Permanence for Nonautonomous Differential Systems with Delays in the Linear and Nonlinear Terms

Faria, Teresa

doi:10.3390/math9030263

Cited by 8 publications

(6 citation statements)

References 30 publications

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“…The results are obtained by means of a suitable generalization of the Lyapunov approach together with a Poincarè-type inequality. The proposed criteria generalize and extend some existing results on the dynamical properties of reaction-diffusion epidemic and virus dynamic models [ 9 , 14 , 15 , 16 , 17 , 18 , 19 , 58 , 59 , 60 ] to the impulsive case considering integral manifolds instead of single steady state. The presented results allow for the application of an impulsive control therapeutic strategy to the considered epidemic model.…”

Section: Discussionmentioning

confidence: 52%

“…It is well known that boundedness and permanence are two qualitative properties of a significant importance for models in biology [ 58 , 59 , 60 ]. In this paper, the concepts will be generalized to the integral manifolds method as follows.…”

Section: Model Description and Preliminariesmentioning

confidence: 99%

“…Numerous excellent boundedness and permanence results for different models in biology have been presented in the existing literature. See, for example, [ 58 , 59 , 60 ]. The results in Theorems 2 and 3 generalize such results using the integral manifold approach, and apply to the impulsive epidemic model (2).…”

Section: Existence Boundedness and Permanence Of Integral Manifoldsmentioning

confidence: 99%

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Impulsive Reaction-Diffusion Delayed Models in Biology: Integral Manifolds Approach

Stamov

Stamova

Spirova

2021

Entropy

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In this paper we study an impulsive delayed reaction-diffusion model applied in biology. The introduced model generalizes existing reaction-diffusion delayed epidemic models to the impulsive case. The integral manifolds notion has been introduced to the model under consideration. This notion extends the single state notion and has important applications in the study of multi-stable systems. By means of an extension of the Lyapunov method integral manifolds’ existence, results are established. Based on the Lyapunov functions technique combined with a Poincarè-type inequality qualitative criteria related to boundedness, permanence, and stability of the integral manifolds are also presented. The application of the proposed impulsive control model is closely related to a most important problems in the mathematical biology—the problem of optimal control of epidemic models. The considered impulsive effects can be used by epidemiologists as a very effective therapy control strategy. In addition, since the integral manifolds approach is relevant in various contexts, our results can be applied in the qualitative investigations of many problems in the epidemiology of diverse interest.

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Section: Discussionmentioning

confidence: 52%

Section: Model Description and Preliminariesmentioning

confidence: 99%

Section: Existence Boundedness and Permanence Of Integral Manifoldsmentioning

confidence: 99%

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Impulsive Reaction-Diffusion Delayed Models in Biology: Integral Manifolds Approach

Stamov

Stamova

Spirova

2021

Entropy

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“…Under the above general assumptions and the stronger requirement (4.12), the nonimpulsive version of system (4.1) was proven to be permanent [10]. Moreover, in this setting, a fixed point argument allows to conclude that the sufficient conditions for permanence also imply the existence of a positive periodic solution [8,35].…”

Section: Systems With Bounded Nonlinearitiesmentioning

confidence: 95%

Positive periodic solutions for systems of impulsive delay differential equations

Faria¹,

Figueroa²

2021

Preprint

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A class of periodic differential n-dimensional systems with patch structure with (possibly infinite) delay and nonlinear impulses is considered. These systems incorporate very general nonlinearities and impulses whose signs may vary. Criteria for the existence of at least one positive periodic solution are presented, extending and improving previous ones established for the scalar case. Applications to systems inspired in mathematical biology models, such as impulsive hematopoiesis and Nicholson-type systems, are also included.

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“…It is worth recalling that Nicholson's equation and extensions have been studied by different authors. Earlier work includes: [1], [2], [5], [7], [8], [10], [14], [15], [17], [19] and references therein. A number of cases involving multiple delays, nonlinear coefficients and linear harvesting terms have been also explored.…”

Section: Introductionmentioning

confidence: 99%

On persistence of a Nicholson-type system with multiple delays and nonlinear harvesting

Amster¹,

Bondorevsky²

2021

Preprint

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An N -dimensional generalization of Nicholson's equation is analyzed. We consider a model including multiple delays, nonlinear coefficients and a nonlinear harvesting term. Inspired by previous results in this subject, we obtain sufficient conditions to guarantee strong and uniform persistence. Furthermore, under extra suitable hypotheses we prove the existence of T -periodic solutions and, reversing the prior conditions in a convenient manner, we show that the zero is a global attractor.

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Permanence for Nonautonomous Differential Systems with Delays in the Linear and Nonlinear Terms

Cited by 8 publications

References 30 publications

Impulsive Reaction-Diffusion Delayed Models in Biology: Integral Manifolds Approach

Impulsive Reaction-Diffusion Delayed Models in Biology: Integral Manifolds Approach

Positive periodic solutions for systems of impulsive delay differential equations

On persistence of a Nicholson-type system with multiple delays and nonlinear harvesting

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