Bifurcation of Nontrivial Periodic Solutions for a Food Chain Beddington–DeAngelis Interference Model with Impulsive Effect

2018 ◽  
Vol 28 (11) ◽  
pp. 1850131 ◽  
Author(s):  
Wang Shuai ◽  
Huang Qingdao
Keyword(s):  
Periodic Solution ◽  
Periodic Solutions ◽  
Release Rate ◽  
Food Chain ◽  
Interference Model ◽  
Impulsive Effect ◽  
Asymptotically Stable ◽  
Nontrivial Periodic Solutions

In this paper, a food chain Beddington–DeAngelis interference model with impulsive effect is studied. The trivial periodic solution is locally asymptotically stable if the release rate or the release period is suitable. Conditions for permanence of the model are obtained. The existence of nontrivial periodic solutions and semi-trivial periodic solutions are established when the trivial periodic solution loses its stability under different conditions.

scholarly journals Periodic Solutions of a System of Delay Differential Equations for a Small Delay

2002 ◽  
Vol 7 (2) ◽  
pp. 295
Author(s):  
Adu A.M. Wasike ◽  
Wandera Ogana
Keyword(s):  
Periodic Solution ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Delay Differential Equations ◽  
Original System ◽  
System Of Equations ◽  
Asymptotically Stable ◽  
Stable Periodic Solution ◽  
Stable Periodic ◽  
Delay Differential

We prove the existence of an asymptotically stable periodic solution of a system of delay differential equations with a small time delay t > 0. To achieve this, we transform the system of equations into a system of perturbed ordinary differential equations and then use perturbation results to show the existence of an asymptotically stable periodic solution. This approach is contingent on the fact that the system of equations with t = 0 has a stable limit cycle. We also provide a comparative study of the solutions of the original system and the perturbed system.  This comparison lays the ground for proving the existence of periodic solutions of the original system by Schauder's fixed point theorem.   

ANALYSIS OF A CHEMOSTAT MODEL WITH PULSED INPUT

2006 ◽  
Vol 14 (04) ◽  
pp. 583-598 ◽  
Author(s):  
XIANGYUN SHI ◽  
XINYU SONG
Keyword(s):  
Periodic Solution ◽  
Periodic Solutions ◽  
Continuous System ◽  
Critical Value ◽  
Asymptotically Stable ◽  
Chemostat Model ◽  
Globally Asymptotically Stable

In this paper, we consider a chemostat model with pulsed input. We find a critical value of the period of pulses. If the period is more than the critical value, the microorganism-free periodic solution is globally asymptotically stable. If less, the system is permanent. Moreover, the nutrient and the microorganism can co-exist on a periodic solution of period τ. Finally, by comparing the corresponding continuous system, we find that the periodically pulsed input destroys the equilibria of the continuous system and initiates periodic solutions. Our results are valuable for the manufacture of products by genetically altered organisms.

scholarly journals Dynamics of an Almost Periodic Food Chain System with Impulsive Effects

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Yaqin Li ◽  
Wenquan Wu ◽  
Tianwei Zhang
Keyword(s):  
Periodic Solution ◽  
Differential Equations ◽  
Food Chain ◽  
Impulsive Differential Equations ◽  
Almost Periodic Solution ◽  
Asymptotically Stable ◽  
Almost Periodic ◽  
Chain System ◽  
Positive Almost Periodic Solution ◽  
Impulsive Perturbations

In order to obtain a more accurate description of the ecological system perturbed by human exploitation activities such as planting and harvesting, we need to consider the impulsive differential equations. Therefore, by applying the comparison theorem and the Lyapunov method of the impulsive differential equations, this paper gives some new sufficient conditions for the permanence and existence of a unique uniformly asymptotically stable positive almost periodic solution in a food chain system with almost periodic impulsive perturbations. The method used in this paper provides a possible method to study the permanence and existence of a unique uniformly asymptotically stable positive almost periodic solution of the models with impulsive perturbations in biological populations. Finally, an example and numerical simulations are given to illustrate that our results are feasible.

scholarly journals Dynamics of a Nonautonomous Leslie-Gower Type Food Chain Model with Delays

2011 ◽  
Vol 2011 ◽  
pp. 1-19 ◽  
Author(s):  
Hongying Lu ◽  
Weiguo Wang
Keyword(s):  
Periodic Solution ◽  
Asymptotic Stability ◽  
Food Chain ◽  
Time Delays ◽  
Sufficient Conditions ◽  
Positive Periodic Solution ◽  
Chain Model ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Food Chain Model

A nonautonomous Leslie-Gower type food chain model with time delays is investigated. It is proved the general nonautonomous system is permanent and globally asymptotically stable under some appropriate conditions. Furthermore, if the system is periodic one, some sufficient conditions are established, which guarantee the existence, uniqueness, and global asymptotic stability of a positive periodic solution of the system. The conditions for the permanence, global stability of system, and the existence, uniqueness of positive periodic solution depend on delays; so, time delays are profitless.

