scholarly journals Homoclinic tangencies with infinitely many asymptotically stable single-round periodic solutions

2019 ◽  
Vol 0 (0) ◽  
pp. 0-0
Author(s):  
Sishu Shankar Muni ◽  
◽  
Robert I. McLachlan ◽  
David J. W. Simpson
Keyword(s):  
Periodic Solutions ◽  
Asymptotically Stable ◽  
Homoclinic Tangencies

Bifurcation analysis of a phage-bacteria interaction model with prophage induction

2020 ◽  
Author(s):  
H M Ndongmo Teytsa ◽  
B Tsanou ◽  
S Bowong ◽  
J M-S Lubuma
Keyword(s):  
Periodic Solutions ◽  
Basin Of Attraction ◽  
Disease Outbreaks ◽  
Interaction Model ◽  
Life Cycles ◽  
Asymptotically Stable ◽  
Prophage Induction ◽  
Globally Asymptotically Stable ◽  
Offspring Number ◽  
Predator Prey Model

Abstract A predator-prey model is used to investigate the interactions between phages and bacteria by considering the lytic and lysogenic life cycles of phages and the prophage induction. We provide answers to the following conflictual research questions: (1) what are conditions under which the presence of phages can purify a bacterial infected environment? (2) Can the presence of phages triggers virulent bacterial outbreaks? We derive the basic offspring number $\mathcal N_0$ that serves as a threshold and the bifurcation parameter to study the dynamics and bifurcation of the system. The model exhibits three equilibria: an unstable environment-free equilibrium, a globally asymptotically stable (GAS) phage-free equilibrium (PFE) whenever $\mathcal N_0<1$, and a locally asymptotically stable environment-persistent equilibrium (EPE) when $\mathcal N_0>1$. The Lyapunov–LaSalle techniques are used to prove the GAS of the PFE and estimate the EPE basin of attraction. Through the center manifold approximation, topological types of the PFE are precised. Existence of transcritical and Hopf bifurcations are established. Precisely, when $\mathcal N_0>1$, the EPE loses its stability and periodic solutions arise. Furthermore, increasing $\mathcal N_0$ can purify an environment where bacteriophages are introduced. Purposely, we prove that for large values of $\mathcal N_0$, the overall bacterial population asymptotically approaches zero, while the phage population sustains. Ecologically, our results show that for small values of $\mathcal N_0$, the existence of periodic solutions could explain the occurrence of repetitive bacteria-borne disease outbreaks, while large value of $\mathcal N_0$ clears bacteria from the environment. Numerical simulations support our theoretical results.

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.   

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.

Freely-Moving Delay Induces Periodic Oscillations in a Structured SEIR Model

2017 ◽  
Vol 27 (08) ◽  
pp. 1750122 ◽  
Author(s):  
Yanfei Du ◽  
Yuxiao Guo ◽  
Peng Xiao
Keyword(s):  
Periodic Solutions ◽  
Disease Transmission ◽  
Time Lag ◽  
Stage Structure ◽  
Transmission Model ◽  
Positive Equilibrium ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Normal Form Theory ◽  
Freely Moving

In this paper, a disease transmission model of SEIR type with stage structure is proposed and studied. Two kinds of time delays are considered: the first one is the mature delay which divides the population into two stages; the second one is the time lag between birth and being able to move freely, which we call the freely-moving delay. Our mathematical analysis establishes that the global dynamics are determined by the basic reproduction number [Formula: see text]. If [Formula: see text], then the disease free equilibrium [Formula: see text] is globally asymptotically stable, and the disease will die out. If [Formula: see text], then a unique positive equilibrium [Formula: see text] exists, and [Formula: see text] is locally asymptotically stable when the freely-moving delay is less than the critical value. We show that increasing this delay can destabilize [Formula: see text] and lead to Hopf bifurcations and stable periodic solutions. By using the normal form theory and the center manifold theory, we derive the formulae for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions. Finally, some numerical simulations are carried out to verify the theoretical analysis and some biological implications are discussed.

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.

Multistability in Spiking Neuron Models of Delayed Recurrent Inhibitory Loops

2007 ◽  
Vol 19 (8) ◽  
pp. 2124-2148 ◽  
Author(s):  
Jianfu Ma ◽  
Jianhong Wu
Keyword(s):  
Periodic Solutions ◽  
Asymptotically Stable ◽  
Spiking Neuron ◽  
Phase Resetting ◽  
Neuron Models ◽  
Absolute Refractory Period ◽  
Integrate And Fire ◽  
Stable Periodic ◽  
Spiking Neuron Models ◽  
Integrate And Fire Neuron

We consider the effect of the effective timing of a delayed feedback on the excitatory neuron in a recurrent inhibitory loop, when biological realities of firing and absolute refractory period are incorporated into a phenomenological spiking linear or quadratic integrate-and-fire neuron model. We show that such models are capable of generating a large number of asymptotically stable periodic solutions with predictable patterns of oscillations. We observe that the number of fixed points of the so-called phase resetting map coincides with the number of distinct periods of all stable periodic solutions rather than the number of stable patterns. We demonstrate how configurational information corresponding to these distinct periods can be explored to calculate and predict the number of stable patterns.

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