating periodic orbits, and the period of the periodic orbits by using Poincare’ normal form and center manifold theory. We have performed numerical simulation to support the analytical results.     

Top-predator interference and gestation delay as determinants of the dynamics of a realistic model food chain

An attempt has been made to understand the role of top predator interference and gestation delay on the dynamics of a three species food chain model involving intermediate and top predator populations. Interaction between the prey and an intermediate predator follows the Volterra scheme (with Holling type IV functional response), while that between the top predator and its prey depends on Beddington–DeAngelis type functional response. Stability switches and Hopf-bifurcation occurs when the delay crosses some critical value. Model system exhibits irregular behavior when the interference is high or gestation period is larger than its critical value. Furthermore, the direction of Hopf-bifurcation and the stability of the bifurcating periodic solutions are determined using the center manifold theorem and normal form theory. Computer simulations have been carried out to illustrate the analytical findings. Different diagnostic tests, like, initial sensitivity, Lyapunov exponent, recurrence plot tests ensure the complex dynamical behavior of the model system. Finally, we observed the subcritical Hopf-bifurcation phenomena in the designed model system and the bifurcating periodic solution is unstable for the considered set of parameter values.     

Dynamical behavior of a delay virus dynamics model with CTL immune response

In this paper, the dynamical behavior of a virus dynamics model with CTL immune response and time delay is studied. Time delay is used to describe the time between the infected cell and the emission of viral particles on a cellular level. The effect of time delay on stability of the equilibria of the CTL immune response model has been studied and sufficient criteria for local asymptotic stability of the disease-free equilibrium, immune-free equilibrium and endemic equilibrium and global asymptotic stability of the disease-free equilibrium are given. Some conditions for Hopf bifurcation around immune-free equilibrium and endemic equilibrium to occur are also obtained by using the time delay as a bifurcation parameter. Numerical simulation with some hypothetical sets of data has been done to support the analytical findings.     

A delayed prey–predator model with Crowley–Martin‐type functional response including prey refuge

In this paper, we have studied a prey–predator model living in a habitat that divided into two regions: an unreserved region and a reserved (refuge) region. The migration between these two regions is allowed. The interaction between unreserved prey and predator is Crowley–Martin‐type functional response. The local and global stability of the system is discussed. Further, the system is extended to incorporate the effect of time delay. Then the dynamical behavior of the system is analyzed, taking delay as a bifurcation parameter. The direction of Hopf bifurcation and the stability of the bifurcated periodic solution are determined with the help of normal form theory and centre manifold theorem. We have also discussed the influence of prey refuge on densities of prey and predator species. The analytical results are supplemented with numerical simulations. Copyright © 2017 John Wiley & Sons, Ltd.     

The dynamics of Bowley's model with bounded rationality

A nonlinear dynamical system which describe the time evolution of n-competitors in a Cournot game (Bowley's model) with bounded rationality is analyzed. The existence and stability of the equilibria of this system is studied. The stability conditions of the steady states for two and three players are explicitly computed. Complex behavior such as cycles and chaotic behavior are observed by numerical simulation. Delayed Bowley's with bounded rationality in monopoly is studied. We show that firms using bounded rationality with delay has a higher chance of reaching Nash equilibrium.     

Analysis of a mathematical predator-prey model with delay

We study the behavior of dynamic processes in a mathematical predator-prey model and show that the dynamical system may have a periodic solution whose period coincides with the delay. By the bifurcation method for stability analysis of periodic solutions, we establish that this periodic solution is unstable.     

Analysis of a viral infection model with delayed immune response

It is well known that the immune response plays an important role in eliminating or controlling the disease after human body is infected by virus. In this paper, we investigate the dynamical behavior of a viral infection model with retarded immune response. The effect of time delay on stability of the equilibria of the system has been studied and sufficient condition for local asymptotic stability of the infected equilibrium and global asymptotic stability of the infection-free equilibrium and the immune-exhausted equilibrium are given. By numerical simulating,we observe that the stationary solution becomes unstable at some critical immune response time, while the delay time and birth rate of susceptible host cells increase, and we also discover the occurrence of stable periodic solutions and chaotic dynamical behavior. The results can be used to explain the complexity of the immune state of patients.     

