Research

15 pages, 3293 KiB

Open AccessArticle

Stability Analysis and Hopf Bifurcation of a Delayed Diffusive Predator–Prey Model with a Strong Allee Effect on the Prey and the Effect of Fear on the Predator

by Yining Xie, Jing Zhao and Ruizhi Yang

Mathematics 2023, 11(9), 1996; https://0-doi-org.brum.beds.ac.uk/10.3390/math11091996 - 23 Apr 2023

Cited by 2 | Viewed by 1038

Abstract

In this paper, we propose a diffusive predator–prey model with a strong Allee effect and nonlocal competition in the prey and a fear effect and gestation delay in the predator. We mainly study the local stability of the coexisting equilibrium and the existence [...] Read more.

In this paper, we propose a diffusive predator–prey model with a strong Allee effect and nonlocal competition in the prey and a fear effect and gestation delay in the predator. We mainly study the local stability of the coexisting equilibrium and the existence and properties of Hopf bifurcation. We provide bifurcation diagrams with the fear effect parameter (s) and the Allee effect parameter (a), showing that the stable region of the coexisting equilibrium increases (or decreases) with an increase in the fear effect parameter (s) (or the Allee effect parameter (a)). We also show that gestation delay (

τ

) can affect the local stability of the coexisting equilibrium. When the delay (

τ

) is greater than the critical value, the coexistence equilibrium loses its stability, and bifurcating periodic solutions appear. Whether the bifurcated periodic solution is spatially homogeneous or inhomogeneous depends on the fear effect parameter (s) and the Allee effect parameter (a). These results show that the fear effect parameter (s), the Allee effect parameter (a), and gestation delay (

τ

) can be used to control the growth of prey and predator populations. Full article

(This article belongs to the Special Issue Recent Advances in Theory and Application of Dynamical Systems)

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16 pages, 2711 KiB

Open AccessArticle

Diffusion-Induced Instability of the Periodic Solutions in a Reaction-Diffusion Predator-Prey Model with Dormancy of Predators

by Mi Wang

Mathematics 2023, 11(8), 1875; https://0-doi-org.brum.beds.ac.uk/10.3390/math11081875 - 15 Apr 2023

Cited by 1 | Viewed by 784

Abstract

A reaction-diffusion predator-prey model with the dormancy of predators is considered in this paper. We are concerned with the long-time behaviors of the solutions of this system. We divided our investigations into two cases: for the ODEs system, we study the existence and [...] Read more.

A reaction-diffusion predator-prey model with the dormancy of predators is considered in this paper. We are concerned with the long-time behaviors of the solutions of this system. We divided our investigations into two cases: for the ODEs system, we study the existence and stability of the equilibrium solutions and derive precise conditions on system parameters so that the system can undergo Hopf bifurcations around the positive equilibrium solution. Moreover, the properties of Hopf bifurcation are studied in detail. For the reaction-diffusion system, we are able to derive conditions on the diffusion coefficients so that the spatially homogeneous Hopf bifurcating periodic solutions can undergo diffusion-triggered instability. To support our theoretical analysis, we also include several numerical results. Full article

(This article belongs to the Special Issue Recent Advances in Theory and Application of Dynamical Systems)

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13 pages, 5987 KiB

Open AccessArticle

Neimark–Sacker Bifurcation of a Discrete-Time Predator–Prey Model with Prey Refuge Effect

by Binhao Hong and Chunrui Zhang

Mathematics 2023, 11(6), 1399; https://0-doi-org.brum.beds.ac.uk/10.3390/math11061399 - 14 Mar 2023

Cited by 7 | Viewed by 1473

Abstract

In this paper, we deduce a predator–prey model with discrete time in the interior of

R_{+}^{2}

using a new discrete method to study its local dynamics and Neimark–Sacker bifurcation. Compared with continuous models, discrete ones have many unique properties that help [...] Read more.

