scholarly journals Effects of Additional Foods to Predators on Nutrient-Consumer-Predator Food Chain Model

2012 ◽  
Vol 2012 ◽  
pp. 1-8 ◽  
Author(s):  
Banshidhar Sahoo
Keyword(s):  
Death Rate ◽  
Local Stability ◽  
Equilibrium Points ◽  
Chain Model ◽  
Additional Food ◽  
Boundedness Property ◽  
Interior Equilibrium ◽  
Predator Model ◽  
Food Chain Model ◽  
Unstable Dynamics

We have proposed a nutrient-consumer-predator model with additional food to predator, at variable nutrient enrichment levels. The boundedness property and the conditions for local stability of boundary and interior equilibrium points of the system are derived. Bifurcation analysis is done with respect to quality and quantity of additional food and consumer’s death rate for the model. The system has stable as well as unstable dynamics depending on supply of additional food to predator. This model shows that supply of additional food plays an important role in the biological controllability of the system.

scholarly journals Density Dependent Predator Death Prevalence Chaos in a Tri-trophic Food Chain Model

2008 ◽  
Vol 13 (3) ◽  
pp. 305-324 ◽  
Author(s):  
M. Bandyopadhyay ◽  
S. Chatterjee ◽  
S. Chakraborty ◽  
J. Chattopadhyay
Keyword(s):  
Food Chain ◽  
Death Rate ◽  
Chaotic Dynamics ◽  
Equilibrium Points ◽  
Model System ◽  
Chain Model ◽  
Predator Population ◽  
Predator Species ◽  
Density Dependent ◽  
Food Chain Model

Ecological systems have all the properties to produce chaotic dynamics. To predict the chaotic behavior in an ecological system and its possible control mechanism is interesting. Aziz-Alaoui [1] considered a tri-trophic food-chain model with modified Leslie-Gower type growth rate for top-predator population and established the chaotic dynamics exhibited by the model system for a certain choice of parameter values. We have modified the said model by incorporating density dependent death rate for predator population. Our mathematical findings reveal the fact that there are two coexisting equilibrium points one of which is a source and the other one is a sink. The positive equilibrium point which is sink is actually globally asymptotically stable under certain parametric conditions. Numerical experiment analysis shows that the model system are capable to produce chaotic dynamics when the rate of intra specific completion is very low and chaotic dynamics disappears for a certain value of the rate of intra specific completion for predator species. Our results suggest that the consideration of density dependent death rate for predator species have the ability to control the chaotic dynamics.

scholarly journals DETERMINISTIC CHAOS VERSUS STOCHASTIC OSCILLATION IN A PREY-PREDATOR-TOP PREDATOR MODEL

2011 ◽  
Vol 16 (3) ◽  
pp. 343-364 ◽  
Author(s):  
Ranjit Kumar Upadhyay ◽  
Malay Banerjee ◽  
Rana Parshad ◽  
Sharada Nandan Raw
Keyword(s):  
Deterministic Model ◽  
Driving Forces ◽  
Equilibrium Points ◽  
Chaotic Oscillation ◽  
Three Dimensional ◽  
Deterministic Chaos ◽  
Model System ◽  
Dynamical Analysis ◽  
Interior Equilibrium ◽  
Predator Model

The main objective of the present paper is to consider the dynamical analysis of a three dimensional prey-predator model within deterministic environment and the influence of environmental driving forces on the dynamics of the model system. For the deterministic model we have obtained the local asymptotic stability criteria of various equilibrium points and derived the condition for the existence of small amplitude periodic solution bifurcating from interior equilibrium point through Hopf bifurcation. We have obtained the parametric domain within which the model system exhibit chaotic oscillation and determined the route to chaos. Finally, we have shown that chaotic oscillation disappears in presence of environmental driving forces which actually affect the deterministic growth rates. These driving forces are unable to drive the system from a regime of deterministic chaos towards a stochastically stable situation. The stochastic stability results are discussed in terms of the stability of first and second order moments. Exhaustive numerical simulations are carried out to validate the analytical findings.

