Mathematics

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14 pages, 308 KiB

Open AccessArticle

Global Stability of Traveling Waves for the Lotka–Volterra Competition System with Three Species

by Shulin Hu, Chaohong Pan and Lin Wang

Mathematics 2023, 11(9), 2189; https://doi.org/10.3390/math11092189 - 6 May 2023

Viewed by 1294

The stability of traveling waves for the Lotka–Volterra competition system with three species is investigated in this paper. Specifically, we first show the asymptotic behavior of traveling wave solutions and then establish the local stability and the global stability under the weighted functional [...] Read more.

The stability of traveling waves for the Lotka–Volterra competition system with three species is investigated in this paper. Specifically, we first show the asymptotic behavior of traveling wave solutions and then establish the local stability and the global stability under the weighted functional space. For local stability, the spectrum approach is used, while for global stability, the comparison principle and squeezing theorem are combined. Full article

(This article belongs to the Special Issue Difference and Differential Equations and Applications)

22 pages, 394 KiB

Open AccessArticle

The Dynamical Behavior of a Three-Dimensional System of Exponential Difference Equations

by Abdul Khaliq, Stephen Sadiq, Hala M. E. Ahmed, Batul A. A. Mahmoud, Bushra R. Al-Sinan and Tarek Fawzi Ibrahim

Mathematics 2023, 11(8), 1808; https://doi.org/10.3390/math11081808 - 11 Apr 2023

Cited by 2 | Viewed by 1415

Abstract

The boundedness nature and persistence, global and local behavior, and rate of convergence of positive solutions of a second-order system of exponential difference equations, is investigated in this work. Where the parameters [...] Read more.

The boundedness nature and persistence, global and local behavior, and rate of convergence of positive solutions of a second-order system of exponential difference equations, is investigated in this work. Where the parameters

A, B, C, α, β, γ, δ, η

, and

ξ

are constants that are positive, and the initials

U_{- 1}, U_{0}, V_{- 1}, V_{0}, W_{- 1}

, and

W_{0}

are non-negative real numbers. Some examples are provided to support our theoretical results. Full article

(This article belongs to the Special Issue Difference and Differential Equations and Applications)

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13 pages, 487 KiB

Open AccessArticle

Wolbachia Invasion Dynamics by Integrodifference Equations

by Yijie Li and Zhiming Guo

Mathematics 2022, 10(22), 4253; https://doi.org/10.3390/math10224253 - 14 Nov 2022

Cited by 1 | Viewed by 1218

Abstract

Releasing mosquitoes infected with the endosymbiotic bacterium Wolbachia to invade and replace the wild populations can effectively interrupt dengue transmission. Recently, a reasonable discrete competitive non-spatial model was developed and the conditions for the successful invasion of Wolbachia were given. However, Wolbachia propagation [...] Read more.

Releasing mosquitoes infected with the endosymbiotic bacterium Wolbachia to invade and replace the wild populations can effectively interrupt dengue transmission. Recently, a reasonable discrete competitive non-spatial model was developed and the conditions for the successful invasion of Wolbachia were given. However, Wolbachia propagation is a matter of spatial dynamics. In this paper, we introduce a dispersal kernel and establish integrodifference equations, a class of discrete-time spatial diffusion systems that have recently gained much attention as an important tool for spatial ecology. We analyzed the spatial model by average dispersal success approximation to find the criteria for the successful spread of Wolbachia, and then compared it with the non-spatial model to discuss the effect of spatial parameters. Full article

(This article belongs to the Special Issue Difference and Differential Equations and Applications)

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12 pages, 267 KiB

Open AccessArticle

Inviscid Limit of 3D Nonhomogeneous Navier–Stokes Equations with Slip Boundary Conditions

by Hongmin Li, Yuanxian Hui and Zhong Zhao

Mathematics 2022, 10(21), 3999; https://doi.org/10.3390/math10213999 - 28 Oct 2022

Viewed by 1030

Abstract

In this paper, we consider the inviscid limit of a nonhomogeneous incompressible Navier–Stokes system with a slip-without-friction boundary condition. We study the convergence in strong norms for a solution and obtain the convergence rate in space

W^{2, p} (Ω)

[...] Read more.