In this paper, we deduce a predator–prey model with discrete time in the interior of

R_{+}^{2}

using a new discrete method to study its local dynamics and Neimark–Sacker bifurcation. Compared with continuous models, discrete ones have many unique properties that help to understand the changing patterns of biological populations from a completely new perspective. The existence and stability of the three equilibria are analyzed, and the formation conditions of Neimark–Sacker bifurcation around the unique positive equilibrium point are established using the center manifold theorem and bifurcation theory. An attracting closed invariant curve appears, which corresponds to the periodic oscillations between predators and prey over a long period of time. Finally, some numerical simulations and their biological meanings are given to reveal the complex dynamical behavior. Full article

(This article belongs to the Special Issue Recent Advances in Theory and Application of Dynamical Systems)

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12 pages, 1162 KiB

Open AccessArticle

Hopf Bifurcation in a Predator–Prey Model with Memory Effect in Predator and Anti-Predator Behaviour in Prey

by Wenqi Zhang, Dan Jin and Ruizhi Yang

Mathematics 2023, 11(3), 556; https://0-doi-org.brum.beds.ac.uk/10.3390/math11030556 - 20 Jan 2023

Cited by 2 | Viewed by 1328

Abstract

In this paper, a diffusive predator–prey model with a memory effect in predator and anti-predator behaviour in prey is studied. The stability of the coexisting equilibrium and the existence of Hopf bifurcation are analysed by analysing the distribution of characteristic roots. The property [...] Read more.

In this paper, a diffusive predator–prey model with a memory effect in predator and anti-predator behaviour in prey is studied. The stability of the coexisting equilibrium and the existence of Hopf bifurcation are analysed by analysing the distribution of characteristic roots. The property of Hopf bifurcation is investigated by the theory of the centre manifold and normal form method. Through the numerical simulations, it is observed that the anti-predator behaviour parameter

η

, the memory-based diffusion coefficient parameter d, and memory delay

τ

can affect the stability of the coexisting equilibrium under some parameters and cause the spatially inhomogeneous oscillation of prey and predator’s densities. Full article

(This article belongs to the Special Issue Recent Advances in Theory and Application of Dynamical Systems)

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31 pages, 7916 KiB

Open AccessArticle

A New Fractal-Fractional Version of Giving up Smoking Model: Application of Lagrangian Piece-Wise Interpolation along with Asymptotical Stability

by Sina Etemad, Albert Shikongo, Kolade M. Owolabi, Brahim Tellab, İbrahim Avcı, Shahram Rezapour and Ravi P. Agarwal

Mathematics 2022, 10(22), 4369; https://0-doi-org.brum.beds.ac.uk/10.3390/math10224369 - 20 Nov 2022

Cited by 12 | Viewed by 1180

Abstract

In this paper, a new kind of mathematical modeling is studied by providing a five-compartmental system of differential equations with respect to new hybrid generalized fractal-fractional derivatives. For the first time, we design a model of giving up smoking to analyze its dynamical [...] Read more.

In this paper, a new kind of mathematical modeling is studied by providing a five-compartmental system of differential equations with respect to new hybrid generalized fractal-fractional derivatives. For the first time, we design a model of giving up smoking to analyze its dynamical behaviors by considering two parameters of such generalized operators; i.e., fractal dimension and fractional order. We apply a special sub-category of increasing functions to investigate the existence of solutions. Uniqueness property is derived by a standard method based on the Lipschitz rule. After proving stability property, the equilibrium points are obtained and asymptotically stable solutions are studied. Finally, we illustrate all analytical results and findings via numerical algorithms and graphs obtained by Lagrangian piece-wise interpolation, and discuss all behaviors of the relevant solutions in the fractal-fractional system. Full article

(This article belongs to the Special Issue Recent Advances in Theory and Application of Dynamical Systems)

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10 pages, 1526 KiB

Open AccessArticle

Modeling Nonlinear Hydroelastic Response for the Endwall of the Plane Channel Due to Its Upper-Wall Vibrations

by Marina Barulina, Loredana Santo, Victor Popov, Anna Popova and Dmitry Kondratov

Mathematics 2022, 10(20), 3844; https://0-doi-org.brum.beds.ac.uk/10.3390/math10203844 - 17 Oct 2022

Cited by 2 | Viewed by 1075

Abstract

A mathematical model for studying the nonlinear response of the endwall of a narrow channel filled with a viscous fluid to the vibration of the channel’s upper wall was formulated. The channel, formed by two parallel, rigid walls, was investigated. The right end-channel [...] Read more.