Dynamics of an autonomous food chain model and existence of global attractor of the associated non-autonomous system

2019 ◽  
Vol 12 (08) ◽  
pp. 1950082 ◽  
Author(s):  
Jyotirmoy Roy ◽  
Shariful Alam
Keyword(s):  
Hopf Bifurcation ◽  
Food Chain ◽  
Migration Rate ◽  
Equilibrium Points ◽  
Model System ◽  
Seasonal Effect ◽  
Chain Model ◽  
Upward Migration ◽  
Downward Migration ◽  
Food Chain Model

In this paper, we have analyzed a tri-trophic food chain model consisting of phytoplankton, zooplankton and fish population in an aquatic environment. Here, the pelagic water column is divided into two layers namely, the upper layer and the lower layer. The zooplankton population makes a diel vertical migration (DVM) from lower portion to upper portion and vice-versa to trade-off between food source and fear from predator (Fish). Here, mathematical model has been developed and analyzed in a rigorous way. Apart from routine calculations like boundedness and positivity of the solution, local stability of the equilibrium points, we performed Hopf bifurcation analysis of the interior equilibrium point of our model system in a systematic way. It is observed that the migratory behavior of zooplankton plays a crucial role in the dynamics of the model system. Both the upward and downward migration rates of DVM leads the system into Hopf bifurcation. The upward migration rate of zooplankton deteriorates the stable coexistence of all the species in the system, whereas the downward migration rate enhance the stability of the system. Further, we analyze the non-autonomous version of the system to capture seasonal effect of environmental variations. We have shown that under certain parametric restrictions periodic coexistence of all the species of our system is possible. Finally, extensive numerical simulation has been performed to support our analytical findings.

Analysis of a harvested tritrophic food chain model in the presence of additional food for top predator

2018 ◽  
Vol 11 (04) ◽  
pp. 1850059 ◽  
Author(s):  
Prabir Panja ◽  
Swarup Poria ◽  
Shyamal Kumar Mondal
Keyword(s):  
Extinction Risk ◽  
Predation Rate ◽  
Optimal Harvesting ◽  
Chain Model ◽  
Predator Population ◽  
Top Predator ◽  
Additional Food ◽  
Revenue Collection ◽  
Existence And Stability ◽  
Food Chain Model

In this paper, we propose and analyze a three-species predator–prey system in the presence of additional food for top predator. It is assumed that the middle predator is acting as a prey as well as a predator and the top predator consumes both prey as well as middle predator. It is also considered that a constant amount of additional food for top predator exists in the ecosystem. The effects of harvesting of top predator are investigated. The existence and stability conditions of the equilibria have been discussed analytically. The Hopf bifurcation analysis of the system with respect to predation rate of prey to the top predator and the harvesting effort have been analyzed both analytically and numerically. Pontryagin’s maximum principle is used to determine the optimal harvesting of top predator population to maximize the discounted net revenue. From our analysis, it is seen that the additional food has a significant impact to prevent the extinction risk of top predator population and also to increase revenue collection. Finally, some numerical results have been given in support of our analytical findings.

scholarly journals Chaotic Discrete Fractional-Order Food Chain Model and Hybrid Image Encryption Scheme Application

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 161
Author(s):  
Sameh Askar ◽  
Abdulrahman Al-khedhairi ◽  
Amr Elsonbaty ◽  
Abdelalim Elsadany
Keyword(s):  
Fractional Order ◽  
Food Chain ◽  
Chaotic Behavior ◽  
Equilibrium Points ◽  
High Sensitivity ◽  
Phase Portraits ◽  
Chain Model ◽  
Encryption Scheme ◽  
Dynamical Behaviors ◽  
Food Chain Model

Using the discrete fractional calculus, a novel discrete fractional-order food chain model for the case of strong pressure on preys map is proposed. Dynamical behaviors of the model involving stability analysis of its equilibrium points, bifurcation diagrams and phase portraits are investigated. It is demonstrated that the model can exhibit a variety of dynamical behaviors including stable steady states, periodic and quasiperiodic dynamics. Then, a hybrid encryption scheme based on chaotic behavior of the model along with elliptic curve key exchange scheme is proposed for colored plain images. The hybrid scheme combines the characteristics of noise-like chaotic dynamics of the map, including high sensitivity to values of parameters, with the advantages of reliable elliptic curves-based encryption systems. Security analysis assures the efficiency of the proposed algorithm and validates its robustness and efficiency against possible types of attacks.

scholarly journals Erratum to the paper “Extinction and ergodic stationary distribution of a Markovian-switching prey-predator model with additional food for predator”

2020 ◽  
Vol 15 ◽  
pp. 56
Author(s):  
Xiaoxia Guo ◽  
Dehao Ruan
Keyword(s):  
Stationary Distribution ◽  
Carrying Capacity ◽  
Death Rate ◽  
Sufficient Conditions ◽  
Additional Food ◽  
Predator Prey ◽  
Predator Prey Model ◽  
Predator Model ◽  
White Noises ◽  
Prey Populations

