Journal of Nonlinear Modeling and Analysis
The Journal of Nonlinear Modeling and Analysis (JNMA), started in 2019, aims to publish original research papers, communications as well as survey papers on theory and application of nonlinear analysis related to differential equations, dynamical systems and other areas in mathematics and mathematical sciences, including mathematical biology. The journal was published quarterly in March, June, September and December between 2019 and 2024. From 2025 on, it will be published bimonthly with its issues released in February, April, June, August, October and December.
ESS in a River Network with Two Branches
In a recent work by Jiang et al. [Bull. Math. Biol., 2020, Paper No. 131, 42pp], to discuss the impact of differing topology of river networks on the evolution of dispersal, the authors proposed three different models and found the winning of slower/faster diffuser or evolutionarily singular strategies. However, it is unknown whether there is an evolutionarily stable strategy (ESS, a central concept in evolution game theory) for these three models. In this paper, focusing on the Model (III) proposed by them (the most complicated one as they stated), we give a confirmed answer to this issue, that is, there does exist ESS.
Copyright (c) 2026 Journal of Nonlinear Modeling and Analysis
Cross-Invariant Sets of the Cubic Nonlinear Schrödinger System with Partial Confinement
This paper studies the cubic nonlinear Schrödinger system with partial confinement: \begin{cases}-i\varphi_t + (x_1^2 + x_2^2)\varphi = \Delta\varphi +\mu_1|\varphi|^2\varphi + \beta|\psi|^2\varphi, & (t,x) \in \mathbb{R}^+\times \mathbb{R}^3, \\ -i\psi_t + (x_1^2 + x_2^2)\psi =\Delta\psi + \mu_2|\psi|^2\psi + \beta|\varphi|^2\psi, & (t,x) \in\mathbb{R}^+ \times \mathbb{R}^3,\end{cases} which models the Bose-Einstein condensates with multiple states or the propagation of mutually incoherent wave packets in nonlinear optics. The cross-invariant sets of the evolution flow are obtained by constructing the cross-constrained variational problem. Furthermore, the sharp condition for global existence and blowup of the solutions is derived.
Analysis of a Fractional Order Eco-Epidemiological Model with Prey Infection and Predator Cannibalism
This study proposes a fractional-order prey-predator model that integrates disease dynamics within prey populations, fear induced behavioral responses, and predator cannibalism. The prey population is divided into susceptible and infected classes to represent disease dynamics effectively, while predator cannibalism is modeled as a survival strategy during prey scarcity. Essential properties such as non-negativity, boundedness, and the existence and uniqueness of solutions are thoroughly examined. Stability of equilibrium points is established, and global stability is proved using Lyapunov functions. Numerical simulations validate the theoretical results, highlighting how variations in cannibalism rates and fractional-order parameters influence population dynamics and system behavior.
Optimal Control Approach for Bilateral Elastic Contact Problem with Power-Law Friction
We use the control variational technique to examine an elastic contact model, subject to a non-penetration condition in the normal direction and to power-law friction, proving the unique existence of the solution. This method uses optimal control theory to minimize the energy functional of the nonlinear equation. A multivalued equation $f\in \mathcal{F}y+∂\Phi(y)$ for the displacement field describes the problem in a weak formulation, where a linear mapping is represented by $\mathcal{F},$ and the Clarke’s subdifferential of the mapping $\Phi$ is indicated by $∂\Phi.$ We employ abstract existence theorems to verify the unique weak solution to the contact model.
Coulomb dry friction (power-law friction)
Nonlinear Dynamics and Pull-in Phenomena in a Magneto-Electro MEMS Actuator with Hardening Spring
This paper investigates the dynamic behavior and pull-in instability of a magneto-electro Micro-Electro-Mechanical System (MEMS) actuator, focusing on the nonlinear effects arising from magnetic forces and spring stiffness. The system consists of a movable wire attracted toward a stationary wire due to magnetic forces generated by applied currents. A critical equilibrium, known as the pull-in point, is reached when the currents exceed a threshold, leading to instability. We consider the governing equation based on Newton’s Second Law, incorporating a nonlinear restoring force for the spring, which exhibits hardening behavior. The resulting second-order differential equation is analyzed using qualitative and bifurcation theories, revealing the critical bifurcation values determined by the currents and spring stiffness. Through a dynamical systems approach, we characterize the phase portraits and solutions, identifying distinct dynamical behaviors and the conditions for pull-in instability. Numerical simulations are performed to validate the analytical predictions, demonstrating excellent agreement with the theoretically derived threshold.