Dynamical analysis of a delay model of phytoplankton-zooplankton interaction

The interaction of toxic-phytoplankton-zooplankton systems and their dynamical behavior will be considered in this paper based upon nonlinear ordinary differential equation model system. We induced a discrete time delay to the both of the consume response function and distribution of toxic substance term to describe the delay in the conversion of nutrient consumed to species and the fact that the time required for the phytoplankton species to mature before they can produce toxic substances. We generalized the model in [1] and explicit results are derived for globally asymptotically stability of the boundary equilibrium. Using numerical simulation method, we determine there is a parameter range for the delay parameter τ where more complicated dynamics occurs, and this appears to be a new result. Significant outcomes of our numerical findings and their interpretations from ecological point of view are provided in this paper.     

Analysis of a delayed predator-prey model with ratio-dependent functional response and quadratic harvesting

In this paper, we investigate the dynamics of a ratio dependent predator-prey model with quadratic harvesting. We examine the existence of the positive equilibria, the related dynamical behaviors of the model, as well as the boundedness and permanence property of the system. We also study the global stability of the interior equilibrium without time delay. Finally some bifurcation analysis is carried out for the system with delay and the results are illustrated numerically.     

Challenges of living in the harsh environments: A mathematical modeling study

We propose and analyze a mathematical model, which mimics community dynamics of plants and animals in harsh environments. The mathematical model exploits type IV functional responses whose idiosyncrasies have been recognized only in recent years. The interaction of the middle predator with the top predator is cast into Leslie-Gower scheme. Linear and non-linear stability analyses are performed to get an idea of the stability behavior of the model food chain. It turns out that carrying capacity of the prey and the immunity parameter of the middle predator are two crucial parameters governing the model. Availability of alternative food options to the generalist predator also plays a key role in deciding the model dynamics.Simulation runs performed on this model provide insight into population dynamics of monkeys of macaque family found in northern Japan. These monkeys are social animals which reproduce sexually. The characteristic feature of the model dynamics is that the generalist predator (macaque monkeys) is able to avoid impending extinction frequently and recovers at a rate which falsify threats from exogenous external forces; extreme weather conditions, etc.     

Bursting mechanism in a time-delayed oscillator with slowly varying external forcing

This paper investigates the generation of complex bursting patterns in the Duffing oscillator with time-delayed feedback. We present the bursting patterns, including symmetric fold–fold bursting and symmetric Hopf–Hopf bursting when periodic forcing changes slowly. We make an analysis of the system bifurcations and dynamics as a function of the delayed feedback and the periodic forcing. We calculate the conditions of fold bifurcation and Hopf bifurcation as well as its stability related to external forcing and delay. We also identify two regimes of bursting depending on the magnitude of the delay itself and the strength of time delayed coupling in the model. Our results show that the dynamics of bursters in delayed system are quite different from those in systems without any delay. In particular, delay can be used as a tuning parameter to modulate dynamics of bursting corresponding to the different type. Furthermore, we use transformed phase space analysis to explore the evolution details of the delayed bursting behavior. Also some numerical simulations are included to illustrate the validity of our study.     

Stability and Hopf bifurcation of a Nonlinear oscillator with multiple time-delays

This paper examines dynamical behavior of a nonlinear oscillator which models a quarter-car forced by the road profile. The effect of multiple time delays is studied in detail. The focus is on the influence of delay in the system. This naturally gives rise to a delay differential equation (DDE) model of the system. The domain where the control is efficient in reducing the amplitude of vibration is found by the harmonic balance method. Technical stability within definite time and asymptotic stability is derived for selected gain control parameters. The control gain parameters are chosen according to technical and asymptotic stability. The energy analysis is a combination of Lyapunov’s function and the averaging technique, and is used to analyze the Hopf bifurcation.     

Stability analysis of the continuous ethanol fermentation process with a delayed product inhibition

In the paper, we propose and analyze a mathematical model of the continuous ethanol fermentation process to study the mechanisms of the self-sustained oscillations of ethanol concentration. The model is based on the assumption that microorganism cells response to the inhibitory effect of product (ethanol) concentration with a delay. From the local stability analysis of the system, we show that the delay time is one of the crucial factors for the occurrence of oscillations and for a critical delay time the fermentation process undergoes a Hopf bifurcation. Further analysis shows that the operating variables and kinetic parameters have also a significant effect on the dynamical behavior of the fermentation system. A proper manipulation of the operating variables allow us to eliminate the oscillatory behavior.     

HOPF BIFURCATION ANALYSIS OF A 3D CHAOTIC SYSTEM

In this paper,a 3D chaotic system with multi-parameters is introduced. The dynamical systems of the original ADVP circuit and the modified ADVP model are regarded as special examples to the system.Some basic dynamical behaviors such as the stability of equilibria,the existence of Hopf bifurcation are investigated.We analyse the Hopf bifurcation of the system comprehensively using the first Lyapunov coefficient by precise symbolic computation.As a result,in fact we have studied the further dynamical behaviors.     

