scholarly journals Dynamical complexity in a delayed Plankton-Fish model with alternative food for predators

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Rajinder Pal Kaur ◽  
Amit Sharma ◽  
Anuj Kumar Sharma
Keyword(s):  
Hopf Bifurcation ◽  
Ecological Model ◽  
Fish Population ◽  
Alternative Food ◽  
Predator Population ◽  
Additional Food ◽  
Fish Model ◽  
Main Motive ◽  
The Stability ◽  
The Impact

<p style='text-indent:20px;'>The present manuscript deals with a 3-D food chain ecological model incorporating three species phytoplankton, zooplankton, and fish. To make the model more realistic, we include predation delay in the fish population due to the vertical migration of zooplankton species. We have assumed that additional food is available for both the predator population, viz., zooplankton, and fish. The main motive of the present study is to analyze the impact of available additional food and predation delay on the plankton-fish dynamics. The positivity and boundedness (with and without delay) are proved to make the system biologically valid. The steady states are determined to discuss the stability behavior of non-delayed dynamics under certain conditions. Considering available additional food as a control parameter, we have estimated ranges of alternative food for maintaining the sustainability and stability of the plankton-fish ecosystem. The Hopf-bifurcation analysis is carried out by considering time delay as a bifurcation parameter. The predation delay includes complexity in the system dynamics as it passes through its critical value. The direction of Hopf-bifurcation and stability of bifurcating periodic orbits are also determined using the centre manifold theorem. Numerical simulation is executed to validate theoretical results.</p>

THE ROLE OF ADDITIONAL FOOD IN A PREDATOR–PREY MODEL WITH A PREY REFUGE

2016 ◽  
Vol 24 (02n03) ◽  
pp. 345-365 ◽  
Author(s):  
SUDIP SAMANTA ◽  
RIKHIYA DHAR ◽  
IBRAHIM M. ELMOJTABA ◽  
JOYDEV CHATTOPADHYAY
Keyword(s):  
Stable Equilibrium ◽  
Predator Population ◽  
Stable Limit ◽  
Prey Refuge ◽  
Additional Food ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
The Stability ◽  
The Impact

In this paper, we propose and analyze a predator–prey model with a prey refuge and additional food for predators. We study the impact of a prey refuge on the stability dynamics, when a constant proportion or a constant number of prey moves to the refuge area. The system dynamics are studied using both analytical and numerical techniques. We observe that the prey refuge can replace the predator–prey oscillations by a stable equilibrium if the refuge size crosses a threshold value. It is also observed that, if the refuge size is very high, then the extinction of the predator population is certain. Further, we observe that enhancement of additional food for predators prevents the extinction of the predator and also replaces the stable limit cycle with a stable equilibrium. Our results suggest that additional food for the predators enhances the stability and persistence of the system. Extensive numerical experiments are performed to illustrate our analytical findings.

Fear Induced Stabilization in an Intraguild Predation Model

2020 ◽  
Vol 30 (04) ◽  
pp. 2050053
Author(s):  
Mainul Hossain ◽  
Nikhil Pal ◽  
Sudip Samanta ◽  
Joydev Chattopadhyay
Keyword(s):  
Growth Rate ◽  
Hopf Bifurcation ◽  
Intraguild Predation ◽  
Equilibrium Points ◽  
Stable Limit ◽  
Interior Equilibrium ◽  
The Rich ◽  
The Stability ◽  
The Cost ◽  
The Impact