On the Numerical Solution of Solidification of Water
The objective of this paper is to find an economical computer algorithm using a third-degree polynomial for the free boundary problem corresponding to phase change problem. We solve the problem to obtain the approximate analytical expression for the temperature distribution and the movement of the interface using the constrained integral method. Results are validated against existing numerical methods, demonstrating strong agreement with other techniques.
Generalized Variational Inclusion Problem Involving Averaged Operator and Cayley Operator
In this paper, we consider and study a variational inclusion problem involving averaged and Cayley operators in the context of a Hilbert space. It is shown that our problem is equivalent to a fixed-point equation. We define an iterative algorithm based on a fixed-point formulation to obtain the solution. Our result is supported by a numerical example, computation tables, and a convergence graph are also provided.
Stability Analysis of a Diffusive Ratio-Dependent Predator-Prey Model with Nonlocal Perception
In this paper, we propose a diffusive ratio-dependent predator-prey model with a nonlocal perceptual term. We first discuss the well-posedness of solutions. Then for the model without the nonlocal perception term, we obtain stability conditions of the spatially homogeneous steady state. Under these conditions, when considering the effects of nonlocal perception, our result indicates that if the system without nonlocal perception term is stable, and even after introducing nonlocal perception term, the system remains stable. However, if the system without nonlocal perception term is unstable, and the introduction of nonlocal perception term will make the system become stable.
The Impact of Educational Campaigns on the Global Dynamics of Cholera Epidemics
In this study, we examine a cholera epidemic model that incorporates educational transmission and delve into the long-term dynamics of the epidemic equilibrium point, particularly in relation to the spread of information about disease transmission. We find that if the basic reproduction number $R_0$ is less than or equal to 1, the disease is destined to be eradicated, and the unique disease-free equilibrium point becomes globally stable. It is noteworthy that the stability of this disease-free equilibrium is not influenced by the extent of disease education, the variety of infections, or the rate of education. On the other hand, if the basic reproduction number is greater than 1, we study the existence of the endemic equilibrium, and demonstrate that the disease will persist if $β_0$ > $D_0$. To assess the impact of educational campaigns on disease control, we further establish the asymptotic behavior of both infected and susceptible populations in response to educational interventions. Our findings indicate that well-executed educational campaigns can substantially contribute to the management and mitigation of the disease’s effects.
Existence Results for a Generalized $p$($x$)-Biharmonic Problem Type with No-Flux Boundary Condition
This paper aims to study the existence and multiplicity of weak solutions for a problem involving a generalized $p$($x$)-biharmonic operator with no flux boundary condition. By using the variational techniques and the theory of the variable exponent Lebesgue spaces, we obtain the existence of at least one nontrivial solution and at least $n$ distinct pairs of nontrivial weak solutions to this problem, respectively.
MEK Inhibition May Prevent the Recruitment of NK Cells in the Tumor Microenvironment: A Mathematical Model of Glioma Treatment
In this paper, we analyze the dynamics of oncolytic virotherapy combined with MEK inhibitors for the treatment of glioma. Although the therapeutic effect of oncolytic viruses (OV) depends on the recruitment of NK cells, these viruses are frequently subject to NK cell clearance, which may reduce the efficacy of oncolytic virotherapy. Furthermore, it is more challenging for OV to enter glioma cells because these cells lack Coxsackievirus and Adenovirus Receptors (CAR). MEK inhibitors, however, are able to compensate for this deficiency in CAR molecules. Herein, we propose and analyze a mathematical model of the dynamics of cancer cells, oncolytic viruses, activated NK cells, and CAR molecules while combining oncolytic virotherapy with MEK inhibition. Our goal is to identify the circumstances in which the treatment may be effective. In this research, we investigated the existence of equilibrium points. The two endemic equilibriums and the virus infection-free equilibrium stability conditions are given. The findings demonstrate that, in a dynamical system, NK cell activation can either establish or destroy equilibrium points and that substantial recruitment of activated NK cells might have detrimental effects on oncolytic virotherapy. However, MEK inhibitors boost OV effectiveness and may prevent NK cell recruitment.
Local Ground-State and Mountain Pass Solutions for a $p$-Kirchhoff Equation with Critical Exponent
We study a Kirchhoff-type equation where the diffusion coefficient is non-locally affected, the nonlinear diffusion phenomenon is governed by the $p$-Laplace operator and the population supply presents critical growth. The energy functional associated with the equation is not bounded from below so that there is no global ground-state; however, we prove the existence of a positive local ground-state. We also prove that the equation has a positive solution of mountain pass type. The concentration-compactness principle is a main tool in our approach.
