In this paper, we consider a predator-prey model with Cosner type functional response and combined harvesting. First, we explore the existence and stability of the equilibria. Then using the center manifold theorem and normal form theory, we investigate codimension one and codimension two bifurcations of the model. The analysis shows that the system has a variety of bifurcation phenomena including transcritical bifurcation, saddle-node bifurcation, Hopf bifurcation, Bogdanov-Takens bifurcation and homoclinic bifurcation. Our findings indicate that the dynamics with harvesting are significantly richer than the system without harvesting. Finally, numerical simulations are provided to support the analytical results.

A general approach is developed for discriminating strong and hereditary symmetric operators. The recursion operator of the Blaszak-Marciniak (BM) equation hierarchy is proved to be strong and hereditary symmetric. As an example of discrete soliton equations related to 3×3 matrix spectral problems, the τ-symmetries and Lie algebra structure of the BM equation are built firstly.

We address the problem of existence, uniqueness and approximation of the solution of a large class of fractional integral differential equations with boundary conditions. Based on fixed point techniques with Boyd-Wong type conditions, the existence and uniqueness of the solution is investigated. This, together with the use of certain types of biorthogonal systems, allows us to propose a method to approximate the solution which is tested with several examples.

In this article, we attempt to provide a more general method based on Petryshyn’s fixed-point theorem to ensure the existence of solutions to implicit functional equations. These implicit functional equations include fractional, non-fractional, (fractional) stochastic integral equations, etc., and any combination of them in C( I). Some results regarding the existence of fixed points in implicit functional integral equations will be reviewed in the literature. We show that this general result unifies and improves many of the main results in the literature. To illustrate that our approach is more general than other methods, we present some concrete examples. Also, we apply our method to create new functional equations in practice and check the existence of solutions.

The well-posedness and regularity properties of diffusion-aggregation equations, emerging from interacting particle systems, are established on the whole space for bounded interaction force kernels by utilizing a compactness convergence argument to treat the non-linearity as well as a Moser iteration. Moreover, we prove a quantitative estimate in probability with arbitrary algebraic rate between the approximative interacting particle systems and the approximative McKean--Vlasov SDEs, which implies propagation of chaos for the interacting particle systems.

In this paper, we develop and analyze a mathematical model for spreading malaria, including treatment with Transmission Blocking Drugs (TBDs). The paper’s main aim is to demonstrate the impact the chosen model for demographic growth has on the disease’s transmission and the effect of its treatment with TBDs. We calculate the model’s control reproduction number and equilibria, and perform a global stability analysis of the disease-free equilibrium point. The mathematical analysis reveals that, depending on the model’s demography, the model can exhibit forward, backward and even some unconventional types of bifurcation, where disease elimination can occur for both small and large values of the reproduction number. We also conduct a numerical analysis to explore the short-time behavior of the model. A key finding is that for one type of demographic growth, the population experienced a significantly higher disease burden than the others, and when exposed to high levels of treatment with TBDs, only this population succeeded in effectively eliminating the disease within a reasonable timeframe.

This paper proposes a dynamic programming approach targeted to solve a natural resource problem of water storage management for irrigation in an environmentally and socially sustainable way. The problem we examine in our formulation, which deals with the control of water storage in tanks, incorporates assumptions that are milder compared to those previously discussed in the literature concerning similar approaches. Specifically, we consider a time periodic optimal control problem whose performance functional to be maximized merely satisfies a contraction assumption (in the sense of Boyd and Wong) weaker than the one usually considered in the pertinent literature. By using an appropriate fixed point theorem, a time periodic value function can be constructed to enable the definition of optimal feedback control strategies. After showing the insufficiency of Banach contraction frameworks, and proving the underlying auxiliary mathematical results, we show our main result under conditions that are weaker than the previous related ones. Simulation results illustrate the performance of our approach.

