Nontrivial Equilibrium Research Articles

Theoretical framework for biological control of pest: a mathematical modeling approach.

Crop losses to pests were the main obstacle to food security globally. Pest control was a laudable exercise, but the exercise could be hindered by the inevitable adjustment between pest reductions, operation costs as well as impacts on the environment and human health. The pest could be controlled by many methods, but biological control was the most popular technique because it addressed inevitable trade-offs between costs and side effects. In this paper, a mathematical model was developed to quantify intricate biological procedures in the context of biological control using prey-predator mechanisms. Three equilibrium points (one trivial and two non-trivial) were derived, and the stability of each equilibrium point was examined. The stability results indicated that the adoption of biological control might neutralize pest infestation but the situation might not persist (unstable trivial equilibrium). It was also discovered that pest control through biological means might fail if the predator was wrongly selected or if the population of the predator vanished while the pest remained in existence (unstable non-trivial equilibrium). The analytical results were finally justified by a means of simulation via a computer-in-built maple program.

Mathematical model of mosquito population dynamics with constants and periodic releases of Wolbachia-infected males

In this article, we propose three mathematical models of mosquito dynamics. The first model describes the dynamics of wild mosquitoes in combination with mechanical control and larvicide. For this model, we derive a threshold parameter N , called the basic number of offspring, and show that the trivial equilibrium is globally asymptotically stable when N ≤ 1 , while if N > 1 , the non-trivial equilibrium is globally asymptotically stable. In the second model, we evaluated a constant release of Wolbachia-infected males in the wild mosquito population. Our analysis identified a critical threshold for releasing Wolbachia-infected male mosquitoes M i crit to reduce or eliminate the wild mosquito population. In the third model, we assessed wild mosquito dynamics by periodically releasing Wolbachia-infected male mosquitoes. The analysis of this model produced two thresholds, T ∗ the threshold on the release period and Λ ∗ the threshold on the release quantity. The existence of trivial periodic solutions is obtained, and the corresponding local and global stability conditions are proved by Floquet's theory and Lyapunov's stability theorem, respectively. We also prove the existence conditions for non-trivial periodic solutions and their local stability. Finally, we perform some numerical simulations to illustrate our theoretically pertinent results.

Study of chronic myeloid leukemia with T-cell under fractal-fractional order model

Abstract This research work is devoted to investigate myeloid leukemia mathematical model. We give some details about the existence of trivial and nontrivial equilibrium points and their stability. Also, local asymptotical stability of disease-free and endemic equilibrium points is discussed. Also, positivity of the solution has been discussed. Some sufficient results are achieved to study the local existence and uniqueness of solution to the considered model for Mittag–Leffler kernel using the Banach contraction theorem. Three numerical algorithms are derived in obtaining the numerical solution of suggested model under three different kernels using Adams–Basforth technique. Numerical results have been presented for different fractals and fractional orders to show the behavior of the proposed model.

A mathematical model with uncertainty quantification for allelopathy with applications to real-world data

We revisit a deterministic model for studying the dynamics of allelopathy. The model is formulated in terms of a non-homogeneous linear system of differential equations whose forcing or source term is a piecewise constant function (square wave). To account for the inherent uncertainties present in this natural phenomenon, we reformulate the model as a system of random differential equations where all model parameters and the initial condition are assumed to be random variables, while the forcing term is a stochastic process. Taking extensive advantage of the so-called Random Variable Transformation (RVT) method, we obtain the solution of the randomized model by providing explicit expressions of the first probability density function of the solution under very general assumptions on the model data. We also determine the joint probability density function of the non-trivial equilibrium point, which is a random vector. If the source term is a time-dependent stochastic process, the RVT method might not be applicable since no explicit solution of the model is available. We then show an alternative approach to overcome this drawback by applying the Liouville–Gibbs partial differential equation. All the theoretical findings are illustrated through several examples, including the application of the randomized model to real-world data on alkaloid contents from leaching thornapple seed.

Modelling the adverse impacts of urbanization on human health

Abstract Urbanization has been a growing trend for decades, but the Coronavirus disease 2019 (COVID-19) pandemic has accelerated this trend in several ways. In many countries, the pandemic has prompted significant investments in urban health infrastructure, including hospitals, clinics, and public health facilities. Also, there are many studies suggesting that COVID-19, urbanization, and chronic diseases are interconnected in several ways, and addressing one aspect requires a comprehensive approach that considers the interconnectedness of these factors. To date, there are no studies depicting COVID-19 as a cause of emerging chronic diseases within the urban population. To address this sensitive issue, we propose a mathematical model in which we divide the total human population into three compartments: susceptible, diseased, and recovered. We have obtained the equilibrium points of the model and the conditions for their local stabilities. A suitable Lyapunov function has been used to derive the condition for the global stability of the nontrivial equilibrium point. A detailed numerical simulation of the model has been carried out, which is followed by sensitivity analysis of the model system.

