One and three limit cycles in a cubic predator–prey system

Huang, Xiaolin; Wang, Yuanming; Zhu, Linsheng

doi:10.1002/mma.791

Cited by 14 publications

(11 citation statements)

References 15 publications

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“…The other equilibrium point(s) are defined by the system of Equations (7) and (8) Thus, consider the roots of the Cubic (9). Note that it can be assumed that 0 > c , since all the parameters of the System (4) are assumed to be positive.…”

Section: Non-trivial Equilibrium Pointsmentioning

confidence: 99%

A Predator-prey Model with Predator Population Saturation

Hoff¹,

Fay²

2016

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In this article, a new predator-prey model having predator saturation is proposed. The model resembles a classical Rosenzweig-MacArthur type model, but comes with an added function, the population saturation function of the predator. This function of the predator population is a factor in the predator fertility term in the model. Consequently the model behaves better than the Rosenzweig-MacArthur model since all solutions are bounded within the population quadrant. An invariant region arises where the Poincaré-Bendixon theorem can be applied. In most cases there is but a single critical value, either an attracting spiral point suggesting a stable population pair or an unstable node, resulting in a unique limit cycle. This model is fully described and an analysis of the stability of critical values is provided. The robustness of the model is demonstrated based on the classification of Gunawardena [8].

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Section: Non-trivial Equilibrium Pointsmentioning

confidence: 99%

A Predator-prey Model with Predator Population Saturation

Hoff¹,

Fay²

2016

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“…Various generalized predator-prey models that involve quadratic functions which exhibit logistic behaviour [1][2][3], cubic functions which show different rates of reproduction [4,5], Holling type II functions which state constant consumption [6,7] and Beddington-DeAngelis functions which indicate mutual interference and extinction [8,9] have been shown to overcome some of the biological problems of the original Lotka-Volterra model [10,11]. Population carrying capacities are introduced into these generalizations by adding the proposed functions to the self-interaction and the coupling terms [12].…”

Section: Introductionmentioning

confidence: 99%

Stability and Hopf Bifurcation in Three-Dimensional Predator-Prey Models with Allee Effect

Aybar

2019

Sakarya University Journal of Science

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In this study, we perform the stability and Hopf bifurcation analysis for two population models with Allee effect. The population models within the scope of this study are the one prey-two predator model with Allee growth in the prey and the two prey-one predator model with Allee growth in the preys. Our procedure for investigating each model is as follows. First, we investigate the singular points where the system is stable. We provide the necessary parameter conditions for the system to be stable at the singular points. Then, we look for Hopf bifurcation at each singular point where a family of limit cycles cycle or oscillate. We provide the parameter conditions for Hopf bifurcation to occur. We apply the algebraic invariants method to fully examine the system. We investigate the algebraic properties of the system by finding all algebraic invariants of degree two and three. We give the conditions for the system to have a first integral.

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“…Another extended formation of logistic source term is the cubic source term = ( 1 + 2 − 3 2 ), where 1 ≥ 0 is the intrinsic growth rate, the sign of 2 is undetermined, 3 > 0 is a positive constant, and 2 − 3 2 is the density restriction term (see [10,11] for more information and references). Recently, Cao and Fu in [11] studied global existence and convergence of solutions to a cross-diffusion cubic predatorprey system with stage structure for the prey.…”

Section: Introductionmentioning

confidence: 99%

A Mathematical Characterization for Patterns of a Keller-Segel Model with a Cubic Source Term

Liu²

2013

Advances in Mathematical Physics

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This paper deals with a Neumann boundary value problem for a Keller-Segel model with a cubic source term in ad-dimensional box(d=1,2,3), which describes the movement of cells in response to the presence of a chemical signal substance. It is proved that, given any general perturbation of magnitudeδ, its nonlinear evolution is dominated by the corresponding linear dynamics along a finite number of fixed fastest growing modes, over a time period of the order ln(1/δ). Each initial perturbation certainly can behave drastically differently from another, which gives rise to the richness of patterns. Our results provide a mathematical description for early pattern formation in the model.

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One and three limit cycles in a cubic predator–prey system

Cited by 14 publications

References 15 publications

A Predator-prey Model with Predator Population Saturation

A Predator-prey Model with Predator Population Saturation

Stability and Hopf Bifurcation in Three-Dimensional Predator-Prey Models with Allee Effect

A Mathematical Characterization for Patterns of a Keller-Segel Model with a Cubic Source Term

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