Memory effect on Bazykin’s prey-predator model: Stability and bifurcation analysis

Ghosh, Uttam; Pal, Swadesh; Banerjee, Malay

doi:10.1016/j.chaos.2020.110531

Cited by 41 publications

(12 citation statements)

References 37 publications

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“…For any time t > 0, the concentration of the substances depends on the reaction terms on the righthand side of the equation ( 1). This is known as the Markovian process, but in reality, these concentrations not only depend on the time instance t; rather, they depend on some weighted average concentrations of past time interval, say [t p , t] for t p < t. This is usually termed collectively as the memory effect (e.g., [49,50,51,53,55]). The distribution of the weights is proportional to some power of the length of the time interval, i.e., (t − t p ), following the power law correlation function [48,51].…”

Section: Model Formulationmentioning

confidence: 99%

“…A time-fractional reaction-diffusion equation is used to analyze such processes in the AD context, which we call collectively the "memory effect". This type of memory effect has been seen in the disease progression [51,54,55]. In this work, we construct a time-fractional network model to describe the AD progression in the brain connectome and analyze the damage dynamics concerning the memory effect.…”

Section: Introductionmentioning

confidence: 99%

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Non-Markovian behaviour and the dual role of astrocytes in Alzheimer's disease development and propagation

Pal¹,

Melnik²

2022

Preprint

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Alzheimer's disease (AD) is a common neurodegenerative disorder nowadays. Amyloid-beta (Aβ) and tau proteins are among the main contributor to the development or propagation of AD. In AD, Aβ proteins clump together to form plaques and disrupt cell functions. On the other hand, the abnormal chemical change in the brain helps to build sticky tau tangles that block the neuron's transport system. Astrocytes generally maintain a healthy balance in the brain by clearing the Aβ plaques (toxic Aβ). But, over-activated astrocytes release chemokines and cytokines in the presence of Aβ and also react to pro-inflammatory cytokines further increasing the production of Aβ. In this paper, we construct a mathematical model that can capture astrocytes' dual behaviour. Furthermore, we reveal that the disease propagation does not depend only on the current time instance; rather, it depends on the disease's earlier status, called the "memory effect". We consider a fractional order network mathematical model to capture the influence of such memory effect on AD propagation. We have integrated brain connectome data in the network model and studied the memory effect and the dual role of astrocytes together. Depending on toxic loads in the brain, we have also analyzed the neuronal damage in the brain. Based on the pathology, primary, secondary, and mixed tauopathies parameters have been used in the model. Due to the mixed tauopathy, different brain nodes or regions in the brain connectome accumulate different toxic concentrations of toxic Aβ and toxic tau proteins. Finally, we explain how the memory effect can slow down the propagation of such toxic proteins in the brain and hence decreases the rate of neuronal damage.

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Section: Model Formulationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Non-Markovian behaviour and the dual role of astrocytes in Alzheimer's disease development and propagation

Pal¹,

Melnik²

2022

Preprint

Self Cite

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“…so as to obtain a Hopf bifurcation about coexistence equilibria E * (x * , y * , z * ). The characteristic equation of Jacobian matrix of model system (4) at the coexistence equilibria E * (x * , y * , z * ) is given in (6). Expressions of C i ; i = 1, 2, 3 and C 1 C 2 − C 3 are depends on k 1 if we fix other parameters at constants.…”

Section: Hopf Bifurcation Analysismentioning

confidence: 99%

“…Many mathematical models of prey-predator interactions have been formulated and considered to investigate the consumption and survival dynamics of species [3,4]. Prey-predator interactions may be governed by ordinary differential equations [3,5], fractional differential equations [6,7], partial differential equations [4,8] and stochastic differential equations [9], delay differential equations [10] and difference equations also. These mathematical models are constructed and studied to identify different environmental effects such as the interaction between species, Allee effect, refuge of species, harvesting, pattern formation, environmental fluctuations etc.…”

Section: Introductionmentioning

confidence: 99%

Chaotic dynamics of a tri-topic food chain model with Beddington–DeAngelis functional response in presence of fear effect

et al. 2021

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The most important fact in the field of theoretical ecology and evolutionary biology is the strategy of predation for predators and the avoidance of prey from predator attack. A lot of experimental works suggest that the reduction of prey depends on both direct predation and fear of predation. We explore the impact of fear effect and mutual interference into a three-species food chain model. In this manuscript, we have considered a tri-topic food web model with Beddington-DeAngelis functional response between interacting species, incorporating the reduction of prey and intermediate predator growth because of the fear of intermediate and top predator respectively. We have provided parametric conditions on the existence of biologically feasible equilibria as well as their local and global stability also. We have established conditions of transcritical, saddle-node and Hopf bifurcation in vicinity of different equilibria. Finally, we performed some numerical investigations to justify analytical findings.

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“…Bazykin [4] presented a variation of Volterra's classical predator-prey model under the help of Michaelis-Menton equation, which can be called Bazykin's predator-prey model according to Bazykin's modi cation. Since then, the Bazykin's predator-prey system has received the attention of a large number of scholars and obtained some excellent research results [5][6][7][8][9][10]. e paper [5] investigated bifurcations of equilibria in Bazykin's predator-prey model.…”

Section: Introductionmentioning

confidence: 99%

Equilibria and Bogdanov‐Takens Bifurcation Analysis in the Bazykin’s Predator‐Prey System

Wang

2022

Discrete Dynamics in Nature and Society

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In the paper, we proposed a Bazykin’s predator-prey system to explore the equilibrium point and Bogdanov-Takens bifurcation problems. Firstly, we derived some key parameter threshold conditions to ensure that the Bazykin’s predator-prey system had a multiple focus of multiplicity one, weak focus of order 2, cusps of codimension 2 and a degenerate Bogdanov-Takens singularity (focus or center case) of codimension 3. Furthermore, the distinction of two types of codimension 2 cusps was also discussed, which showed that the threshold of the two types of cusps could exhibit a cusp, which was a special case of the mentioned degenerate Bogdanov-Takens singularity (focus or center case) of codimension 3. Secondly, we systematically calculated that the Bazykin’s predator-prey system could undergo two types of Bogdanov-Takens bifurcations of codimension 2 and a degenerate focus type Bogdanov-Takens bifurcation of codimension 3. Finally, some numerical examples were implemented to verify the correctness and feasibility of mathematical theory derivation, which also directly showed all possible equilibrium points and Bogdanov-Takens bifurcations of Bazykin’s predator-prey system. In a word, all the research results could play an important theoretical support role in the study of controlling cyanobacteria bloom.

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Memory effect on Bazykin’s prey-predator model: Stability and bifurcation analysis

Cited by 41 publications

References 37 publications

Non-Markovian behaviour and the dual role of astrocytes in Alzheimer's disease development and propagation

Non-Markovian behaviour and the dual role of astrocytes in Alzheimer's disease development and propagation

Chaotic dynamics of a tri-topic food chain model with Beddington–DeAngelis functional response in presence of fear effect

Equilibria and Bogdanov‐Takens Bifurcation Analysis in the Bazykin’s Predator‐Prey System

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