In this paper, we consider the Ricker model with delay and constant or periodic stocking. The impact of delay and stocking on stability is known to reflect opposing effect, which motivates investigating the interaction between these factors and their influence on overall stability. We found that the high stocking density tends to neutralize the delay effect. Conditions are established on the parameters in order to guarantee the global stability of the equilibrium solution in the case of constant stocking, as well as the global stability of the 2-periodic solution in the case of 2-periodic stocking. Our approach extensively relies on the utilization of the embedding technique. Whether constant stocking or periodic stocking, the mode has the potential to undergo a Neimark-Sacker bifurcation in both cases. However, the Neimark-Sacker bifurcation in the 2-periodic case results in the emergence of two invariant curves that collectively function as a single attractor. Finally, we pose open questions in the form of conjectures about global stability for certain choices of the parameters.

Synovial joints are the joints that act as excellent bearings under suitable biological conditions. Among them, the hip joint is one of the most essential synovial joints and plays a very vital role in joint movement. These joints have synovial fluid, which acts as a lubrication agent, helps in mobility, and prevents wear and tear. Lubrication is vital to reduce friction between cartilages during activities. A mathematical model for the hip joint squeeze film lubrication action has been represented geometrically by a sphere approaching a flat plate and developed to estimate the pressure distribution, load capacity, and squeeze time film thickness. Finite element stress analysis of the bone-implant system was carried out assuming micro-motion between the stem and the femur. Here, we aimed to study squeeze film lubrication of the hip joint for a Newtonian fluid flow. Lubrication of the human hip joint is the focus of our attention in this paper. Using a modified Reynolds equation, we aim to study squeeze-film lubrication for a Newtonian fluid. It was applied to present the human hip joint's pressure distribution, squeeze time, and load carry capacity in squeeze action. Squeeze film pressure rises when the squeeze velocity of articular cartilages and synovial fluid increases. Squeeze time is affected by the thickness of the articular cartilage. The squeeze film traits may be critical in determining the presence of sick synovial joint diseases. The early identification of joint illness and the creation of biomedical implants can benefit from research on synovial fluid and articular cartilage characteristics. Squeeze film alterations can signify joint disease, with changes in synovial fluid composition or viscosity suggesting rheumatoid arthritis and osteoarthritis. We may infer that an increase in pressure leads to a diseased condition in which the human hip joint is less flexible. As a result, it is possible to conclude that exercise increases the human hip joint's capacity to hold weights, whereas illness diminishes it.

This paper investigates an optimal investment-reinsurance problem for an insurer who possesses extra information regarding the future realizations of the claim process and risky asset process. The insurer sells insurance contracts, has access to proportional reinsurance business, and invests in a financial market consisting of three assets: one risk-free asset, one bond and one stock. Here, the nominal interest rate is characterized by the Vasicek model; and the stock price is driven by the Heston’s stochastic volatility model. Applying the enlargement of filtration techniques, we establish the optimal control problem in which an insurer maximizes the expected power utility of the terminal wealth. By using the dynamic programming principle, the problem can be changed to four-dimensional Hamilton-Jacobi-Bellman equation. In addition, we adopt a deep neural network method by which the partial differential equation is converted to two backward stochastic differential equations and solved by a stochastic gradient descent-type optimization procedure. Numerical results obtained using TensorFlow in Python and the economic behavior of the approximate optimal strategy and the approximate optimal utility of the insurer are analyzed.

This paper discusses the Maxwell system of electrodynamics in the context of distributions. It is used to establish boundary conditions for fields at the interface when the charge and current densities are concentrated measures on the interface. The paper derives the generalized Snell's law from this analysis.