A Mathematical Modeling of Two-Strain Tuberculosis Transmission: Deterministic and Stochastic Approaches to Screening, Treatment, and Quarantine Strategies
This research observes the transmission dynamics of drug-sensitive tuberculosis (DS-TB), drug-resistant tuberculosis (DR-TB) and undiagnosed tuberculosis infections using deterministic and stochastic models. The study focuses on the impacts of contact rates, screening of latent individuals or high-risk groups, treatment, and quarantine measures on the basic reproduction rate ($R_0$) and the estimated number of infected individuals ($m$). The results show that undiagnosed infected individuals are the greatest factor on the spread of tuberculosis within the system, which emphasizes the importance of screening on latent individuals. Analysis on the effects of screening on latent individuals also emphasizes the importance of treatment and quarantine for both DS-TB and DR-TB infections, respectively. However, analysis on the effects of both treatment and quarantine states that relying on either treatment or quarantine efforts is not enough to stop the spread of the two-strain TB within the system; thus a combined strategies is required to help eradicate the disease.
Bullen-Type Fractional Integral Inequalities Involving Twice-Differentiable Mappings with Applications
In this paper, we first introduce a novel extension of the Bullen-type inequalities using Riemann-Liouville integral operators and establish a generalized fractional integral identity of the Bullen type. Next, we derive integral inequalities related to the Bullen-type inequalities for twice-differentiable mappings, utilizing this novel extended identity. To demonstrate these inequalities, we provide examples along with corresponding graphs. Furthermore, we investigate how these inequalities can be practically applied to mean inequalities, contributing to a deeper comprehension and wider usefulness of these mathematical ideas. Finally, the inequalities given in this work extend several known inequalities in the literature, which is the main advantage of the newly found inequalities.
New Refinements of Hermite-Hadamard Inequalities for Quantum Integrals
In this study, we first introduce two functions including quantum integrals. Then, we prove some properties of these mappings such as convexity and monotony. Moreover, by using the newly defined mappings, we prove some refinements of the left hand side of Hermite-Hadamard inequality for left and right quantum integrals.
The Second-Order Melnikov Function and Chaotic Tangles on Toral van der Pol Equation
The van der Pol equation on the torus is considered. This equation contains a heteroclinic cycle consisting of four symmetric heteroclinic orbits. By high order Melnikov method, the periodic forced van der Pol equation is investigated and chaotic dynamics is obtained. An explicit formula of the second-order Melnikov function is derived for the splitting heteroclinic connection. It is used to detect chaotic dynamics when the first-order(classical) Melnikov function is degenerate. By the second-order Melnikov function, we deduce chaotic heteroclinic tangles with rigorously theoretical analysis as well as numerical simulations.
Copyright (c) 2025 Journal of Nonlinear Modeling and Analysis
Application of the Functional Variable Method to Some Nonlinear Evolution Equations
The functional variable method is a highly effective approach for deriving exact solutions to nonlinear evolution equations. It offers broad applicability in addressing nonlinear wave equations. In this paper, the functional variable method is employed to obtain soliton solutions for the Radhakrishnan-Kundu-Lakshmanan (RKL) equation and the Landau-Ginzburg-Higgs equation. Exact solutions play a crucial role in uncovering the internal mechanisms of physical phenomena. Graphical representations of the obtained optical soliton solutions are provided to illustrate some of their physical parameters.
Bifurcation Analysis of Predator-Prey Mathematical Model with Resource-Limited Growth and Predator Switching Behavior
Ecology examines the interactions between organisms and their environment, with a particular focus on population dynamics, resource availability, and predator-prey relationships. This study presents a mathematical model designed to investigate the interactions between two prey populations, one in an unprotected region and the other in a protected region, along with a predator population and shared resource availability. The model employs nonlinear differential equations to capture processes such as prey growth, predation, and resource utilization. By identifying equilibrium points and conducting eigenvalue analysis, we assess the system’s stability. Numerical simulations demonstrate a range of outcomes, including stable states, cyclic behavior, and population collapse, depending on ecological conditions (parametric values). Biologically, predator-prey coexistence equilibria may lose stability, shifting to extinction or dominance scenarios. This makes bifurcation point. Bifurcation analysis reveals how competition and predation impact stability, with critical points marking transitions between coexistence, oscillations, or extinction. These results underscore the intricate balance of ecological forces and emphasize the significance of resource management and conservation in preserving ecosystem stability.