In the present paper, we investigate the impact of fear in an intraguild predation model. We consider that the growth rate of intraguild prey (IG prey) is reduced due to the cost of fear of intraguild predator (IG predator), and the growth rate of basal prey is suppressed due to the cost of fear of both the IG prey and the IG predator. The basic mathematical results such as positively invariant space, boundedness of the solutions, persistence of the system have been investigated. We further analyze the existence and local stability of the biologically feasible equilibrium points, and also study the Hopf-bifurcation analysis of the system with respect to the fear parameter. The direction of Hopf-bifurcation and the stability properties of the periodic solutions have also been investigated. We observe that in the absence of fear, omnivory produces chaos in a three-species food chain system. However, fear can stabilize the chaos thus obtained. We also observe that the system shows bistability behavior between IG prey free equilibrium and IG predator free equilibrium, and bistability between IG prey free equilibrium and interior equilibrium. Furthermore, we observe that for a suitable set of parameter values, the system may exhibit multiple stable limit cycles. We perform extensive numerical simulations to explore the rich dynamics of a simple intraguild predation model with fear effect.

scholarly journals Analysis of a Fishery Model with two competing prey species in the presence of a predator species for Optimal Harvesting

2018 ◽  
Vol 31 ◽  
pp. 08008 ◽  
Author(s):  
Sutimin ◽  
Siti Khabibah ◽  
Dita Anis Munawwaroh
Keyword(s):  
Ecological Model ◽  
Dynamical Behavior ◽  
Fish Population ◽  
Optimal Harvesting ◽  
Globally Asymptotically Stable ◽  
Predator Species ◽  
Optimal Harvest ◽  
The Stability ◽  
Fishery Model ◽  
Maximal Principle

A harvesting fishery model is proposed to analyze the effects of the presence of red devil fish population, as a predator in an ecosystem. In this paper, we consider an ecological model of three species by taking into account two competing species and presence of a predator (red devil), the third species, which incorporates the harvesting efforts of each fish species. The stability of the dynamical system is discussed and the existence of biological and bionomic equilibrium is examined. The optimal harvest policy is studied and the solution is derived in the equilibrium case applying Pontryagin’s maximal principle. The simulation results is presented to simulate the dynamical behavior of the model and show that the optimal equilibrium solution is globally asymptotically stable. The results show that the optimal harvesting effort is obtained regarding to bionomic and biological equilibrium.

scholarly journals Effects of delayed immune-activation in the dynamics of tumor-immune interactions

2020 ◽  
Vol 15 ◽  
pp. 45 ◽  
Author(s):  
Parthasakha Das ◽  
Pritha Das ◽  
Samhita Das
Keyword(s):  
Hopf Bifurcation ◽  
Immune Activation ◽  
Distributed Delay ◽  
Normal Form Theory ◽  
Effector Cells ◽  
Center Manifold Theorem ◽  
Discrete Delays ◽  
Form Theory ◽  
The Stability ◽  
The Impact

This article presents the impact of distributed and discrete delays that emerge in the formulation of a mathematical model of the human immunological system describing the interactions of effector cells (ECs), tumor cells (TCs) and helper T-cells (HTCs). We investigate the stability of equilibria and the commencement of sustained oscillations after Hopf-bifurcation. Moreover, based on the center manifold theorem and normal form theory, the expression for direction and stability of Hopf-bifurcation occurring at tumor presence equilibrium point of the system has been derived explicitly. The effect of distributed delay involved in immune-activation on the system dynamics of the tumor is demonstrated. Numerical simulations are also illustrated for elucidating the change of dynamic behavior by varying system parameters.

A NONAUTONOMOUS MODEL FOR THE INTERACTIVE EFFECTS OF FEAR, REFUGE AND ADDITIONAL FOOD IN A PREY–PREDATOR SYSTEM

2021 ◽  
pp. 1-39
Author(s):  
NAZMUL SK ◽  
PANKAJ KUMAR TIWARI ◽  
YUN KANG ◽  
SAMARES PAL
Keyword(s):  
Growth Rate ◽  
Hopf Bifurcation ◽  
Seasonal Variations ◽  
Positive Periodic Solution ◽  
Oscillatory System ◽  
Interactive Effects ◽  
Predator Population ◽  
Prey Refuge ◽  
Additional Food ◽  
Predator System