A mathematical model for studying the nonlinear response of the endwall of a narrow channel filled with a viscous fluid to the vibration of the channel’s upper wall was formulated. The channel, formed by two parallel, rigid walls, was investigated. The right end-channel wall was supported by a nonlinear spring. At the end of the left channel, the fluid flowed into a cavity with constant pressure. The upper channel wall oscillated according to a given law. As a result of the interaction between the endwall and the upper wall via a viscous fluid, the forced, nonlinear oscillations of the channel endwall arose. The fluid motion was considered in terms of the hydrodynamic lubrication theory. The endwall was studied as a spring-mass system with a nonlinear cubic restoring force. The coupled hydroelasticity problem was formulated, and it was shown that the problem under consideration was reduced to a single equation in the form of the Duffing equation. The nonlinear hydroelastic response of the end wall was determined by means of the harmonic balance method. The results of numerical experiments on nonlinear hydroelastic response behavior and a comparison with the case when the support spring is linear were presented. The obtained results are of a fundamental nature and can be used in modeling various devices and systems that have narrow channels filled with viscous fluid and are subjected to vibrations on one side of the channel. For example, coolant pipes are subjected to vibrations from the engine. Of particular interest is the application of the presented solution to the mathematical modeling of nano- and micro-spacecraft systems with fluids since the proposed decision allows for the consideration of some boundary effects, which is important for nano- and micro-spacecraft due to their small size. Full article

(This article belongs to the Special Issue Recent Advances in Theory and Application of Dynamical Systems)

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12 pages, 570 KiB

Open AccessArticle

Stability and Optimal Control of Tree-Insect Model under Forest Fire Disturbance

by Xiaoxiao Liu and Chunrui Zhang

Mathematics 2022, 10(15), 2563; https://0-doi-org.brum.beds.ac.uk/10.3390/math10152563 - 22 Jul 2022

Cited by 7 | Viewed by 1178

Abstract

In this article, we propose a mathematical model for insect outbreaks coupled with wildfire disturbances and an optimization model for finding suitable wildfire frequencies. We use a refined Holling II function as a model for the nonlinear response of fire frequency against trees [...] Read more.

In this article, we propose a mathematical model for insect outbreaks coupled with wildfire disturbances and an optimization model for finding suitable wildfire frequencies. We use a refined Holling II function as a model for the nonlinear response of fire frequency against trees and insects. The results show that for the tree–insect–wildfire model, there is a coexistence equilibrium in the system. Sensitivity analysis is performed to determine the effect of wildfire on trees in the optimization model. The results show that forest fires have a significant impact on the equilibrium mechanism of tree–insect coexistence. Numerical simulations suggest that in some areas of high fire intensity, there may be positive feedback between disturbances from wildfires and insect outbreaks. The result is consistent with the present theory in this field. Full article

(This article belongs to the Special Issue Recent Advances in Theory and Application of Dynamical Systems)

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27 pages, 740 KiB

Open AccessArticle

Stability and Numerical Simulations of a New SVIR Model with Two Delays on COVID-19 Booster Vaccination

by Xinyu Liu and Yuting Ding

Mathematics 2022, 10(10), 1772; https://0-doi-org.brum.beds.ac.uk/10.3390/math10101772 - 23 May 2022

Cited by 4 | Viewed by 1749

Abstract

As COVID-19 continues to threaten public health around the world, research on specific vaccines has been underway. In this paper, we establish an SVIR model on booster vaccination with two time delays. The time delays represent the time of booster vaccination and the [...] Read more.

As COVID-19 continues to threaten public health around the world, research on specific vaccines has been underway. In this paper, we establish an SVIR model on booster vaccination with two time delays. The time delays represent the time of booster vaccination and the time of booster vaccine invalidation, respectively. Second, we investigate the impact of delay on the stability of non-negative equilibria for the model by considering the duration of the vaccine, and the system undergoes Hopf bifurcation when the duration of the vaccine passes through some critical values. We obtain the normal form of Hopf bifurcation by applying the multiple time scales method. Then, we study the model with two delays and show the conditions under which the nontrivial equilibria are locally asymptotically stable. Finally, through analysis of official data, we select two groups of parameters to simulate the actual epidemic situation of countries with low vaccination rates and countries with high vaccination rates. On this basis, we select the third group of parameters to simulate the ideal situation in which the epidemic can be well controlled. Through comparative analysis of the numerical simulations, we concluded that the most appropriate time for vaccination is to vaccinate with the booster shot 6 months after the basic vaccine. The priority for countries with low vaccination rates is to increase vaccination rates; otherwise, outbreaks will continue. Countries with high vaccination rates need to develop more effective vaccines while maintaining their coverage rates. When the vaccine lasts longer and the failure rate is lower, the epidemic can be well controlled within 20 years. Full article

(This article belongs to the Special Issue Recent Advances in Theory and Application of Dynamical Systems)

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24 pages, 761 KiB

Open AccessArticle

Dynamic Analysis of a COVID-19 Vaccination Model with a Positive Feedback Mechanism and Time-Delay

by Xin Ai, Xinyu Liu, Yuting Ding and Han Li

Mathematics 2022, 10(9), 1583; https://0-doi-org.brum.beds.ac.uk/10.3390/math10091583 - 07 May 2022

Cited by 1 | Viewed by 1308

Abstract

As the novel coronavirus pandemic has spread globally since 2019, most countries in the world are conducting vaccination campaigns. First, based on the traditional SIR infectious disease model, we introduce a positive feedback mechanism associated with the vaccination rate, and consider the time [...] Read more.