In this work we have studied a stochastic predator-prey model where the prey grows logistically in the absence of predator. All parameters but carrying capacity have been perturbed with telephone noise. The prey’s growth rate and the predator’s death rate have also been perturbed with white noises. Both of these noises have been proved extremely useful to model rapidly fluctuating phenomena Dimentberg (1988). The conditions under which extinction of predator and prey populations occur have been established. We also give sufficient conditions for positive recurrence and the existence of an ergodic stationary distribution of the positive solution, red which in stochastic predator-prey systems means that the predator and prey populations can be persistent, that is to say, the predator and prey populations can be sustain a quantity that is neither too much nor too little. In our analysis, it is found that the environmental noise plays an important role in extinction as well as coexistence of prey and predator populations. It is shown in numerical simulation that larger white noise intensity will lead to the extinction of the population, while telephone noise may delay or reduce the risk of species extinction.

scholarly journals Stability of a fractional order harvested prey–predator model in the existence of infection in prey

2021 ◽  
Vol 1 (1) ◽  
Author(s):  
Subhashis Das ◽  
◽  
Sanat Mahato ◽  
Prasenjit Mahato
Keyword(s):  
Fractional Order ◽  
Equilibrium Points ◽  
Type I ◽  
Interior Equilibrium ◽  
Proposed Model ◽  
Different Types ◽  
Predator Model ◽  
Predictor Corrector ◽  
Uniqueness Criteria ◽  
Type Ii Functional Response

The growing relationship between prey and their predator is one of the important aspects in the field of ecology and mathematical biology. On the other hand, the utility of fractional calculus in different types of mathematical modelling have been applied extensively. In this paper, a fractional order prey–predator model is developed with the consideration of Holling type-I and Holling type-II functional response of the predator. As infection spreads through prey, the prey population is divided into two parts. In addition, we exploit the effect of harvesting to control the excessive spread of the infection. The existence and uniqueness criteria, the boundedness of the solution of the proposed model are investigated. A number of five possible equilibrium points of the proposed model are determined along with the feasibility conditions for each equilibrium points. The local stability at these equilibrium points and global stability at interior equilibrium point are investigated. Numerical simulation is presented with the help of modified Predictor-corrector method in MATLAB software to understand the dynamics of the proposed model.

scholarly journals Dynamical Behavior and Stability Analysis in a Stage-Structured Prey Predator Model with Discrete Delay and Distributed Delay

2014 ◽  
Vol 2014 ◽  
pp. 1-15 ◽  
Author(s):  
Chao Liu ◽  
Qingling Zhang
Keyword(s):  
Stability Analysis ◽  
Hopf Bifurcation ◽  
Local Stability ◽  
Dynamical Behavior ◽  
Distributed Delay ◽  
Stable Limit ◽  
Discrete Delay ◽  
Normal Form Theory ◽  
Interior Equilibrium ◽  
Predator Model

We propose a prey predator model with stage structure for prey. A discrete delay and a distributed delay for predator described by an integral with a strong delay kernel are also considered. Existence of two feasible boundary equilibria and a unique interior equilibrium are analytically investigated. By analyzing associated characteristic equation, local stability analysis of boundary equilibrium and interior equilibrium is discussed, respectively. It reveals that interior equilibrium is locally stable when discrete delay is less than a critical value. According to Hopf bifurcation theorem for functional differential equations, it can be found that model undergoes Hopf bifurcation around the interior equilibrium when local stability switch occurs and corresponding stable limit cycle is observed. Furthermore, directions of Hopf bifurcation and stability of the bifurcating periodic solutions are studied based on normal form theory and center manifold theorem. Numerical simulations are carried out to show consistency with theoretical analysis.

scholarly journals Chaos in a Hybrid Food Chain Model

2021 ◽  
pp. 2362-2368
Author(s):  
Safaa Jawad Ali ◽  
Abed Almohsen Naji Almohasin ◽  
Adwea Naji Atewi ◽  
Raid Kamel Naji ◽  
Norihan Md Arifin
Keyword(s):  
Global Stability ◽  
Food Chain ◽  
Direct Method ◽  
Equilibrium Points ◽  
Chain Model ◽  
Lyapunov Direct Method ◽  
Type Iv ◽  
Controlling Parameter ◽  
Periodic Dynamics ◽  
Food Chain Model

In this paper, chaotic and periodic dynamics in a hybrid food chain system with Holling type IV and Lotka-Volterra responses are discussed. The system is observed to be dissipative. The global stability of the equilibrium points is analyzed using Routh-Hurwitz criterion and Lyapunov direct method. Chaos phenomena is characterized by attractors and bifurcation diagram. The effect of the controlling parameter of the model is investigated theoretically and numerically.

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