In this paper, we consider the inviscid limit of a nonhomogeneous incompressible Navier–Stokes system with a slip-without-friction boundary condition. We study the convergence in strong norms for a solution and obtain the convergence rate in space

W^{2, p} (Ω)

when the boundary is flat. We need to establish the uniform bound of the solution in space

W^{3, p} (Ω)

, and the key of proofs is to obtain a priori estimation of

\partial_{t} u

in space

W^{1, p} (Ω)

. Full article

(This article belongs to the Special Issue Difference and Differential Equations and Applications)

18 pages, 573 KiB

Open AccessArticle

Dynamic Behavior of an Interactive Mosquito Model under Stochastic Interference

by Xingtong Liu, Yuanshun Tan and Bo Zheng

Mathematics 2022, 10(13), 2284; https://doi.org/10.3390/math10132284 - 29 Jun 2022

Cited by 4 | Viewed by 1500

Abstract

For decades, mosquito-borne diseases such as dengue fever and Zika have posed serious threats to human health. Diverse mosquito vector control strategies with different advantages have been proposed by the researchers to solve the problem. However, due to the extremely complex living environment [...] Read more.

For decades, mosquito-borne diseases such as dengue fever and Zika have posed serious threats to human health. Diverse mosquito vector control strategies with different advantages have been proposed by the researchers to solve the problem. However, due to the extremely complex living environment of mosquitoes, environmental changes bring significant differences to the mortality of mosquitoes. This dynamic behavior requires stochastic differential equations to characterize the fate of mosquitoes, which has rarely been considered before. Therefore, in this article, we establish a stochastic interactive wild and sterile mosquito model by introducing the white noise to represent the interference of the environment on the survival of mosquitoes. After obtaining the existence and uniqueness of the global positive solution and the stochastically ultimate boundedness of the stochastic system, we study the dynamic behavior of the stochastic model by constructing a series of suitable Lyapunov functions. Our results show that appropriate stochastic environmental fluctuations can effectively inhibit the reproduction of wild mosquitoes. Numerical simulations are provided to numerically verify our conclusions: the intensity of the white noise has an effect on the extinction and persistence of both wild and sterile mosquitoes. Full article

(This article belongs to the Special Issue Difference and Differential Equations and Applications)

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22 pages, 452 KiB

Open AccessArticle

Approximate Methods for Solving Problems of Mathematical Physics on Neural Hopfield Networks

by Ilya Boykov, Vladimir Roudnev and Alla Boykova

Mathematics 2022, 10(13), 2207; https://doi.org/10.3390/math10132207 - 24 Jun 2022

Cited by 1 | Viewed by 1328

Abstract

A Hopfield neural network is described by a system of nonlinear ordinary differential equations. We develop a broad range of numerical schemes that are applicable for a wide range of computational problems. We review here our study on an approximate solution of the [...] Read more.

A Hopfield neural network is described by a system of nonlinear ordinary differential equations. We develop a broad range of numerical schemes that are applicable for a wide range of computational problems. We review here our study on an approximate solution of the Fredholm integral equation, and linear and nonlinear singular and hypersingular integral equations, using a continuous method for solving operator equations. This method assumes that the original system is associated with a Cauchy problem for systems of ordinary differential equations on Hopfield neural networks. We present sufficient conditions for the Hopfield networks’ stability defined via coefficients of systems of differential equations. Full article

(This article belongs to the Special Issue Difference and Differential Equations and Applications)

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20 pages, 435 KiB

Open AccessArticle

Stochastic Transcription with Alterable Synthesis Rates

by Chunjuan Zhu, Zibo Chen and Qiwen Sun

Mathematics 2022, 10(13), 2189; https://doi.org/10.3390/math10132189 - 23 Jun 2022

Cited by 1 | Viewed by 1227

Abstract

Background: Gene transcription is a random bursting process that leads to large variability in mRNA numbers in single cells. The main cause is largely attributed to random switching between periods of active and inactive gene transcription. In some experiments, it has been observed [...] Read more.