As the novel coronavirus pandemic has spread globally since 2019, most countries in the world are conducting vaccination campaigns. First, based on the traditional SIR infectious disease model, we introduce a positive feedback mechanism associated with the vaccination rate, and consider the time delay from antibody production to antibody disappearance after vaccination. We establish an

U V_{a} V

model for COVID-19 vaccination with a positive feedback mechanism and time-delay. Next, we verify the existence of the equilibrium of the formulated model and analyze its stability. Then, we analyze the existence of the Hopf bifurcation, and use the multiple time scales method to derive the normal form of the Hopf bifurcation, further determining the direction of the Hopf bifurcation and the stability of the periodic solution of the bifurcation. Finally, we collect the parameter data of some countries and regions to determine the reasonable ranges of multiple parameters to ensure the authenticity of simulation results. Numerical simulations are carried out to verify the correctness of the theoretical results. We also give the critical time for controllable widespread antibody failure to provide a reference for strengthening vaccination time. Taking two groups of parameters as examples, the time of COVID-19 vaccine booster injection should be best controlled before

38.5

weeks and

35.3

weeks, respectively. In addition, study the impact of different expiration times on epidemic prevention and control effectiveness. We further explore the impact of changes in vaccination strategies on trends in epidemic prevention and control effectiveness. It could be concluded that, under the same epidemic vaccination strategy, the existence level of antibody is roughly the same, which is consistent with the reality. Full article

(This article belongs to the Special Issue Recent Advances in Theory and Application of Dynamical Systems)

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21 pages, 532 KiB

Open AccessArticle

Bifurcation Analysis of a Synthetic Drug Transmission Model with Two Time Delays

by Hu Zhang, Anwar Zeb, Aying Wan and Zizhen Zhang

Mathematics 2022, 10(9), 1532; https://0-doi-org.brum.beds.ac.uk/10.3390/math10091532 - 03 May 2022

Viewed by 999

Abstract

Synthetic drugs are taking the place of traditional drugs and have made headlines giving rise to serious social issues in many countries. In this work, a synthetic drug transmission model incorporating psychological addicts with two time delays is being developed. Local stability and [...] Read more.

Synthetic drugs are taking the place of traditional drugs and have made headlines giving rise to serious social issues in many countries. In this work, a synthetic drug transmission model incorporating psychological addicts with two time delays is being developed. Local stability and exhibition of Hopf bifurcation are established analytically and numerically by taking the combinations of the two time delays as bifurcation parameters. The exhibition of Hopf bifurcation shows that it is burdensome to eradicate the synthetic drugs transmission in the population. Full article

(This article belongs to the Special Issue Recent Advances in Theory and Application of Dynamical Systems)

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18 pages, 2441 KiB

Open AccessArticle

Dynamical Analysis of a Delayed Diffusive Predator–Prey Model with Additional Food Provided and Anti-Predator Behavior

by Ruizhi Yang, Xiao Zhao and Yong An

Mathematics 2022, 10(3), 469; https://0-doi-org.brum.beds.ac.uk/10.3390/math10030469 - 31 Jan 2022

Cited by 26 | Viewed by 3844

Abstract

We studied a delayed predator–prey model with diffusion and anti-predator behavior. Assume that additional food is provided for predator population. Then the stability of the positive equilibrium is considered. The existence of Hopf bifurcation is also discussed based on the Hopf bifurcation theory. [...] Read more.

We studied a delayed predator–prey model with diffusion and anti-predator behavior. Assume that additional food is provided for predator population. Then the stability of the positive equilibrium is considered. The existence of Hopf bifurcation is also discussed based on the Hopf bifurcation theory. The property of Hopf bifurcation is derived through the theory of center manifold and normal form method. Finally, we analyze the effect of time delay on the model through numerical simulations. Full article

(This article belongs to the Special Issue Recent Advances in Theory and Application of Dynamical Systems)

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