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.

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.

scholarly journals Chaos in Ecology: The Topological Entropy of a Tritrophic Food Chain Model

2008 ◽  
Vol 2008 ◽  
pp. 1-12 ◽  
Author(s):  
Jorge Duarte ◽  
Cristina Januário ◽  
Nuno Martins
Keyword(s):  
Topological Entropy ◽  
Food Chain ◽  
Chaotic Dynamics ◽  
Life Sciences ◽  
Chain Model ◽  
Parameter Region ◽  
Complex Interactions ◽  
Return Maps ◽  
Food Chain Model ◽  
Specific Food

An ecosystem is a web of complex interactions among species. With the purpose of understanding this complexity, it is necessary to study basic food chain dynamics with preys, predators and superpredators interactions. Although there is an elegant interpretation of ecological models in terms of chaos theory, the complex behavior of chaotic food chain systems is not completely understood. In the present work we study a specific food chain model from the literature. Using results from symbolic dynamics, we characterize the topological entropy of a family of logistic-like Poincaré return maps that replicates salient aspects of the dynamics of the model. The analysis of the variation of this numerical invariant, in some realistic system parameter region, allows us to quantify and to distinguish different chaotic regimes. This work is still another illustration of the role that the theory of dynamical systems can play in the study of chaotic dynamics in life sciences.

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 Dynamics of a three species ratio-dependent food chain model with diffusion and double free boundaries

2020 ◽  
Vol 15 ◽  
pp. 62
Author(s):  
Dawei Zhang ◽  
Beiping Duan ◽  
Binxiang Dai
Keyword(s):  
Food Chain ◽  
Dimensional Space ◽  
Chain Model ◽  
Time Behavior ◽  
Free Boundaries ◽  
Top Predator ◽  
Predator Species ◽  
Spreading And Vanishing ◽  
Ratio Dependent ◽  
Food Chain Model

This paper focuses on the dynamics of a three species ratio-dependent food chain model with diffusion and double free boundaries in one dimensional space, in which the free boundaries represent expanding fronts of top predator species. The existence, uniqueness and estimates of the global solution are discussed firstly. Then we prove a spreading–vanishing dichotomy, specifically, the top predator species either successfully spreads to the entire space as time t goes to infinity and survives in the new environment, or fails to establish and dies out in the long run. The long time behavior of the three species and criteria for spreading and vanishing are also obtained. Besides, our simulations illustrate the impacts of initial occupying area and expanding capability on the dynamics of top predator for free boundaries.

scholarly journals Global Stability for a Three-Species Food Chain Model in a Patchy Environment

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Hongli Li ◽  
Yaolin Jiang ◽  
Long Zhang ◽  
Zhidong Teng
Keyword(s):  
Food Chain ◽  
Sufficient Conditions ◽  
Coupled Systems ◽  
Positive Equilibrium ◽  
Chain Model ◽  
Patchy Environment ◽  
Globally Asymptotically Stable ◽  
Predator Species ◽  
Graph Theoretical Approach ◽  
Food Chain Model

We investigate a three-species food chain model in a patchy environment where prey species, mid-level predator species, and top predator species can disperse amongndifferent patches(n≥2). By using the method of constructing Lyapunov functions based on graph-theoretical approach for coupled systems, we derive sufficient conditions under which the positive equilibrium of this model is unique and globally asymptotically stable if it exists.

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 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.

Effects of Additional Food on the Dynamics of a Three Species Food Chain Model Incorporating Refuge and Harvesting

2019 ◽  
Vol 20 (7-8) ◽  
pp. 787-801
Author(s):  
Prabir Panja ◽  
Soovoojeet Jana ◽  
Shyamal Kumar Mondal
Keyword(s):  
Growth Rate ◽  
Food Chain ◽  
Carrying Capacity ◽  
Rate Increase ◽  
Optimal Harvesting ◽  
Chain Model ◽  
Predator Population ◽  
Intrinsic Growth Rate ◽  
Additional Food ◽  
Food Chain Model

