Volterra Type Research Articles

Global dynamics of a periodically forced SI disease model of Lotka–Volterra type

In this paper, we investigate the dynamics of an SI disease model of Lotka–Volterra type in the presence of a periodically fluctuating environment. We give a global analysis of the dynamical behavior of the model. Interestingly, our results show that the permanence guarantees the existence of a unique positive harmonic time-periodic solution which is globally attracting when the horizontal disease transmission has a weaker impact than the intraspecific competition. While for the case when the horizontal disease transmission has a stronger impact than the intraspecific competition, we numerically show that complex dynamics such as chaos can occur in a permanent system. Nonetheless, we provide sufficient conditions for the existence and uniqueness of the positive harmonic time-periodic solution for the latter case. The impact of the environment on the spread of disease is studied by using a bifurcation analysis. We show that in each of the qualitatively different cases of the associated autonomous SI model in a constant environment, an alternative possibility can appear in the periodic model.

Stability and convergence computational analysis of a new semi analytical-numerical method for fractional order linear inhomogeneous integro-partial-differential equations

Abstract The motive behind this research work, is to originate a semi-analytical-numerical approach for the solution of fractional order linear integro partial differential equations (FOLIPDEs), especially in-homogeneous FOLIPDEs of different types, like fractional version of Fredholm and Volterra type IEs etc. To attain the said purpose, we will study some fractional formulations of linear model integral equations from the existing literature. Then we will describe the proposed semi analytical-numerical procedure alongwith its stability analysis and convergence properties. We will then prove with the aid of some particular numerical examples that the said approach is not only lucid and efficient but also precise. The outcomes of the proposed technique will show that this scheme has notable prospectives for solving a large class of FOLIEs. At the end, the contribution of this work in the advancements of semi analytical-numerical approaches for
the solution of fractional order linear integro partial differential equations will be laid down and discussion for future research in this field.

Discretization methods and their extrapolations for two-dimensional nonlinear Volterra-Urysohn integral equations

In this paper, a class of nonlinear two-dimensional (2D) integral equations of Volterra type, i.e. Volterra-Urysohn integral equations, is studied. Following the ideas of [24], and assuming that the kernels of the integral equation are Lipschitz functions with respect to the dependent variable, the existence and uniqueness of a solution to the integral equation is shown by a technique based on the Picard iterative method. Then, the Euler and trapezoidal discretization methods are used to reduce the solution of the integral equation to the solution of a system of nonlinear algebraic equations. It is proved that the solution of the Euler method has first order convergence to the exact solution of the integral equation while the solution of the trapezoidal method has quadratic convergence. To prove the convergence of the trapezoidal method, a new Gronwall inequality is developed. Some numerical examples are given which confirm our theoretical results.

The moving least square method for solving the time fractional partial integro-differential equation of Volterra type and its convergence analysis

The moving least square method for solving the time fractional partial integro-differential equation of Volterra type and its convergence analysis

Mathematical model of the spread of a pandemic like COVID-19

Using the example of the infectious disease called COVID-19, a mathematical model of the spread of a pandemic is considered. The virus that causes this disease emerged at the end of 2019 and spread to most countries around the world over the next year. A mathematical model of the emerging pandemic, called the SEIR-model (from the English words susceptible, exposed, infected, recovered), is described by a system of four ordinary dynamical equations given in §1. The indicated system is reduced to a nonlinear integral equation of Hammerstein–Volterra type with an operator that does not have the property of monotonicity. In §3, we prove a theorem on the existence and uniqueness of a non-negative, bounded and summable solution of this system. Based on real data on the COVID-19 disease in France and Italy, given in §2, numerical calculations are performed showing the absence of a second wave for the obtained solution.

On existence of extremal integrable solutions and integral inequalities for nonlinear Volterra type integral equations

We prove the existence of maximal and minimal integrable solutions of nonlinear Volterra type integral equations. Two basic integral inequalities are obtained in the form of extremal integrable solutions which are further exploited for proving the boundedness and uniqueness of the integrable solutions of the considered integral equation.

Numerical approximation of a generalized time fractional partial integro-differential equation of Volterra type based on a meshless method

The moving least squares (MLS) approximation is used in this study to solve time-fractional partial integro-differential equations (TFPIDE). In our approach, we approximate the time Caputo fractional derivative term and the integral term by using a finite difference scheme and trapezoidal method, respectively, which yields an order of accuracy of O(τ+τ2−α), where 0<α<1. We thoroughly investigate the applicability and validity of the proposed method and provide an error estimate for its performance. Additionally, we solve various numerical problems to demonstrate the accuracy and computational efficiency of our approach, confirming the theoretical findings.

