Qualitative analysis of Holling type II predator–prey systems with prey refuges and predator restricts

Qiu, Xiaoxiao; Xiao, Haibin

doi:10.1016/j.nonrwa.2013.01.001

Cited by 18 publications

(9 citation statements)

References 20 publications

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“…The parameters have the following ecological meanings: is the intrinsic prey growth rate or biotic potential is the prey environmental carrying capacity > 0 is the strong Allee effect threshold or the minimum of viable prey population is the quantity of prey that can be eaten by a predator in each time unit is the efficiency with which predators convert consumed prey into new predators is the natural death rate of predators is the level of the struggle among predators. Particularly, the term − 2 represents interspecific density-restricted effects on the predators [6,28]. System (6) describes a generalized Gause-type predatorprey model [27], which does not obey the mass action principle [29].…”

Section: The Modelmentioning

confidence: 99%

Competition among Predators and Allee Effect on Prey, Their Influence on a Gause‐Type Predation Model

González‐Olivares

Cabrera-Villegas

Córdova–Lepe

et al. 2019

Mathematical Problems in Engineering

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Interference or competition among predators (CAP) has often been ruled out in depredation models, although there are varied mathematical forms to describe and incorporate it into this interaction. In this work, we present the most known of these descriptions and one of them will be used in a modified Volterra model. Moreover, of this ecological phenomenon, a simple and strong Allee effect affecting the prey population will be considered in the relationship. An important feature of the new model is to have until two positive equilibrium points, to the difference with the Volterra model (without Allee effect); hence different and interesting dynamic situations appear in the system. Conditions for the existence and local stability of equilibria are determined. The boundedness of solutions, the existence of a limit cycle and a separatrix curve are also proven. Besides, the main properties of the model are examined from an ecological point of view. To make a comparative discussion of our results, an Appendix is added with the main properties of models, in which neither the Allee effect nor the competition among predators is considered. Some simulations are shown to endorse our results.

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Section: The Modelmentioning

confidence: 99%

Competition among Predators and Allee Effect on Prey, Their Influence on a Gause‐Type Predation Model

González‐Olivares

Cabrera-Villegas

Córdova–Lepe

et al. 2019

Mathematical Problems in Engineering

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“…Comparing to integer order differential equations FODEs have ability to provide precise description of the modeled mathematical problem. Predator -prey models are more significant in the modeling multi species population interactions and these interactions through integer order models have been studied by many authors [2,5,7]. Xu et al [10], have studied the stability with globality and entire Hopf bifurcation by considering the stage structure for predator.…”

Section: Preliminaries and Model Description 21 Fractional Derivatimentioning

confidence: 99%

“…Dynamical analysis of the systems with time delays is more complex due to the non deterministic polynomial time hard (NP-hard) nature of the stability system [1]. Fractional order system describes behaviour of real physical system more truthfully than the integer order system [4,7].…”

Section: Introductionmentioning

confidence: 99%

Fractional Order Delayed Prey-Predator Model with Stage Structure

Sivakumar¹,

Arts²,

Loganathan³

2018

APAM

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This paper is concerned mainly with the study of fractional order prey-predator system with time delay analyzing stage structure system, the characteristic equation which deals with the local stability is calculated and the existence of Hopf bifurcation is derived using Routh Hurwitz strategies. The parameter in this paper are fractional order and time delay. They deal with oscillatory behaviour of solutions and by Lyapunov global stability, dynamical value of complex system which are reviewed with incommensurate order. Atlast the numerical examples are used to validate the effectiveness of derived analytic results.

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“…The mathematical formalism for analysing different aspects of the evolution of the socio-economic system [27,33,36], including network methods [23], has been intensively developed. Moreover, in the studies of the dynamics of socio-economic systems, the various phenomenological approaches based on physical principles [12,16,22] as well as the mathematical models related to the class of so-called predator-prey models [13,20,34,38,39], have been actively developed and applied. In the listed papers, the Lotka-Volterra model is investigated.…”

Section: Introductionmentioning

confidence: 99%

On the validity of use of physical equations and principles in the socio-economic field and on the predictability of socio-economic system dynamics

Андреев

Moscow²

2015

NAMC

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Abstract. Usually, the various empirical and semi-empirical equations and mathematical models are used to study the dynamics of socio-economic systems. In this case, there are very often questions related to the validity of application of such equations and models.In this paper, it is shown that the dynamics of socio-economic systems can be described by mathematical equations analogous to the motion equations that are well known in physics (particularly in classical mechanics). In this connection, it is possible to say that the predictability of the dynamics of a socio-economic system is described by the motion equations (i.e., Newton's equations) to the same degree that these equations predict the dynamics of the physical system. Such models allow us to determine the trends of development in the dynamics of socio-economic systems. The importance of this investigation in using adequate mathematical models is that they allow us to notice those or other adverse trends in the socio-economic systems, therefore, ensuring timely optimal management decisions.

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Qualitative analysis of Holling type II predator–prey systems with prey refuges and predator restricts

Cited by 18 publications

References 20 publications

Competition among Predators and Allee Effect on Prey, Their Influence on a Gause‐Type Predation Model

Competition among Predators and Allee Effect on Prey, Their Influence on a Gause‐Type Predation Model

Fractional Order Delayed Prey-Predator Model with Stage Structure

On the validity of use of physical equations and principles in the socio-economic field and on the predictability of socio-economic system dynamics

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