A Semi-Analytical Study of a Non-Linear Initial Value Problem for the Lassa Fever Model
Mathematical modelling plays a crucial role in comprehending and exercising control over Lassa fever outbreaks. The Homotopy Analysis Method (HAM) is employed to address the system of nonlinear differential equations and generate a semi-analytical solution for the Lassa disease Susceptible-Exposed-Infected-Recovered (SEIR) model. Comparing with numerical simulations, the HAM technique yielded extremely precise approximate solutions, proving its accuracy and efficiency. The numerical simulation is carried out using MATLAB programming. There is a good agreement between the numerical simulation and the approximate analytical results. We also examine the effects of changing model parameters on the different compartments, obtaining insightful knowledge on how the model behaves in various scenarios. This analysis is crucial for understanding how variations in transmission rates, recovery rates, and other key factors affect the overall dynamics of Lassa fever, thereby guiding more effective public health strategies and disease management. Moreover, it underscores the potential of the Homotopy Analysis Method (HAM) as a powerful tool for exploring epidemic models and formulating control strategies.
Fractional Order Solution to Analyze Tumor-Immune Cells by Using Hybrid Natural Transform Method and Residual Power Series Method
The purpose of this study is to examine the impact of fractional-order chemotherapy drug diffusion on tumor-immune cell growth. To address the proposed system of fractional-order tumor-immune cell dynamics, two distinct and robust methods have been employed. Initially, the Hybrid Natural Decomposition method, an analytical approach that merges two effective techniques – Natural Transform and the Adomian Decomposition Method has been applied. Subsequently, the Residual Power Series method, a numerical technique, has been employed to derive the solution in series form. The graphical comparison of the results obtained from both methods with those from classical-order solutions has been provided. The findings clearly demonstrate that the proposed approaches offer a reliable and accurate means of understanding the intricate interactions between tumor growth and immune cell activity.
Properties Around the First Eigenvalue of a Partial Discrete Dirichlet Boundary Value Problem
In this article, we study some results around the first eigenvalue $\lambda_{1,m}$ for the partial discrete Dirichlet boundary value problem, like the constant sign of the eigenfunction associated with $\lambda_{1,m}$, the simplicity of $\lambda_{1,m}$ and the sign change of any eigenfunction associated with $\lambda > \lambda_{1,m}$.
Exact Soliton of Fifth Order (1+1) Dimensional Triple Non-Linear Partial Differential Equations on Modified Truncated Expansion Methods
In this manuscript, modified truncated expansion method-I taking the traveling wave variable $u(x, t)=y(r)=y(kx+wt)$ and modified truncated expansion method-II introducing the traveling wave variable $u(x, t) = y(r)=y(kx − wt)$ are built up to obtain analytical solution in the form of traveling wave solutions with different frequencies and velocities that can be constructed for triple (1+1) dimensional nonlinear partial differential equations (NLPDEs) such as Sawada-Kotera equation (SKE), generalized Korteweg-de Vries equation (GKdVE) and Kaup-Kuperschmidt equation (KKE), which have been widely used in mathematical physics. The present topic minimizes the complex nature and non-integrable characteristics to obtain solutions of NLPDEs. To demonstrate the influence of the parameters, 3D plots are generated for triple NLPDEs. This content is employed in physics such as magneto sound in plasma and nonlinear optics.
triple nonlinear partial differential equations
Dynamical Properties of a Stochastic Tumor-Immune Model with Pulsed Chemotherapeutic Dose Response
Comprehensive pulsed chemotherapy and immunotherapy are widely employed in clinical tumor treatment. Given the periodically pulsed nature of this approach, we propose a stochastic tumor-immune dynamical model with a pulsed chemotherapeutic dose response. The model accounts for the combined effects of pulsed chemotherapy and pulsed immunotherapy, as well as the influence of environmental random disturbances. We prove the existence and uniqueness of a global positive solution to the proposed model. By using comparison theorems for impulsive differential equations, we show the boundedness of the solution’s expectation. Furthermore, we derive sufficient conditions for the extinction and non-persistence in the mean of tumor cells, hunting T-cells, and helper T-cells, as well as for the weak persistence in the mean of tumor cells and helper T-cells, and the stochastic persistence of tumor cells. The results of our study, supported by numerical simulations, demonstrate that random disturbances can effectively inhibit tumor cell growth.