Study of a Diseased Volterra Type Population Model featuring Prey Refuge and Fear Influence

In order to study the local stability characteristics of a predator-prey dynamical model, this work proposes a Volterra-type model that takes into account the fear influence of prey resulting from predator domination. Because of an outbreak of disease in the prey species, the prey gets classified as either healthy or diseased. Both predator and prey species compete for their resources. In addition, the prey sought refuge against the predator. All these factors are addressed when setting up the mathematical model. The biological validity of the model is ensured by testing its boundedness. The equilibrium points have been identified. The short-term behavior of the system is analyzed at all equilibrium points. Routh Hurwitz conditions are employed to examine the local stability property.

Weekly Ill-posed integral geometry problems of Volterra type in three-dimensional space

In this paper considers the problem of recovering a function from families of spheres in space. The uniqueness of the solution of the problem is proved by reducing it to the Volterra integral equation of the first and then the second kind. Fourier transform methods are also used. Uniqueness theorems are proved for some new classes of operator equations of Volterra type in three-dimensional space.

Utilizing Cubic B-Spline Collocation Technique for Solving Linear and Nonlinear Fractional Integro-Differential Equations of Volterra and Fredholm Types

Fractional integro-differential equations (FIDEs) of both Volterra and Fredholm types present considerable challenges in numerical analysis and scientific computing due to their complex structures. This paper introduces a novel approach to address such equations by employing a Cubic B-spline collocation method. This method offers a robust and systematic framework for approximating solutions to the FIDEs, facilitating precise representations of complex phenomena. Within this research, we establish the mathematical foundations of the proposed scheme, elucidate its advantages over existing methods, and demonstrate its practical utility through numerical examples. We adopt the Caputo definition for fractional derivatives and conduct a stability analysis to validate the accuracy of the method. The findings showcase the precision and efficiency of the scheme in solving FIDEs, highlighting its potential as a valuable tool for addressing a wide array of practical problems.

Bounded solutions and exponential stability for linear integro-differential equations of Volterra type

For the following scalar functional–differential equation ẍ(t)+∫t0tG(t,s)ẋ(s)ds+∫t0tH(t,s)x(s)ds=f(t),t≥t0,we obtain sufficient conditions for boundedness of all solutions and uniform exponential stability for the homogeneous equation. As examples, equations with singular kernels and equations with bounded delays ẍ(t)+∫g(t)tG(t,s)ẋ(s)ds+∫h(t)tH(t,s)x(s)ds=f(t),t≥t0,are considered.

THE DEVELOPMENT OF A PIEZOELECTRIC DEFECT DETECTION DEVICE THROUGH MATHEMATICAL MODELING APPLIED TO POLYMER COMPOSITE MATERIALS

In this research article, the focus is on examining the strength characteristics of polymer composite materials under various stress states. The investigation into the material dissolution process is conducted in two distinct stages. Initially, the article employs a criterion approach to derive explicit formulas for the latent decay time (incubation period) of materials, considering singular, exponential and Abelian kernels of the damage operator. In the subsequent stage, the classical strength condition is applied to establish a Volterra type integral equation, which characterizes the damage process for hereditary materials. These resulting differential equations are then solved with appropriate initial conditions. The study utilizes the obtained results to create time-dependent graphs of the damage parameter and performs comparisons. A piezoelectric defect detection device for polymer composite materials has been proposed through the application of mathematical modeling.

Solution of a Cauchy type problem for an integral equation of Volterra type with singular kernels, when the roots of the characteristic equations are complex conjugate

Solution of a Cauchy type problem for an integral equation of Volterra type with singular kernels, when the roots of the characteristic equations are complex conjugate

Dynamics of Quadratic Volterra-Type Stochastic Operators Corresponding to Strange Tournaments

By studying the dynamics of these operators on the simplex, focusing on the presence of an interior fixed point, we investigate the conditions under which the operators exhibit nonergodic behavior. Through rigorous analysis and numerical simulations, we demonstrate that certain parameter regimes lead to nonergodicity, characterized by the convergence of initial distributions to a limited subset of the simplex. Our findings shed light on the intricate dynamics of quadratic stochastic operators with interior fixed points and provide insights into the emergence of nonergodic behavior in complex dynamical systems. Also, the nonergodicity of quadratic stochastic operators of Volterra type with an interior fixed point defined in a simplex introduces additional complexity to the already intricate dynamics of such systems. In this context, the presence of an interior fixed point within the simplex further complicates the exploration of the state space and convergence properties of the operator. In this paper, we give sufficiency and necessary conditions for the existence of strange tournaments. Also, we prove the nonergodicity of quadratic stochastic operators of Volterra type with an interior fixed point, defined in a simplex.