Dynamical behavior in a stage-structured differential-algebraic prey-predator model with discrete time delay and harvesting

A differential-algebraic model system which considers a prey-predator system with stage structure for prey and harvest effort on predator is proposed. By using the differential-algebraic system theory and bifurcation theory, dynamic behavior of the proposed model system with and without discrete time delay is investigated. Local stability analysis of the model system without discrete time delay reveals that there is a phenomenon of singularity induced bifurcation due to variation of the economic interest of harvesting, and a state feedback controller is designed to stabilize the proposed model system at the interior equilibrium; Furthermore, local stability of the model system with discrete time delay is studied. It reveals that the discrete time delay has a destabilizing effect in the population dynamics, and a phenomenon of Hopf bifurcation occurs as the discrete time delay increases through a certain threshold. Finally, numerical simulations are carried out to show the consistency with theoretical analysis obtained in this paper.     

The chaos and control of a food chain model supplying additional food to top-predator

The control and management of chaotic population is one of the main objectives for constructing mathematical model in ecology today. In this paper, we apply a technique of controlling chaotic predator–prey population dynamics by supplying additional food to top-predator. We formulate a three species predator–prey model supplying additional food to top-predator. Existence conditions and local stability criteria of equilibrium points are determined analytically. Persistence conditions for the system are derived. Global stability conditions of interior equilibrium point is calculated. Theoretical results are verified through numerical simulations. Phase diagram is presented for various quality and quantity of additional food. One parameter bifurcation analysis is done with respect to quality and quantity of additional food separately keeping one of them fixed. Using MATCONT package, we derive the bifurcation scenarios when both the parameters quality and quantity of additional food vary together. We predict the existence of Hopf point (H), limit point (LP) and branch point (BP) in the model for suitable supply of additional food. We have computed the regions of different dynamical behaviour in the quantity–quality parametric plane. From our study we conclude that chaotic population dynamics of predator prey system can be controlled to obtain regular population dynamics only by supplying additional food to top predator. This study is aimed to introduce a new non-chemical chaos control mechanism in a predator–prey system with the applications in fishery management and biological conservation of prey predator species.     

具有Monod-Haldane功能反应的一类食物链模型的动力学行为

该文研究了一个由食饵种群、捕食者种群和杂食者种群所构成的食物链系统, 其捕食功能反应为Monod-Haldane功能反应. 应用定性分析和Hopf分支理论, 得到了该系统边界平衡点的全局稳定性和周期解存在性的判别准则. 为了概括和归类这个系统的全局动力学行为,该文得到了具有不同动力学行为的参数区域. 应用MATLAB软件,该文提供了一个例子来展示这些结论, 并且表明: 这个系统能够产生非常复杂的动力学行为.     

On the stability and Hopf bifurcation of a delay-induced predator–prey system with habitat complexity

We study the effect of the degree of habitat complexity and gestation delay on the stability of a predator–prey model. It is observed that there is stability switches, and Hopf bifurcation occurs when the delay crosses some critical value. By applying the normal form theory and the center manifold theorem, the explicit formulae which determine the stability and direction of the bifurcating periodic solutions are determined. The qualitative dynamical behavior of the model system is verified with the published data of Paramecium aurelia (prey) and Didinium nasutum (predator) interaction. It is observed that the quantitative level of abundance of system populations depends crucially on the delay parameter if the gestation period exceeds some critical value. However, the fluctuations in the population levels can be controlled completely by increasing the degree of habitat complexity.     

A statistical approach to the determination of stability for dynamical systems modelling physiological processes

One of the questions involved in the formulation of a new model for a physiological phenomenon, when the model represents a dynamical system, is that concerning its qualitative behavior. The determination of the stability of a particular dynamical system is usually made analytically, from a linearization of the system around an equilibrium point. This analytic proof may often be very complex or impossible, leading to the imposition of conditions on the relative magnitude of the structural model parameters or to other partial results. We discuss a general technique whereby a probabilistic judgment is made on the stability of a dynamical system, and we apply it to the study of a particular delay differential system modelling the relationship between insulin secretion and glucose uptake. This technique is applicable in case experimental material is available from which to estimate the dispersion of the model parameters. A stability criterion is obtained via the usual linearization around an equilibrium point, it is approximated as a Taylor series in the parameters truncated after the first term, and its variance is then computed from the dispersion of the parameters. While the conclusion is probabilistic in nature, it can be obtained for a wide class of models and from either sample or individual experimental subject's parameter estimates.