AbstractIn this paper, a three species food chain model has been developed among the interaction of prey, predator and super predator. It is assumed that the predator shows refuge behavior to the super predator. It is also assumed that a certain amount of additional food will be supplied to the super predator. It is considered that the predator population is benefiting partially from the additional food. To get optimal harvesting of super predator the Pontryagin’s maximum principle has been used. It is found that super predator may be extinct if harvesting rate increase. It is observed that as the refuge rate increases, predator population gradually increases, but super predator population decreases. Also, it is found that our proposed system undergoes oscillatory or periodic behavior as the value of refuge rate (m1), harvesting rate (E), the intrinsic growth rate of prey (r), carrying capacity of prey (k) and conservation rate of prey (c1) varies for some certain range of these parameters. It is found that this study may be useful for the increase of harvesting of a super predator by supplying the additional food to our proposed system.

scholarly journals Pattern Formation in Spatially Extended Tritrophic Food Chain Model Systems: Generalist versus Specialist Top Predator

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Nitu Kumari
Keyword(s):  
Pattern Formation ◽  
Food Chain ◽  
Model System ◽  
Model Systems ◽  
Subsurface Water ◽  
Chain Model ◽  
Top Predator ◽  
Top Predators ◽  
Spatially Extended ◽  
Food Chain Model

The complex dynamics of two types of tritrophic food chain model systems when the species undergo spatial movements, modeling two real situations of marine ecosystem, are investigated in this study analytically and using numerical simulations. The study has been carried out with the objective to explore and compare the competitive effects of fish and molluscs species being the top predators, when phytoplankton and zooplankton species are undergoing spatial movements in the subsurface water. Reaction diffusion systems have been used to represent temporal evolution and spatial interaction among the species. The two model systems differ in an essential way that the top predators are generalist and specialist, respectively, in two models. “Wave of Chaos” mechanism is found to be the responsible factor for the pattern (non-Turing) formation in one dimension seen in the food chain ending with top generalist predator. In the present work we have reported WOC phenomenon, for the first time in the literature, in a three-species spatially extended food chain model system. The numerical simulation leads to spontaneous and interesting pattern formation in two dimensions. Constraints on different parameters under which Turing and non-Turing patterns may be observed are obtained analytically. Diffusion-driven analysis is carried out, and the effect of diffusion on the chaotic dynamics of the model systems is studied. The existence of chaotic attractor and long-term chaotic behavior demonstrate the effect of diffusion on the dynamics of the model systems. It is observed from numerical study that food chain model system with top predator as generalist has very rich dynamics and shows very interesting patterns. An ecosystem having top predator as specialist leads to the stability of the system.

DYNAMIC RELATIONSHIP BETWEEN THE MUTUAL INTERFERENCE AND GESTATION DELAYS OF A HYBRID TRITROPHIC FOOD CHAIN MODEL

2018 ◽  
Vol 59 (3) ◽  
pp. 370-401 ◽  
Author(s):  
RASHMI AGRAWAL ◽  
DEBALDEV JANA ◽  
RANJIT KUMAR UPADHYAY ◽  
V. SREE HARI RAO
Keyword(s):  
Functional Response ◽  
Food Chain ◽  
Model System ◽  
Mutual Interference ◽  
Model Systems ◽  
Chain Model ◽  
Top Predator ◽  
Functional Responses ◽  
Gestation Delay ◽  
Food Chain Model

We have proposed a three-species hybrid food chain model with multiple time delays. The interaction between the prey and the middle predator follows Holling type (HT) II functional response, while the interaction between the top predator and its only food, the middle predator, is taken as a general functional response with the mutual interference schemes, such as Crowley–Martin (CM), Beddington–DeAngelis (BD) and Hassell–Varley (HV) functional responses. We analyse the model system which employs HT II and CM functional responses, and discuss the local and global stability analyses of the coexisting equilibrium solution. The effect of gestation delay on both the middle and top predator has been studied. The dynamics of model systems are affected by both factors: gestation delay and the form of functional responses considered. The theoretical results are supported by appropriate numerical simulations, and bifurcation diagrams are obtained for biologically feasible parameter values. It is interesting from the application point of view to show how an individual delay changes the dynamics of the model system depending on the form of functional response.

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