Global solvability of a two-species chemotaxis–fluid system with Lotka–Volterra type competitive kinetics

In this paper, we study a two-species chemotaxis–fluid system with Lotka–Volterra type competitive kinetics in a bounded and smooth domain [Formula: see text] with no-flux/Dirichlet boundary conditions. We present the global existence of weak energy solution to a two-species chemotaxis Navier–Stokes system, and then the global weak energy solution which coincides with a smooth function throughout [Formula: see text], where [Formula: see text] represents a countable union of open intervals which is such that [Formula: see text]. In such two-species chemotaxis–fluid setting, our existence improves known blow-up prevention by logistic source to blow-up prevention by sub-logistic source, indicating standard logistic source is not the weakest damping source to prevent blow-up. This finding significantly extends previously known ones.

Modelling effects and control of soil contaminants on the dynamics of plant–herbivore species system

In this paper, we have proposed and analysed a mathematical model to study the effects of impulsive input of contaminants in soil on the dynamics of interacting species system consisting of contaminants, soil-nutrients, plant biomass and herbivore population in order to investigate the criteria for the survival or extinction of plant biomass and herbivore population. The uptake function for herbivore population considered in the model of the paper shows saturation effect and this type of uptake function behave linearly with respect to plant biomass for low density. Also, the saturation for large plant biomass is a reflection of the limited herbivore capability or perseverance when the plant biomass is abundant. Therefore, the uptake function considered in this paper is more natural to lotka volterra and holling type functional responses. We have shown that the plant–herbivore extinction periodic solution exists and also the plant–herbivore extinction periodic solution is locally and globally stable if the impulsive depletion rate of nutrients due to contaminants is more than some critical value. Further, it is concluded that if the impulsive input period for contaminants is increased and impulsive depletion rate of nutrients due to contaminants is decreased then both the plant and herbivore population will survive otherwise they will tend to extinction.

Dynamics for a diffusive prey–predator model with advection and free boundaries

This article deals with a free boundary problem of the Lotka–Volterra type prey–predator model with advection in one space dimension. The model considered here may be applied to describe the expanding of an invasive or new predator species adopting a combination of random movement and advection upward or downward along the resource gradient, with the free boundaries representing expanding fronts of predator species. The main purpose of this article is to understand the influence of the advection environment on the dynamics of the model. We provide sufficient conditions for spreading and vanishing of the predator species, and we find a sharp threshold between the spreading and vanishing concerning this model. Moreover, for the case of successful spreading for the predator, we give estimates of asymptotic spreading speeds and nonlocal long‐time behavior of the prey and the predator.

Efficient numerical schemes based on the cubic B-spline collocation method for time-fractional partial integro-differential equations of Volterra type

Efficient numerical schemes based on the cubic B-spline collocation method for time-fractional partial integro-differential equations of Volterra type

Traveling waves of modified Leslie–Gower predator–prey systems

This paper focuses on the spreading phenomena within modified Leslie–Gower reaction–diffusion predator–prey systems. Our main objective is to investigate the existence of two distinct types of traveling waves. Specifically, with the aid of the upper and lower solution methods, we establish the existence of traveling waves connecting the prey-present state and the coexistence state or the prey-present state and the prey-free state by constructing different and appropriate Lyapunov functions. Moreover, for traveling wave connecting the prey-present state and the prey-free state, more information about the monotonicity of the wave profile can be obtained by analyzing its asymptotic behavior at negative infinity. Finally, our results are applied to modified Leslie–Gower system with Holling-II type functional response or Lotka–Volterra type functional response, and a novel Lyapunov function is constructed for the latter, which further enhances our results. Meanwhile, some numerical simulations are carried out to support our results.

Theoretical understanding of evolutionary dosing following tumor dynamics

Recent preclinical and clinical studies have suggested that evolution-based cancer therapies exploiting intratumor competition can significantly delay resistance. Various mathematical and computational models have been used to explain the effectiveness of particular treatment strategies. However, this evolutionary approach often assumes only preexisting resistance, although acquired resistance and cancer phenotype plasticity can dramatically affect treatment outcomes. We theoretically investigate whether an evolutionary therapy that capitalizes on plasticity could delay resistance. In particular, we propose a time-dependent evolutionary dose that can modulate the evolving competition among drug-sensitive, drug-resistant, and plastic cells. We assume a Lotka–Volterra type competition model of cancer cell populations with constant carrying capacity to analyze the dynamics of tumor growth to show that some tumors can be controlled with proper dosing, which balances the competition between different cell species so that they coexist in a stable tumor below a tolerable burden. The emergence of a new resistant species may further increase the tumor volume, which can be nullified by lowering the dose. In contrast, increasing the tumor volume due to new metastatic formation requires a higher dose. Additionally, we propose tumor-dynamics-driven adaptive scheduling consisting of subsequent treatment holidays and low- and high-dose periods. This approach can contain the tumor at variable volumes below the tolerable burden, which may facilitates reiteration of the treatment strategy in the case of any evolutionary changes. Overall, this study provides a theoretical understanding of the evolutionary dosing that can delay resistance.