The importance of fear, refuge and additional food is being increasingly recognized in recent studies, but their combined effects need to be explored. In this paper, we investigate the joint effects of these three ecologically important factors in a prey–predator system with Crowly–Martin type functional response. We find that the fear of predator significantly affects the densities of prey and predator populations whereas the presence of prey refuge and additional food for predator are recognized to have potential impacts to sustain prey and predator in the habitat, respectively. The fear of predator induces limit cycle oscillations while an oscillatory system becomes stable on increasing the refuge. The system first undergoes a supercritical Hopf-bifurcation and then a subcritical Hopf-bifurcation on increasing either the growth rate of prey or growth rate of predator due to additional food. Increasing the quality/quantity of additional food after a certain value causes extinction of prey species and rapid incline in the predator population. An extension is made in the model by considering the seasonal variations in the cost of fear of predator, prey refuge and growth rate of predator due to additional food. The nonautonomous model is shown to exhibit globally attractive positive periodic solution. Moreover, complex dynamics such as higher periodic solutions and bursting patterns are observed due to seasonal variations in the rate parameters.

Hopf Bifurcation in a Delayed Diffusive Leslie–Gower Predator–Prey Model with Herd Behavior

2019 ◽  
Vol 29 (04) ◽  
pp. 1950055
Author(s):  
Fengrong Zhang ◽  
Yan Li ◽  
Changpin Li
Keyword(s):  
Hopf Bifurcation ◽  
Time Delay ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Herd Behavior ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Prey Model ◽  
The Stability ◽  
The Impact

In this paper, we consider a delayed diffusive predator–prey model with Leslie–Gower term and herd behavior subject to Neumann boundary conditions. We are mainly concerned with the impact of time delay on the stability of this model. First, for delayed differential equations and delayed-diffusive differential equations, the stability of the positive equilibrium and the existence of Hopf bifurcation are investigated respectively. It is observed that when time delay continues to increase and crosses through some critical values, a family of homogeneous and inhomogeneous periodic solutions emerge. Then, the explicit formula for determining the stability and direction of bifurcating periodic solutions are also derived by employing the normal form theory and center manifold theorem for partial functional differential equations. Finally, some numerical simulations are shown to support the analytical results.

A diffusive predator–prey system with additional food and intra-specific competition among predators

2018 ◽  
Vol 11 (04) ◽  
pp. 1850060 ◽  
Author(s):  
Ruizhi Yang ◽  
Ming Liu ◽  
Chunrui Zhang
Keyword(s):  
Hopf Bifurcation ◽  
Global Stability ◽  
Neumann Boundary ◽  
Turing Instability ◽  
Delay System ◽  
Additional Food ◽  
Predator Prey ◽  
Prey System ◽  
Boundary Equilibrium ◽  
The Stability

In this paper, a diffusive predator–prey system with additional food and intra-specific competition among predators subject to Neumann boundary condition is investigated. For non-delay system, global stability, Turing instability and Hopf bifurcation are studied. For delay system, instability and Hopf bifurcation induced by time delay and global stability of boundary equilibrium are discussed. By the theory of normal form and center manifold method, the conditions for determining the bifurcation direction and the stability of the bifurcating periodic solution are derived.

scholarly journals 1708 year in the Belarus: an comparative analysis of the three biggest battles blind spots

2019 ◽  
pp. 31-39
Author(s):  
O. N. Slisarenko
Keyword(s):  
Comparative Analysis ◽  
Military Operations ◽  
The South ◽  
Russian Army ◽  
Main Motive ◽  
Long Time ◽  
The Stability ◽  
Real Picture ◽  
The Impact ◽  
Great Northern War