Background: Gene transcription is a random bursting process that leads to large variability in mRNA numbers in single cells. The main cause is largely attributed to random switching between periods of active and inactive gene transcription. In some experiments, it has been observed that variation in the number of active transcription sites causes the initiation rate to vary during elongation. Results: We established a mathematical model based on the molecular reaction mechanism in single cells and studied a stochastic transcription system consisting of two active states and one inactive state, in which mRNA molecules are produced with two different synthesis rates. Conclusions: By calculation, we obtained the average mRNA expression level, the noise strength, and the skewness of transcripts. We gave a necessary and sufficient condition that causes the average mRNA level to peak at a limited time. The model could help us to distinguish an appropriate mechanism that may be employed by cells to transcribe mRNA molecules. Our simulations were in agreement with some experimental data and showed that the skewness can measure the deviation of the distribution of transcripts from the mean value. Especially for mature mRNAs, their distributions were almost able to be determined by the mean, the noise (or the noise strength), and the skewness. Full article

(This article belongs to the Special Issue Difference and Differential Equations and Applications)

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18 pages, 3157 KiB

Open AccessArticle

The Basic Reproduction Number and Delayed Action of T Cells for Patients Infected with SARS-CoV-2

by Yingdong Yin, Yupeng Xi, Cheng Xu and Qiwen Sun

Mathematics 2022, 10(12), 2017; https://doi.org/10.3390/math10122017 - 11 Jun 2022

Cited by 2 | Viewed by 1832

Abstract

COVID-19 has been prevalent for the last two years. The transmission capacity of SARS-CoV-2 differs under the influence of different epidemic prevention policies, making it difficult to measure the infectivity of the virus itself. In order to evaluate the infectivity of SARS-CoV-2 in [...] Read more.

COVID-19 has been prevalent for the last two years. The transmission capacity of SARS-CoV-2 differs under the influence of different epidemic prevention policies, making it difficult to measure the infectivity of the virus itself. In order to evaluate the infectivity of SARS-CoV-2 in patients with different diseases, we constructed a viral kinetic model by adding the effects of T cells and antibodies. To analyze and compare the delay time of T cell action in patients with different symptoms, we constructed a delay differential equation model. Through the first model, we found that the basic reproduction number of severe patients is greater than that of mild patients, and accordingly, we constructed classification criteria for severe and mild patients. Through the second model, we found that the delay time of T cell action in severe patients is much longer than that in mild patients, and accordingly, we present suggestions for the prevention, diagnosis, and treatment of different patients. Full article

(This article belongs to the Special Issue Difference and Differential Equations and Applications)

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10 pages, 273 KiB

Open AccessArticle

Local Stability of Traveling Waves of a Model Describing Host Tissue Degradation by Bacteria

by Xing He, Guansheng He and Chaohong Pan

Mathematics 2022, 10(11), 1805; https://doi.org/10.3390/math10111805 - 25 May 2022

Cited by 1 | Viewed by 1153

Abstract

The focus of this paper is on the local stability of the traveling waves of reaction–diffusion systems that describe host-tissue degradation by bacteria. On the one hand, we discuss the asymptotic behavior of the solutions near the equilibrium points. On the other hand, [...] Read more.

The focus of this paper is on the local stability of the traveling waves of reaction–diffusion systems that describe host-tissue degradation by bacteria. On the one hand, we discuss the asymptotic behavior of the solutions near the equilibrium points. On the other hand, the local stability of traveling waves is proved by the spectrum method based on the appropriate weighted functional space. Full article

(This article belongs to the Special Issue Difference and Differential Equations and Applications)

14 pages, 341 KiB

Open AccessArticle

A Novel Dynamical Regulation of mRNA Distribution by Cross-Talking Pathways

by Qiwen Sun, Zhaohang Cai and Chunjuan Zhu

Mathematics 2022, 10(9), 1515; https://doi.org/10.3390/math10091515 - 2 May 2022

Cited by 3 | Viewed by 1318

Abstract

In this paper, we use a similar approach to the one proposed by Chen and Jiao to calculate the mathematical formulas of the generating function

V (z, t)

and the mass function

P_{m} (t)

of a cross-talking [...] Read more.