Global Attractor and Non-trivial Equilibrium Solution for Neural Fields with External Stimuli

Global Attractor and Non-trivial Equilibrium Solution for Neural Fields with External Stimuli

Thermal transitions in a one-dimensional, finite-size Ising model

We revisit the one-dimensional ferromagnetic Ising spin chain with a finite number of spins and periodic boundaries, deriving analytically and verifying numerically its various stationary and dynamical properties at different temperatures. In particular, we determine the probability distributions of magnetization, the number of domain walls, and the corresponding residence times for different chain lengths and magnetic fields. While we study finite systems at thermal equilibrium, we identify several temperatures similar to the critical temperatures for first-order phase transitions in the thermodynamic limit. We illustrate the utility of our results by their application to structural transitions in biopolymers having non-trivial intermediate equilibrium states.

Modeling and stability analysis of substance abuse in women with control policies

Substance abuse is considered to be predominantly a problem for all in society, and much substance abuse research is done with a male majority. Recent drug addiction studies, however, reveal that there are substantial gender variations in substance-related epidemiology. The epidemiology of women’s substance misuse raises issues distinct from those addressed by men’s substance abuse. Women with addictions are more likely than males to experience several impediments to accessing and entering substance misuse treatment. In this regard, we framed a mathematical model by considering the whole population as a women population. For this system, we found pair of equilibrium points namely the Addiction-free equilibrium point and the Non-trivial equilibrium point. Moreover, the addicts generation number (threshold parameter) RA has been found to investigate the spread of addiction. Also, the local and global stability analysis for both the equilibrium points have investigated. Further, to analyze the decrease and increase of the infected population and recovery population respectively, an optimum control analysis is done with two control parameters using Pontryagin’s maximum principle. The influence of addict generation numbers on significant parameters involved in RA has been shown. An optimum control study projected that social media awareness is substantially more efficient than the fixed control in optimizing drug addiction among women. Through numerical simulations we have exhibited, the effect of with and without control on each population. Finally, it produced positive efficacy by reducing the number of infected and increasing the rehabilitation population.

Structured model conserving biomass for the size-spectrum evolution in aquatic ecosystems.

Mathematical modelling of the evolution of the size-spectrum dynamics in aquatic ecosystems was discovered to be a powerful tool to have a deeper insight into impacts of human- and environmental driven changes on the marine ecosystem. In this article we propose to investigate such dynamics by formulating and investigating a suitable model. The underlying process for these dynamics is given by predation events, causing both growth and death of individuals, while keeping the total biomass within the ecosystem constant. The main governing equation investigated is deterministic and non-local of quadratic type, coming from binary interactions. Predation is assumed to strongly depend on the ratio between a predator and its prey, which is distributed around a preferred feeding preference value. Existence of solutions is shown in dependence of the choice of the feeding preference function as well as the choice of the search exponent, a constant influencing the average volume in water an individual has to search until it finds prey. The equation admits a trivial steady state representing a died out ecosystem, as well as-depending on the parameter-regime-steady states with gaps in the size spectrum, giving evidence to the well known cascade effect. The question of stability of these equilibria is considered, showing convergence to the trivial steady state in a certain range of parameters. These analytical observations are underlined by numerical simulations, with additionally exhibiting convergence to the non-trivial equilibrium for specific ranges of parameters.

Parametric resonance of axially functionally graded pipes conveying pulsating fluid

Based on the generalized Hamilton’s principle, the nonlinear governing equation of an axially functionally graded (AFG) pipe is established. The non-trivial equilibrium configuration is superposed by the modal functions of a simply supported beam. Via the direct multi-scale method, the response and stability boundary to the pulsating fluid velocity are solved analytically and verified by the differential quadrature element method (DQEM). The influence of Young’s modulus gradient on the parametric resonance is investigated in the subcritical and supercritical regions. In general, the pipe in the supercritical region is more sensitive to the pulsating excitation. The nonlinearity changes from hard to soft, and the non-trivial equilibrium configuration introduces more frequency components to the vibration. Besides, the increasing Young’s modulus gradient improves the critical pulsating flow velocity of the parametric resonance, and further enhances the stability of the system. In addition, when the temperature increases along the axial direction, reducing the gradient parameter can enhance the response asymmetry. This work further complements the theoretical analysis of pipes conveying pulsating fluid.