The article is devoted to one of the key moments of the Great Northern war – the campaign of the Swedish king Charles XII to Moscow. The historians of this campaign until now has not come to a consensus about what caused the sudden turn of the Swedish army to the South with a main route to Moscow. In the article on the basis of the analysis of historiographical achievements of recent years, a comparative analysis of military operations that took place on the territory of Belarus in the summer – autumn of 1708. In this period there were three major battles at the crossings of important water boundaries: when Golovchino, Maletichy and Lesnaja. All of them had different outcomes, so it is interesting to analyze in detail the features of these battles. The author believes that the Russian army managed to combine the impact on the Swedish army of different kind of troops: guards infantry, horse dragoon regiments and irregular cavalry (cossacks, kalmyks and tatars). The Russian army tried to weaken the enemy forces by creating difficulties in supply. The main motive for turning to the South of Charles XII was the loss of a military convoy, accompanied by a corps Lewenhaupt. The degree of reliability of the sources covering the events of the Great Northern war on the territory of Belarus in 1708 is analyzed. Based on the method of comparative analysis of sources, the author attempts to restore the real picture of the events. The conclusion is made about deliberate distortion of facts in the official Russian historiography in order to create a positive image of Tsar Peter I and his associates both in the international arena and within the country. The stability of this tradition for a long time is revealed.

scholarly journals A fractional-order love dynamical model with time delay for synergic couple : Stability analysis and Hopf bifurcation

2021 ◽  
Author(s):  
SANTOSHI PANIGRAHI ◽  
Sunita Chand ◽  
S Balamuralitharan
Keyword(s):  
Stability Analysis ◽  
Hopf Bifurcation ◽  
Time Delay ◽  
Fractional Order ◽  
Reproduction Number ◽  
Equilibrium Points ◽  
Order Model ◽  
Fractional Order Model ◽  
The Stability ◽  
The Impact

We investigate the fractional order love dynamic model with time delay for synergic couples in this manuscript. The quantitative analysis of the model has been done where the asymptotic stability of the equilibrium points of the model have been analyzed. Under the impact of time delay, the Hopf bifurcation analysis of the model has been done. The stability analysis of the model has been studied with the reproduction number less than or greater than 1. By using Laplace transformation, the analysis of the model has been done. The analysis shows that the fractional order model with a time delay can sufficiently improve the components and invigorate the outcomes for either stable or unstable criteria. In this model, all unstable cases are converted to stable cases under neighbourhood points. For all parameters, the reproduction ranges have been described. Finally, to illustrate our derived results numerical simulations have been carried out by using MATLAB. Under the theoretical outcomes from parameter estimation, the love dynamical system is verified.

Bifurcation analysis in a stage-structured predator–prey model with maturation delay

2014 ◽  
Vol 07 (04) ◽  
pp. 1450042
Author(s):  
Jia Liu
Keyword(s):  
Hopf Bifurcation ◽  
Periodic Solutions ◽  
Carrying Capacity ◽  
Positive Equilibrium ◽  
Equilibrium Solutions ◽  
Predator Prey ◽  
Maturation Delay ◽  
Predator Prey Model ◽  
The Stability ◽  
The Impact

In this paper, we investigate the impact of maturation delay on the positive equilibrium solutions in a stage-structured predator–prey system. By analyzing the characteristic equation we derive the conditions for the emergence of Hopf bifurcation. By applying the normal form and the center manifold argument, the direction as well as the stability of periodic solutions bifurcating from Hopf bifurcation is explored. Results show that maturation delay can change the nature of the positive equilibrium solutions, and the loss of equilibrium stability occurs as a consequence of Hopf bifurcation. When Hopf bifurcation takes place, periodic solution arises and is further demonstrated to be asymptotically stable. In addition, the periodic solutions appear only for intermediate maturation delay, that is, there exists a delay window, outside of which the positive equilibrium is locally stable. Furthermore, numerical analysis shows that Hopf bifurcation is favored by a superior competition for adult predators to juveniles, a smaller mortality on juvenile and/or adult predators, and a higher resource carrying capacity. Interestingly, increasing food carrying capacity can lead to the emergence of irregular chaotic dynamics and regular limit cycles.

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