A FOOD CHAIN SYSTEM WITH HOLLING IV FUNCTIONAL RESPONSES AND IMPULSIVE EFFECT

2008 ◽  
Vol 01 (03) ◽  
pp. 361-375 ◽  
Author(s):  
ZUOLIANG XIONG ◽  
YING XUE ◽  
SHUNYI LI
Keyword(s):  
Periodic Solution ◽  
Food Chain ◽  
Floquet Theory ◽  
Positive Periodic Solution ◽  
Impulsive Effect ◽  
Functional Responses ◽  
Globally Asymptotically Stable ◽  
Chain System ◽  
Maximum Value ◽  
Periodic Constant

In the paper, according to biological and chemical control strategy for pest control, our main purpose is to construct a three trophic level food chain system with Holling IV functional responses and periodic constant impulsive effect concerning integrated pest management (IPM), and investigate the dynamic behaviors of this system. By using the Floquet theory and comparison theorem of impulsive differential equation and analytic method, we prove that there exists a globally asymptotically stable pest-eradication periodic solution when the impulsive period is less than some critical value. Further, condition for permanence of the system is established. Finally, numerical simulation shows that there exists a stable positive periodic solution with a maximum value no larger than a given level.

Periodic Solutions For ẋ = Ax + G(x, t) + ∊p(t)

1971 ◽  
Vol 14 (4) ◽  
pp. 575-577
Author(s):  
Peter J. Ponzo
Keyword(s):  
Periodic Solution ◽  
Periodic Solutions ◽  
Stable Equilibrium ◽  
Asymptotically Stable ◽  
Constant Matrix ◽  
Scalar Parameter ◽  
Locally Lipschitz ◽  
Image Position

We wish to establish the existence of a periodic solution to1where x, g and p are n-vectors, A is an n × n constant matrix, and ∊ is a small scalar parameter. We assume that g and p are locally Lipschitz in x and continuous and T-periodic in t, and that the origin is a point of asymptotically stable equilibrium, when ∊ = 0.

On nearly-commensurable periods in the restricted problem of three bodies, with calculations of the long-period variations in the interior 2:1 case

1966 ◽  
Vol 25 ◽  
pp. 197-222 ◽  
Author(s):  
P. J. Message
Keyword(s):  
Periodic Solution ◽  
Periodic Solutions ◽  
Large Amplitude ◽  
Restricted Problem ◽  
Small Amplitude ◽  
Minor Planets ◽  
Long Period ◽  
Numerical Integrations ◽  
Period Variations ◽  
The Mean

An analytical discussion of that case of motion in the restricted problem, in which the mean motions of the infinitesimal, and smaller-massed, bodies about the larger one are nearly in the ratio of two small integers displays the existence of a series of periodic solutions which, for commensurabilities of the typep+ 1:p, includes solutions of Poincaré'sdeuxième sortewhen the commensurability is very close, and of thepremière sortewhen it is less close. A linear treatment of the long-period variations of the elements, valid for motions in which the elements remain close to a particular periodic solution of this type, shows the continuity of near-commensurable motion with other motion, and some of the properties of long-period librations of small amplitude.To extend the investigation to other types of motion near commensurability, numerical integrations of the equations for the long-period variations of the elements were carried out for the 2:1 interior case (of which the planet 108 “Hecuba” is an example) to survey those motions in which the eccentricity takes values less than 0·1. An investigation of the effect of the large amplitude perturbations near commensurability on a distribution of minor planets, which is originally uniform over mean motion, shows a “draining off” effect from the vicinity of exact commensurability of a magnitude large enough to account for the observed gap in the distribution at the 2:1 commensurability.

scholarly journals Multiplicity of positive periodic solutions to second order differential equations

2006 ◽  
Vol 73 (2) ◽  
pp. 175-182 ◽  
Author(s):  
Jifeng Chu ◽  
Xiaoning Lin ◽  
Daqing Jiang ◽  
Donal O'Regan ◽  
R. P. Agarwal
Keyword(s):  
Periodic Solution ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Point Theorem ◽  
Positive Periodic Solution ◽  
Second Order ◽  
Positive Periodic Solutions ◽  
Second Order Differential Equations

In this paper, we study the existence of positive periodic solutions to the equation x″ = f (t, x). It is proved that such a equation has more than one positive periodic solution when the nonlinearity changes sign. The proof relies on a fixed point theorem in cones.

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