Analysis of a class of distributed delay logistic differential equations

Rasmussen, H.; Wake, G. C.; Donaldson, J. D.

doi:10.1016/s0895-7177(03)90010-0

Cited by 14 publications

(6 citation statements)

References 2 publications

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“…In some applications, the delay is distributed over some past time period [81], dufalse(tfalse)dt=f(u(t),∫t−τ2t−τ1g(u(τ),τ) normaldτ,t),1emt∈false(0,Tfalse], with ufalse(tfalse)=hfalse(tfalse),1emt≤0. It should be noted that the discrete DDEs can be written as a special case of distributed DDEs using dirac-delta functions. We can approximate the two functions f and g using two different NNs, and rewrite the above equations (3.10) as our new coupled discrete DDEs, dufalse(tfalse)dt= fNNfalse(ufalse(tfalse),yfalse(tfalse),t;θfalse),1emt∈false(0,Tfalse]dyfalse(tfalse)dt=gNN...…”

Section: Theory and Methodologymentioning

confidence: 99%

Neural closure models for dynamical systems

Gupta

Lermusiaux

2021

Proc. R. Soc. A.

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Complex dynamical systems are used for predictions in many domains. Because of computational costs, models are truncated, coarsened or aggregated. As the neglected and unresolved terms become important, the utility of model predictions diminishes. We develop a novel, versatile and rigorous methodology to learn non-Markovian closure parametrizations for known-physics/low-fidelity models using data from high-fidelity simulations. The new neural closure models augment low-fidelity models with neural delay differential equations (nDDEs), motivated by the Mori–Zwanzig formulation and the inherent delays in complex dynamical systems. We demonstrate that neural closures efficiently account for truncated modes in reduced-order-models, capture the effects of subgrid-scale processes in coarse models and augment the simplification of complex biological and physical–biogeochemical models. We find that using non-Markovian over Markovian closures improves long-term prediction accuracy and requires smaller networks. We derive adjoint equations and network architectures needed to efficiently implement the new discrete and distributed nDDEs, for any time-integration schemes and allowing non-uniformly spaced temporal training data. The performance of discrete over distributed delays in closure models is explained using information theory, and we find an optimal amount of past information for a specified architecture. Finally, we analyse computational complexity and explain the limited additional cost due to neural closure models.

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Section: Theory and Methodologymentioning

confidence: 99%

Neural closure models for dynamical systems

Gupta

Lermusiaux

2021

Proc. R. Soc. A.

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“…Otherwise, the last few decades have seen an increase in using equations with distributed delays to gain insights into real-world phenomena [20,24,26,47]. Even though the interior properties of such models are difficult to analyse, many researchers have contributed on this point [7,8,15,42,44].…”

Section: Introductionmentioning

confidence: 99%

Analysis of Schistosomiasis Global Dynamics with General Incidence Functions and Two Delays

Koutou

Traoré

Sangaré

2021

Int. J. Appl. Comput. Math

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As most communicable diseases, schistosomiasis transmission mechanism involves some delay due to the incubation period. In this study, we seek to investigate the impact of incubation period on schistosomiasis global transmission dynamics. For that, starting from our previous work and using delay differential equations, we have proposed a more sophisticated model in which the new feature we accounted is the latency period in both human and snail hosts. Then, under some suitable assumptions, the mathematical analysis of the model has been done. Specifically, we calculated the basic reproduction number and showed its dependence on the delays in both humans and snails as follows: when the delays in humans and snails increase, R 0 decreases. We also proved that when R 0 < 1, the no-disease point is globally asymptotically stable and when R 0 > 1, the system is uniformly persistent and admits a unique endemic equilibrium. Besides, by formulating a suitable Lyapunov function, we established that the unique endemic steady state of the model is globally asymptotically stable when the threshold parameter R 0 exceeds 1. Furthermore, sensitivity analysis has been carried out to show how parameters variations affect the model global behaviour. We finished by illustrating some numerical results that are well in keeping with our theoretical findings.

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“…The effect of the distributed delay on the stability of the model has been investigated by Shi et al (2013). In last few decades, several investigators have proposed and analyzed mathematical models considering distributed delays in the growth dynamics of biological species (Arino et al 2007;He and Ruan 1998;Bernard et al 2001;Rasmusse et al 2003;Ma and Takeuchi 1998;Gakkhar and Sahani 2009;Khare et al 2011). Almost no studies have been conducted to investigate the modelling effect of toxicant in a three-species food-chain system incorporating delay in toxicant uptake process by prey.…”

Section: Introductionmentioning

confidence: 99%

Modelling effect of toxicant in a three-species food-chain system incorporating delay in toxicant uptake process by prey

Misra

Babu

2016

Model. Earth Syst. Environ.

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In this paper, a mathematical model is proposed and analyzed to study the effect of toxicant in a threespecies food-chain system incorporating delay in toxicant uptake process by prey population. The model is formulated by using the system of non linear ordinary differential equations. In the model, it is assumed that the growth rate of prey population is affected by organismal toxicant. In this paper, we have introduced distributed delay in the environmental toxicant in the model. The distributed delay differential equations, though simple in structure, possess a rich array of solutions. The models are being analyzed by using variational matrix and Liapunov functions. The conditions for local and global stability of the equilibrium points are obtained. A region of attraction is being found for global asymptotic stability of the equilibrium points. Also, a Hopf bifurcation analysis has been performed with respect to key parameters for non-trivial equilibrium points. Furthermore, we support our analytical findings with numerical simulations.

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Analysis of a class of distributed delay logistic differential equations

Cited by 14 publications

References 2 publications

Neural closure models for dynamical systems

Neural closure models for dynamical systems

Analysis of Schistosomiasis Global Dynamics with General Incidence Functions and Two Delays

Modelling effect of toxicant in a three-species food-chain system incorporating delay in toxicant uptake process by prey

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