The Study of Stability Analysis of Modified Leslie-Gower Herbivore Model with Allee Effect in Plants

A modified Leslie-Gower plant-herbivore model is studied under the Allee effect. Holling-type II functional response is used to modify the model. Delay differential equations play an essential role in making this model more realistic and complicated. The non-trivial equilibrium E* (P* ≠ 0, H* ≠ 0) of the proposed model is calculated. Moreover, the stability and instability of the state variables, which include plant population P and herbivore population H are described graphically. It is shown that the system represents absolute stability when it has no time parameter (τ). When the time parameter is less than the threshold value, then the system exhibits asymptotic stability. In addition, the system surrenders its stability, and Hopf-bifurcation occurs when the time parameter surpasses the threshold value. The timeseries graphs are also represented. It is demonstrated that the system becomes more stable with the maximum rate of predation. MATLAB software is used to perform the graphs to justify the theoretical results.

Vibration of solid and thin-walled slender structures made of soft materials by high-order beam finite elements

In this work, high-order beam (1D) finite element models for the modal analysis of structures made of compressible and nearly-incompressible hyperelastic soft materials are presented in the well-established framework of Carrera Unified Formulation (CUF). In this investigation, the modal behavior of soft structures subjected to progressively increasing loads can be correctly predicted by higher-order structural theories, and the influence of pre-stress conditions applied on the modal response of structures is investigated. The mathematical formulation of hyperelastic isotropic materials is presented in terms of invariants of the right Cauchy–Green strain tensor, obtaining the most general expression of the Piola–Kirchoff 2 stress tensor and tangent elasticity tensor, both independent of the model adopted for the material strain energy function. Governing equations in matrix forms for the static nonlinear analysis and subsequent vibration problem around non-trivial equilibrium states are derived through the Principle of Virtual Displacements (PVD) under a total Lagrangian formulation, defining the fundamental nuclei of stiffness matrices and internal and external forces vector, all independent of expansion theories and kinematic models adopted in the mathematical modeling of finite elements. Actual numerical results are obtained by an iterative Newton–Raphson linearized scheme coupled with line-search algorithms, and they are compared with results obtained by the commercial code ABAQUS. Our proposed models are tested with large strain problems involving hyperelastic slender and thin-walled structures, for which mode aberration such as crossing or bearing are observed.

Mathematical analysis and simulation of Ebola virus disease spread incorporating mitigation measures

In order to understand the dynamics of the Ebola virus disease (EVD), this research developed a mathematical model that includes quarantine and public education campaigns as control measures. The model’s equilibrium points are displayed and the effective basic reproduction number Reff is estimated. Bifurcation theory is used for the stability analysis of endemic equilibrium state and general bifurcation theory is used to prove the existence of endemic equilibrium state. We evaluated the nature of the endemic equilibrium state of the model’s equation near the disease-free equilibrium, Reff=1 and introduce a bifurcation parameter. The results of the centre manifold theory are used to demonstrate that there is nontrivial endemic equilibrium near the disease-free equilibrium. We demonstrated the characteristics of the endemic equilibrium state that is close to the disease-free equilibrium as well as the fact that there is a potentially unstable endemic equilibrium condition. In other words, the sickness is slowly declining and will eventually disappear, as in the case of West Africa. Finally, we simulated the model developed to study the dynamics of the diseases with varying parameters using Homotopy Perturbation Method (HPM) to validate the qualitative analysis of the model. The result confirmed the hypothesis of our research that if quarantine and public enlightenment is properly used for the mitigation of the disease, the disease outcomes will drastically reduced. The results presented in this research will be useful for public health experts to contain the Ebola disease spread most especially in Africa.

