Stability and bifurcation analysis of a stage structured predator prey model with time delay

Kar, Tapan Kumar; Jana, Soovoojeet

doi:10.1016/j.amc.2012.10.007

Cited by 29 publications

(10 citation statements)

References 29 publications

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“…Further studies show that the stage structures for both predator and prey should be taken into consideration in modelling [9]. Some interesting results on the dynamical behaviors of predator-prey systems can be found in [10][11][12][13][14][15].…”

Section: Introductionmentioning

confidence: 99%

“…Remark 2 Since we consider a three-species-food-chain model, the dynamical behaviors are more complicated and the system has more equilibria than those in [4,10,12]. Although these conditions of (C2), (C3) and (C4) seem to be intricate, take Case I (please see the section of Numerical simulation (Sect.…”

Section: Existence Of Equilibriamentioning

confidence: 99%

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Dynamical behaviors of a food-chain model with stage structure and time delays

et al. 2018

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Incorporating two delays (τ 1 represents the maturity of predator, τ 2 represents the maturity of top predator), we establish a novel delayed three-species food-chain model with stage structure in this paper. By analyzing the characteristic equations, constructing a suitable Lyapunov functional, using Lyapunov-LaSalle's principle, the comparison theorem and iterative technique, we investigate the existence of nonnegative equilibria and their stability. Some interesting findings show that the delays have great impacts on dynamical behaviors for the system: on one hand, if τ 1 ∈ (m 1 , m 2 ) and τ 2 ∈ (m 4 , +∞), then the boundary equilibrium E 2 (x 0 , y 0 1 , y 0 2 , 0, 0) is asymptotically stable (AS), i.e., the prey species and the predator species will coexist, the top-predator species will go extinct; on the other hand, if τ 1 ∈ (m 2 , +∞), then the axial equilibrium E 1 (k, 0, 0, 0, 0) is AS, i.e., all predators will go extinct. Numerical simulations are great well agreement with the theoretical results.

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“…The modeling of predator-prey populations is an intensive research area where a wide variety of models are proposed to describe and to analyze different mechanisms such as control of populations, oscillations of populations, delay maturation time effects, and pattern formation in predator-prey models with diffusion: see, for example, [1][2][3][4][5][6][7][8][9][10]. One relevant feature in these models is the predator-prey coexistence, which is of vital importance to obtain realistic models of predatorprey interactions.…”

Section: Introductionmentioning

confidence: 99%

Birth Rate Effects on an Age-Structured Predator-Prey Model with Cannibalism in the Prey

Solis

Ku-Carrillo²

2015

Abstract and Applied Analysis

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We develop a family of predator-prey models with age structure and cannibalism in the prey population. It consists of systems of ordinary differential equations, where is a parameter associated with new proposed prey birth rates. We discuss how these new birth rates give the required flexibility to produce differential systems with well-behaved solutions. The main feature required in these models is the coexistence among the involved species, which translates mathematically into stable equilibria and periodic solutions. The search for such characteristics is based on heuristic predation functions that account for cannibalism in the prey.

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“…In [20], the stability and bifurcation analysis of a stage structured predator-prey model with time delay were investigated. Yongzhen et al [34] formulated a three-species ecosystem with a delay digestion process and Holling functional response, the dynamical behaviors of the system are studied.…”

Section: Introductionmentioning

confidence: 99%

Permanence and partial extinction in a delayed three-species food chain model with stage structure and time-varying coefficients

Xi¹,

Huang²,

Qiao³

et al. 2017

J. Nonlinear Sci. Appl.

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By taking full consideration of maturity (τ 1 represents the maturity of predator and τ 2 represents the maturity of top predator) and the effects of environmental parameters, a new delayed three-species food chain model with stage structure and time-varying coefficients is established. With the help of the comparison theorem and the technique of mathematical analysis, the positivity and boundedness of solutions of the model are investigated. Furthermore, some sufficient conditions on the permanence and partial extinction of the system are derived. Some interesting findings show that the delays have great impacts on the permanence for the system. More precisely, if τ 2 ∈ (n, +∞), then the system is partially extinct: on one hand, if τ 1 ∈ (0, n 1 ) and τ 2 ∈ (n, +∞), then the prey and predator species will coexist, i.e., both the prey and predator species are always permanent, yet the top predator species will go extinct eventually. On the other hand, if τ 1 ∈ (n 4 , +∞) and τ 2 ∈ (n, +∞), where n 4 is greater than n 1 , then all predator species will become extinct eventually. Numerical simulations are great well agreement with the theoretical results.

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Stability and bifurcation analysis of a stage structured predator prey model with time delay

Cited by 29 publications

References 29 publications

Dynamical behaviors of a food-chain model with stage structure and time delays

Birth Rate Effects on an Age-Structured Predator-Prey Model with Cannibalism in the Prey

Permanence and partial extinction in a delayed three-species food chain model with stage structure and time-varying coefficients

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