Complex dynamics of sexually reproductive generalist predator and gestation delay in a food chain model: double Hopf-bifurcation to Chaos

Agrawal, Rashmi; Jana, Dipak Kumar; Upadhyay, Ranjit Kumar; Rao, V. Sree Hari

doi:10.1007/s12190-016-1048-1

Cited by 16 publications

(14 citation statements)

References 44 publications

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“…To compare the computed parameter values and periodic orbits on the branch lpcbr with the predictor, we again use the function C1branch_from_C2point, but with the additional argument predictor set to 1…”

Section: S19 Calculate Predicted Periodic Orbitsmentioning

confidence: 99%

“…We use the function C1branch_from_C2point to start to continue the branches emanating from the Hopf-Hopf point. (1) , suc ]= br_contn ( funcs , hbr (1) , nop ) ; assert ( suc >0) hbr (1) = br_rvers ( hbr (1) ) ; [ hbr (1) , suc ]= br_contn ( funcs , hbr (1) , nop ) ; assert ( suc >0) nop =10; [ hbr (2) , suc ]= br_contn ( funcs , hbr (2) , nop ) ; assert ( suc >0) hbr (2) = br_rvers ( hbr (2) ) ; [ hbr (2) , suc ]= br_contn ( funcs , hbr (2) , nop ) ; assert ( suc >0)…”

Section: S34 Continuing Hopf and Neimark-sacker Bifurcation Curvesmentioning

confidence: 99%

“…point ( i ) . profile (2 ,:) , ' Color ' , cm (1 ,:) ) ; end for i =1:12 plot3 ( genpars ( nsbr_pred (1) ,i ) ,... nsbr_pred (1) . point ( i ) .…”

Section: S48 Plot Comparing Computed and Predicted Periodic Orbitsmentioning

confidence: 99%

“…Great interest has recently been shown in the analysis of degenerate Hopf bifurcations in delay differential equations (DDEs), see e.g. [1,17,18,25,35,36,39,40,48,50,51,54,57,61,64,65,66]. In the simplest case, often encountered in applications, such DDEs have the forṁ…”

Section: Introductionmentioning

confidence: 99%

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Switching to Nonhyperbolic Cycles from Codimension Two Bifurcations of Equilibria of Delay Differential Equations

Bosschaert¹,

Janssens²,

Kuznetsov³

2020

SIAM J. Appl. Dyn. Syst.

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In this paper we perform the parameter-dependent center manifold reduction near the generalized Hopf (Bautin), fold-Hopf, Hopf-Hopf and transcritical-Hopf bifurcations in delay differential equations (DDEs). This allows us to initialize the continuation of codimension one equilibria and cycle bifurcations emanating from these codimension two bifurcation points. The normal form coefficients are derived in the functional analytic perturbation framework for dual semigroups (sun-star calculus) using a normalization technique based on the Fredholm alternative. The obtained expressions give explicit formulas which have been implemented in the freely available numerical software package DDE-BifTool. While our theoretical results are proven to apply more generally, the software implementation and examples focus on DDEs with finitely many discrete delays. Together with the continuation capabilities of DDE-BifTool, this provides a powerful tool to study the dynamics near equilibria of such DDEs. The effectiveness is demonstrated on various models.

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Section: S19 Calculate Predicted Periodic Orbitsmentioning

confidence: 99%

Section: S34 Continuing Hopf and Neimark-sacker Bifurcation Curvesmentioning

confidence: 99%

“…point ( i ) . profile (2 ,:) , ' Color ' , cm (1 ,:) ) ; end for i =1:12 plot3 ( genpars ( nsbr_pred (1) ,i ) ,... nsbr_pred (1) . point ( i ) .…”

Section: S48 Plot Comparing Computed and Predicted Periodic Orbitsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Switching to Nonhyperbolic Cycles from Codimension Two Bifurcations of Equilibria of Delay Differential Equations

Bosschaert¹,

Janssens²,

Kuznetsov³

2020

SIAM J. Appl. Dyn. Syst.

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“…Yu [14] studied the global stability of a modified LG model with Beddington-DeAngelis type functional response. Recently, Agrawal et al [15] investigated the occurence of double Hopf bifurcation at positive equilibrium point when they choose appropriate measure of the tolerance of the prey. Furthermore, some dynamic behaviors, such as stability switches, chaos, bifurcation and double Hopf bifurcation scenarios are observed using numerical simulations.…”

Section: Introductionmentioning

confidence: 99%

Bifurcation analysis of a modified Leslie–Gower model with Holling type-IV functional response and nonlinear prey harvesting

2018

Self Cite

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In this work, an attempt is made to understand the dynamics of a modified Leslie-Gower model with nonlinear harvesting and Holling type-IV functional response. We study the model system using qualitative analysis, bifurcation theory and singular optimal control. We show that the interior equilibrium point is locally asymptotically stable and the system under goes a Hopf bifurcation with respect to the ratio of intrinsic growth of the predator and prey population as bifurcation parameter. The existence of bionomic equilibria is analyzed and the singular optimal control strategy is characterized using Pontryagin's maximum principle. The existence of limit cycles appearing through local Hopf bifurcation and its stability is also examined and validated numerically by computing the first Lyapunov number. Optimal singular equilibrium points are obtained numerically for various discount rates.

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Complex dynamics induced by multiple gestation delays in a Leslie–Gower‐type system with competitive interference

Ojha

Thakur

2023

Math Methods in App Sciences

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The dynamics of food consumption by higher trophic level is complicated as its availability, choice, and consequently their development. The top predator interference has a strong influence on the dynamic structure of the tritrophic food chain system. The proposed article deals with the characteristic features of a three‐tire modified Leslie–Gower model with a one‐prey two‐predator system with the assumption of multiple gestation delays. The intermediate predator of the food chain is taken as a specialist type, whereas the growth of top predator is assumed by sexual reproduction. Two different response functions named Monod–Haldane (M‐H) and Beddington–DeAngelis (B‐D) are considered for the modeling of system dynamics. The population densities of intermediate and top predators are assumed to be affected by two different gestation delays. The boundedness and positive equilibria and their stability conditions are examined theoretically for the delay‐free system. The local and global stability conditions of the delayed system are also determined. With the help of normal form theory and central manifold arguments, the properties of bifurcating periodic solutions are carried out. The numerical computation is performed to validate our analytical findings that show the different dynamical outcomes such as periodic and chaotic dynamics. The Hopf‐bifurcation scenario for different parameters of the model system is well studied. Further, the stability behavior of multiple delays for the different cases is investigated. The system shows chaotic behavior when the gestation delay of intermediate predators is large enough. It is also observed that the top predator density is highly fluctuating through the creation of chaos by the inclusion of time delay. Finally, the time delay destabilizing effect is noticed for three‐way interactions among prey, intermediate predator, and top predator, highlighting its significance in chaotic dynamics.

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Complex dynamics of sexually reproductive generalist predator and gestation delay in a food chain model: double Hopf-bifurcation to Chaos

Cited by 16 publications

References 44 publications

Switching to Nonhyperbolic Cycles from Codimension Two Bifurcations of Equilibria of Delay Differential Equations

Switching to Nonhyperbolic Cycles from Codimension Two Bifurcations of Equilibria of Delay Differential Equations

Bifurcation analysis of a modified Leslie–Gower model with Holling type-IV functional response and nonlinear prey harvesting

Complex dynamics induced by multiple gestation delays in a Leslie–Gower‐type system with competitive interference

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