Mathematical Model of Phytoplankton Interspecific Competition for Food Resource

The phytoplankton in an aquatic ecosystem is the basis of its life activity and the main producing link. The functioning of phytoplankton in same time depends on environmental factors: mineral nutrition, photosynthetically active solar radiation, water temperature and other less significant ones. Sunlight is a stable factor, varying predictably over time and space. The water temperature is the small regulatory factor. Concentrations of mineral substances can change quite quickly and significantly, this much influences on plant organisms. Thus, mineral nutrition is a basic environmental factor of influence to phytoplankton. On the other hand, in large aquatic basin such as seas and oceanic areas the distribution of living organisms is very heterogeneous in space. These two aspects – nutrient and spatial heterogeneity – are the focus of this article. A model of competitive interaction is considered using the example of two species of phytoplankton. The phytoplankton move passively in water what is simulated by the diffusion process. The model contains one non-trivial stationary and spatially homogeneous equilibrium and two trivial ones, i.e. degenerate in at least one species of phytoplankton. Trivial equilibria are stable only in some “degenerate” situations. The non-trivial equilibrium in “normal” conditions is stable to temporal and spatial disturbances. The behavior of solutions near a nontrivial equilibrium for a stationary living environment and in cases of its nonstationary is studied. Perturbation of a nontrivial equilibrium in a stationary environment leads to relatively long-term deviations from equilibrium and a slow return to it. The instability of trivial equilibria increases the spatial heterogeneity of solutions. At the same time, the nontrivial equilibrium computationally demonstrates weak properties of global stability in time. The unsteadiness of the environment is simulated by the unsteadiness of the influx of nutrients. It has been shown that the distribution of nutrients can lead to significant heterogeneity in the distribution of individuals across the spatial habitat.

Bifurcación de Hopf en un sistema autónomo presa-predador con crecimiento logístico y respuesta funcional de Holling tipo II

The autonomous prey-predator system with logistic growth and Holling type II functional response, which describes the population dynamics of two species. In this study, the equilibrium points of the system (1.1) were identified. Two saddle-node points and one non-trivial P3 equilibrium point were found. For this latter point, the conditions were determined for the Jacobian matrix of (1.1), evaluated at P3, to have a pair of purely complex eigenvalues (necessary condition for the Hopf bifurcation). Through this analysis, values c0, δ0, k0 were found that satisfied these conditions. Subsequently, each of these values is considered as a bifurcation parameter value, and the remaining two are considered as control parameters, under the assumptions of the normal form theorem for the Hopf bifurcation, it’s concluded that by varying these values slightly, the system undergoes the Hopf bifurcation. Finally, the first Lyapunov coefficient was calculated to determine the conditions under which the system exhibits supercritical, subcritical, and degenerate Hopf bifurcation. The analysis was supported by using MAPLE and MATLAB software, which enabled graphical visualization of the obtained results.

An Investigation of Dynamical Behavior of a Wing Model

AbstractBifurcations of equilibria of a wing model have been investigated in this paper. It is shown that the quintic nonlinear terms in the pitch and the plunge coordinate have affected the bifurcation structure of nontrivial equilibria in different degree. In contrast with the quintic stiffening parameter in plunge, the quintic parameter in pitch has a relatively significant effect, which will affect the number, position and stability of nontrivial equilibria. Therein two pairs of nontrivial equilibria with opposite stability coexist or disappear by two fold bifurcations. If the freestream velocity has been taken as a continuation parameter, it will affect the bifurcation structure of all the equilibria, including the trivial and the nontrivial. Wherein the equilibria vary from a trivial to two nontrivial ones by a pitchfork bifurcation. Then one of nontrivial equilibria experiences a supercritical Hopf bifurcation and the bifurcated limit cycles form an ellipsoidal structure with the limit cycles bifurcated from the trivial equilibrium.

Limit Cycles from Perturbed Center on the Invariant Algebraic Surface of Unified Lorenz-Type System

For a three-dimensional chaotic system, little seems to be known about the perturbation of invariant algebraic surface and the center on this surface. This question is very interesting and worth investigating. This paper is devoted to analyzing the limit cycles from perturbed center (trivial and nontrivial equilibria) on the invariant algebraic surface of the unified Lorenz-type system (ULTS), which contains some common chaotic systems as its particular cases. First, based on the parameter-dependent center manifold, we obtain the approximate two-dimensional center manifold from the perturbation of invariant algebraic surface, as well as the two-dimensional system on this center manifold. Second, by applying the averaging method of third order to the above two-dimensional system, we show that under suitable perturbation of parameters of the ULTS, there is one limit cycle bifurcating from the perturbed center on the invariant algebraic surface of the ULTS, and the stability of this limit cycle is determined as well. By using the averaging method of fourth order, we show the same results with the averaging method of third order. Finally, numerical simulation is used to verify the theoretical analyses.