impulsive stochastic differential equation
Three Solutions for a Perturbed Integral Equation with Homogeneous Dirichlet Condition
We consider the integral equation $-\mathcal{L}_K^p u = \lambda f(x,u) + \mu g(x,u),$ with homogeneous Dirichlet condition on a bounded Lipschitz domain of $\mathbb{R}^N$ where $\lambda, \mu \in \mathbb{R},$ $p \geq 2,$ $s \in ]0,1[,$ $N > ps,$ $f,g : \mathbb{R}^N \rightarrow \mathbb{R}$ are Carathéodory functions with subcritical growth and $-\mathcal{L}^p$ denotes a class of operators that includes $(-\Delta)_p^s,$ the fractional $p$-Laplacian. Here $\mu g$ represents a small perturbation of $\lambda f.$ Applying an abstract critical point theorem due to Ricceri, a variational setting developed by Xiang et al. and a Minti-Browder's theorem, we prove the existence of three weak solutions.
On Dynamics of Certain Models via Generalized Conformable Fractional Derivative
This paper deals with the generalized conformable fractional derivative and certain interesting properties which are not compatible with Riemann-Liouville and Caputo fractional derivatives. The newly defined derivative is more efficient than other conformable fractional derivatives and the nonlocal fractional derivatives from a time perspective. To justify the claim, we provide some direct applications, such as population growth, Newton’s body cooling, heat equation and susceptible-infected-removed models. Solutions obtained from models and comparison with respective previous data are demonstrated with the help of graphs or stems.
Minimizing the Eigenvalue Ratio for the $p$-Laplacian Operator
We focus on the minimization problem of the eigenvalue ratio for the $p$-Laplacian operator with Robin boundary conditions on an interval $[0, \hat{π}],$ where $\hat{\pi} = \frac{2\pi}{p \sin(\pi/p)}.$ Using variational techniques and Prüfer-type transformations, we show that the constant weight is not minimizing for the class of concave weights.
A New Study on Common Fixed Points for ($α$-$β$-$F$)-Contraction Mappings
This paper explores the existence of common fixed points for a newly introduced class of mappings called ($α$-$β$-$F$)-contraction mappings in metric spaces. We establish several common fixed point theorems under specific conditions, incorporating ($α$-$β$-$F$)-weak contractions to extend and generalize existing results in fixed point theory. Unlike many classical approaches, our framework retains flexibility while accommodating key structural properties such as compactness and continuity. To support the theoretical findings, illustrative examples are provided. These results enhance the understanding and applicability of fixed point theory.
Bifurcation Analysis of a Discrete Predator-Prey Model with Gompertz Growth and Increased Functional Response
This study examines a discrete predator-prey model that employs a Gompertz growth function for the prey and a Holling type I functional response. Initially, the research explores the existence and local stability of fixed points within the system, employing a fundamental lemma. Subsequently, the conditions necessary for the emergence of transcritical and Neimark-Sacker bifurcations of the system are established through the application of the center manifold theorem and bifurcation theory. Finally, numerical simulations are performed to confirm the existence of the Neimark-Sacker bifurcation.
Spatiotemporal Dynamics of a Memory-Diffusion Predator-Prey System with Two Delays and Nonlocal Competition
Predator gestation delay and memory period, along with nonlocal competition, play key roles in controlling population density and maintaining the stability of ecosystems. To control the population density of the Dendrolimus superans, which causes significant damage to forests, we propose a pest control system incorporating both two delays and nonlocal competition considering Holling II type functional response. We analyze the conditions for the existence of Hopf bifurcation and derive the normal form of Hopf bifurcation using multiple time scales method. Considering the biological significance, we employ appropriate parameters for numerical simulations. Furthermore, we find that varying habitat complexity parameters leads to different moduli of bifurcation periodic solutions. Especially, both two delays contribute to maintaining the stability of steady states.
A New Approach to Hermite-Hadamard-Type Inequality with Proportional Caputo-Hybrid Operator
Fractional calculus plays a crucial role in mathematics and applied sciences as it extends classical analysis, overcoming many of its constraints. Moreover, using the innovative hybrid fractional operator, which merges the proportional and Caputo operators, is beneficial in numerous domains of computer science and mathematics. In this study, we focus on the proportional Caputo-hybrid operator due to its wide range of applications. Firstly, we present a new extension of Hermite-Hadamard-type inequalities for the proportional Caputo-hybrid operator and derive an identity. Then, utilizing this novel generalized identity, we establish significant integral inequalities associated with the right-hand side of Hermite-Hadamard-type inequalities for the proportional Caputo-hybrid operator. Furthermore, we provide illustrative examples accompanied by the graphs to demonstrate the newly established inequalities.