In this paper, we use a similar approach to the one proposed by Chen and Jiao to calculate the mathematical formulas of the generating function

V (z, t)

and the mass function

P_{m} (t)

of a cross-talking pathways model in large parameter regions. Together with kinetic rates from yeast and mouse genes, our numerical examples reveal novel bimodal mRNA distributions for intermediate times, whereby the mode of distribution

P_{m} (t)

displays unimodality with the peak at

m = 0

for initial and long times, which has not been obtained in previous works. Such regulation of mRNA distribution exactly matches the transcriptional dynamics for the osmosensitive genes in Saccharomyces cerevisiae, which has not been generated by those models with one single pathway or feedback loops. This paper may provide us with a novel observation on transcriptional distribution dynamics regulated by multiple signaling pathways in response to environmental changes and genetic perturbations. Full article

(This article belongs to the Special Issue Difference and Differential Equations and Applications)

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12 pages, 275 KiB

Open AccessArticle

Infinite Homoclinic Solutions of the Discrete Partial Mean Curvature Problem with Unbounded Potential

by Yanshan Chen and Zhan Zhou

Mathematics 2022, 10(9), 1436; https://doi.org/10.3390/math10091436 - 24 Apr 2022

Viewed by 1327

Abstract

The mean curvature problem is an important class of problems in mathematics and physics. We consider the existence of homoclinic solutions to a discrete partial mean curvature problem, which is tied to the existence of discrete solitons. Under the assumptions that the potential [...] Read more.

The mean curvature problem is an important class of problems in mathematics and physics. We consider the existence of homoclinic solutions to a discrete partial mean curvature problem, which is tied to the existence of discrete solitons. Under the assumptions that the potential function is unbounded and that the nonlinear term is superlinear at infinity, we obtain the existence of infinitely many homoclinic solutions to this problem by means of the fountain theorem in the critical point theory. In the end, an example is given to illustrate the applicability of our results. Full article

(This article belongs to the Special Issue Difference and Differential Equations and Applications)

18 pages, 617 KiB

Open AccessArticle

Dynamic Behaviors of a Stochastic Eco-Epidemiological Model for Viral Infection in the Toxin-Producing Phytoplankton and Zooplankton System

by Xiaomei Feng, Yuan Miao, Shulin Sun and Lei Wang

Mathematics 2022, 10(8), 1218; https://doi.org/10.3390/math10081218 - 8 Apr 2022

Viewed by 1507

Abstract

It is well known that the evolution of natural populations is almost inevitably disturbed by various environmental factors. Various experiments have shown that the growth of phytoplankton might be affected by nutrient availability, water temperature, and light, while the development of zooplankton is [...] Read more.

It is well known that the evolution of natural populations is almost inevitably disturbed by various environmental factors. Various experiments have shown that the growth of phytoplankton might be affected by nutrient availability, water temperature, and light, while the development of zooplankton is more disturbed by the pH value of the seawater, water temperature, and water movement. However, it is not clear how these environmental fluctuations affect the dynamical behavior of the phytoplankton and zooplankton system. In this paper, a stochastic eco-epidemiological model for viral infection in the toxin-producing phytoplankton and zooplankton system is proposed. Firstly, the existence and uniqueness of globally positive solutions for this model is shown. Secondly, the stochastic boundedness of solutions for the model is proved. Moreover, sufficient conditions for the extinction and persistence in the mean for the phytoplankton and zooplankton are obtained by constructing appropriate stochastic Lyapunov functions and using analytical techniques. Numerical simulations are carried out to demonstrate different dynamical behaviors including coexistence, extinction of the whole plankton system, partial persistence and extinction, and their corresponding probability density curves. Full article

(This article belongs to the Special Issue Difference and Differential Equations and Applications)

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30 pages, 1540 KiB

Open AccessArticle

Dynamics of a Predator–Prey Model with the Additive Predation in Prey

by Dingyong Bai and Xiaoxuan Zhang

Mathematics 2022, 10(4), 655; https://doi.org/10.3390/math10040655 - 20 Feb 2022

Cited by 7 | Viewed by 2127

Abstract

In this paper, we consider a predator–prey model, in which the prey’s growth is affected by the additive predation of its potential predators. Due to the additive predation term in prey, the model may exhibit the cases of the strong Allee effect, weak [...] Read more.