Parametric resonance for pipes conveying fluid in thermal environment

Parametric resonance is a kind of specific fluid-induced vibration for pipes conveying fluid. This paper focuses on revealing the qualitative characteristics of parametric resonances of the pipe with pulsating fluid speed in thermal environment, and compares the differences between the resonance characteristics in subcritical and supercritical regions. According to the generalized Hamilton's principle, the partial-differential-integral governing equation of a straight pipe is established. Under simply-supported boundary conditions, the non-trivial equilibrium configuration is obtained analytically. The governing equation of a supercritical pipe is derived by coordinate substitution on the basis of the new equilibrium configuration. The approximate analytical solution of parametric resonance for a pipe conveying fluid in a thermal environment is obtained by using the direct multi-scale method. The approximate analytical solution is verified to be reliable by using the Runge-Kutta method. The stability bounds of the parameters inducing parametric resonance are investigated via the introduced analytical method. The results show that the influence of temperature increment on sub-harmonic resonance is non-monotonic. In the subcritical region, when the temperature increment increases, the unstable region widens. This means that the system is more prone to parametric resonance, and the response amplitude decreases. In the supercritical region, when the temperature increment increases, the unstable bandwidth decreases, and the response amplitude increases. When viscous damping is increased or the average velocity is decreased, the sub-critical instability region decreases, and the supercritical instability region increases. With the increase in the pulsation velocity, the unstable bandwidth of subcritical and supercritical regions will increase, and parametric resonance is more likely to occur.

A well-balanced and exactly divergence-free staggered semi-implicit hybrid finite volume / finite element scheme for the incompressible MHD equations

We present a new exactly divergence-free and well-balanced hybrid finite volume/finite element scheme for the numerical solution of the incompressible viscous and resistive magnetohydrodynamics (MHD) equations on staggered unstructured mixed-element meshes in two and three space dimensions. The equations are split into several subsystems, each of which is then discretized with a particular scheme that allows to preserve some fundamental structural features of the underlying governing PDE system also at the discrete level. The pressure is defined on the vertices of the primary mesh, while the velocity field and the normal components of the magnetic field are defined on an edge-based/face-based dual mesh in two and three space dimensions, respectively. This allows to account for the divergence-free conditions of the velocity field and of the magnetic field in a rather natural manner. The non-linear convective and the viscous terms in the momentum equation are solved at the aid of an explicit finite volume scheme, while the magnetic field is evolved in an exactly divergence-free manner via an explicit finite volume method based on a discrete form of the Stokes law in the edges/faces of each primary element. The latter method is stabilized by the proper choice of the numerical resistivity in the computation of the electric field in the vertices/edges of the 2D/3D elements. To achieve higher order of accuracy, a piecewise linear polynomial is reconstructed for the magnetic field, which is guaranteed to be exactly divergence-free via a constrained L2 projection. Finally, the pressure subsystem is solved implicitly at the aid of a classical continuous finite element method in the vertices of the primary mesh and making use of the staggered arrangement of the velocity, which is typical for incompressible Navier-Stokes solvers. In order to maintain non-trivial stationary equilibrium solutions of the governing PDE system exactly, which are assumed to be known a priori, each step of the new algorithm takes the known equilibrium solution explicitly into account so that the method becomes exactly well-balanced. We show numerous test cases in two and three space dimensions in order to validate our new method carefully against known exact and numerical reference solutions. In particular, this paper includes a very thorough study of the lid-driven MHD cavity problem in the presence of different magnetic fields and the obtained numerical solutions are provided as free supplementary electronic material to allow other research groups to reproduce our results and to compare with our data. We finally present long-time simulations of Soloviev equilibrium solutions in several simplified 3D tokamak configurations, showing that the new well-balanced scheme introduced in this paper is able to maintain stationary equilibria exactly over very long integration times even on very coarse unstructured meshes that, in general, do not need to be aligned with the magnetic field.

The asymptotic behavior of solutions for stochastic evolution equations with pantograph delay

The polynomial stability problem of stochastic delay differential equations has been studied in recent years. In contrast, there are relatively few works on stochastic partial differential equations with pantograph delay. The present paper is devoted to investigating large-time asymptotic properties of solutions for stochastic pantograph delay evolution equations with nonlinear multiplicative noise. We first show that the mild solutions of stochastic pantograph delay evolution equations with nonlinear multiplicative noise tend to zero with general decay rate (including both polynomial and logarithmic rates) in the [Formula: see text]th moment and almost sure senses. The analysis is based on the Banach fixed point theorem and various estimates involving the gamma function. Moreover, by using a generalized version of the factorization formula and exploiting an approximation technique and a convergence analysis, we construct the nontrivial equilibrium solution, defined for [Formula: see text], for stochastic pantograph delay evolution equations with nonlinear multiplicative noise. In particular, the uniqueness, Hölder regularity in time and general stability, in the [Formula: see text]th moment and almost sure senses, of the nontrivial equilibrium solution are established.