In this paper, we consider a predator–prey model, in which the prey’s growth is affected by the additive predation of its potential predators. Due to the additive predation term in prey, the model may exhibit the cases of the strong Allee effect, weak Allee effect and no Allee effect. In each case, the dynamics of global features of the model are investigated. Compared to the well-known Lotka–Volterra type model, the model proposed in this paper exhibits much richer and more complex dynamic behaviors, such as the Allee effect, the sensitivity to the initial conditions caused by the strong Allee effect, the oscillatory behavior and the Hopf and heteroclinic bifurcations. Furthermore, the stability and Hopf bifurcation of the model with the density dependent feedback time delay in prey are investigated. By the normal form method and center manifold theory, the explicit formulas are presented to determine the direction of Hopf bifurcation and the stability and period of Hopf-bifurcating periodic solutions. Theoretical analysis and numerical simulation indicate that the delay may destabilize the model, and cause the Hopf bifurcation not only at the interior equilibrium but also at a boundary equilibrium. Full article

(This article belongs to the Special Issue Difference and Differential Equations and Applications)

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21 pages, 997 KiB

Open AccessArticle

Periodic Orbits of a Mosquito Suppression Model Based on Sterile Mosquitoes

by Zhongcai Zhu, Yantao Shi, Rong Yan and Linchao Hu

Mathematics 2022, 10(3), 462; https://doi.org/10.3390/math10030462 - 31 Jan 2022

Cited by 4 | Viewed by 2199

Abstract

In this work, we investigate the existence and stability of periodic orbits of a mosquito population suppression model based on sterile mosquitoes. The model switches between two sub-equations as the actual number of sterile mosquitoes in the wild is assumed to take two [...] Read more.

In this work, we investigate the existence and stability of periodic orbits of a mosquito population suppression model based on sterile mosquitoes. The model switches between two sub-equations as the actual number of sterile mosquitoes in the wild is assumed to take two constant values alternately. Employing the Poincaré map method, we show that the model has at most two T-periodic solutions when the release amount is not sufficient to eradicate the wild mosquitoes, and then obtain some sufficient conditions for the model to admit a unique or exactly two T-periodic solutions. In particular, we observe that the model displays bistability when it admits exactly two T-periodic solutions: the origin and the larger periodic solution are asymptotically stable, and the smaller periodic solution is unstable. Finally, we give two numerical examples to support our lemmas and theorems. Full article

(This article belongs to the Special Issue Difference and Differential Equations and Applications)

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18 pages, 805 KiB

Open AccessArticle

A Novel Approach for Calculating Exact Forms of mRNA Distribution in Single-Cell Measurements

by Jiaxin Chen and Feng Jiao

Mathematics 2022, 10(1), 27; https://doi.org/10.3390/math10010027 - 22 Dec 2021

Cited by 11 | Viewed by 2582

Abstract

Gene transcription is a stochastic process manifested by fluctuations in mRNA copy numbers in individual isogenic cells. Together with mathematical models of stochastic transcription, the massive mRNA distribution data that can be used to quantify fluctuations in mRNA levels can be fitted by [...] Read more.

Gene transcription is a stochastic process manifested by fluctuations in mRNA copy numbers in individual isogenic cells. Together with mathematical models of stochastic transcription, the massive mRNA distribution data that can be used to quantify fluctuations in mRNA levels can be fitted by

P_{m} (t)

, which is the probability of producing m mRNA molecules at time t in a single cell. Tremendous efforts have been made to derive analytical forms of

P_{m} (t)

, which rely on solving infinite arrays of the master equations of models. However, current approaches focus on the steady-state (

t \to \infty

) or require several parameters to be zero or infinity. Here, we present an approach for calculating

P_{m} (t)

with time, where all parameters are positive and finite. Our approach was successfully implemented for the classical two-state model and the widely used three-state model and may be further developed for different models with constant kinetic rates of transcription. Furthermore, the direct computations of

P_{m} (t)

for the two-state model and three-state model showed that the different regulations of gene activation can generate discriminated dynamical bimodal features of mRNA distribution under the same kinetic rates and similar steady-state mRNA distribution. Full article

(This article belongs to the Special Issue Difference and Differential Equations and Applications)

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