We consider an alternative approach to investigate three-dimensional exterior mixed boundary value problems (BVP) for the steady state oscillation equations of the elasticity theory for isotropic bodies. The unbounded domain occupied by an elastic body, Ω − ⊂ R 3 , has a compact boundary surface S = ∂ Ω − , which is divided into two disjoint parts, the Dirichlet part S D and the Neumann part S N , where the displacement vector (the Dirichlet type condition) and the stress vector (the Neumann type condition) are prescribed respectively. Our new approach is based on the classical potential method and has several essential advantages compared with the existing approaches. We look for a solution to the mixed boundary value problem in the form of a linear combination of the single layer and double layer potentials with densities supported on the Dirichlet and Neumann parts of the boundary respectively. This approach reduces the mixed BVP under consideration to a system of boundary integral equations, which contain neither extensions of the Dirichlet or Neumann data nor the Steklov-Poincaré type operator involving the inverse of a special boundary integral operator, which is not available explicitly for arbitrary boundary surface. Moreover, the right-hand sides of the resulting boundary integral equations system are vector-functions coinciding with the given Dirichlet and Neumann data of the problem in question. We show that the corresponding matrix integral operator is bounded and coercive in the appropriate L 2 -based Bessel potential spaces. Consequently, the operator is invertible, which implies unconditional unique solvability of the mixed BVP in the class of vector-functions belonging to the Sobolev space [ W 2 , loc 1 ( Ω − ) ] 3 and satisfying the Sommerfeld-Kupradze radiation conditions at infinity. We also show that the obtained matrix boundary integral operator is invertible in the L p -based Besov spaces and prove that under appropriate boundary data a solution to the mixed BVP possesses C α -Hölder continuity property in the closed domain Ω − ‾ with α = 1 2 − ε , where ε>0 is an arbitrarily small number.

The Moore-Gibson-Thompson equation with a viscoelastic memory and a forcing is considered. The existence and uniqueness of a local solution is obtained via the Faedo-Galerkin's method. Furthermore, blowing-up solutions with or without a positive initial energy exist due to the nonlinear forcing.

The investigation of linear periodic systems is a prominent subject of research within the domain of linear systems theory. In this context, the state-space realization of such systems is of particular significance. In fact, after successfully treating the realization problem for time-invariant systems, various researchers have directed their attention toward investigating the case of linear periodic systems. Meanwhile, in the late 1980s, Jan C. Willems proposed an approach that broadened the range of systems studied, now referred to as the behavioral approach. This approach views the behavior of a system as its fundamental element, including all signals that adhere to the system laws (also called system trajectories). More recently, the behavioral framework has also been extended separately to periodic and quaternionic behavioral systems. Our work consists of considering linear periodic input/output quaternionic behavioral systems and using recent developments in order to obtain quasi-minimal and uniform state-space realizations.

In this paper, Hopf and zero-Hopf bifurcations are investigated for a class of three-dimensional cubic Kolmogorov systems with one positive equilibrium. Firstly, by computing the singular point quantities and figuring out center conditions, we determined that the highest order of the positive equilibrium is eight as a fine focus, which yields that there exist at most seven small amplitude limit cycles restricted to one center manifold and Hopf cyclicity 8 at the positive equilibrium. Secondly, by using the normal form algorithm, we discuss the existence of stable periodic solution via zero-Hopf bifurcation around the positive equilibrium. At the same time, the relevance between zero-Hopf bifurcation and Hopf bifurcation is analyzed via its special case, which is rarely considered. Finally, some related illustrations are given by means of numerical simulation.

In this paper, we study the uniqueness of nodal radial solutions to nonlinear elliptic equations in the unit ball in R 3 . Under suitable conditions, we prove that, for any given positive integer k, the problem we considered has at most one solution possessing exactly k−1 nodes. Together with the results presented by Nagasaki [J. Fac. Sci. Univ. Tokyo Sect. IA Math. 36 (2): 211–232, 1989] and Tanaka [Proc. Roy. Soc. Edinburgh Sect. A. 138 (6): 1331–1343, 2008], we can prove that more types of nonlinear elliptic equations have the uniqueness of nodal radial solutions.

This manuscript is devoted to a derivative-free parametric iterative step to obtain roots simultaneously for both nonlinear systems and equations. We prove that when it is added to an arbitrary scheme, it doubles the convergence order of the original procedure and defines a new algorithm that obtains several solutions simultaneously. Different numerical experiments are carried out to check the behaviour of the iterative methods by changing the value of the parameter and the initial guesses. Also, it is perform a numerical example where the dynamical planes are carried out to see and compare the basins of attraction.

The models for symmetric stochastic matrices that we consider in this study are developed using the spectral analysis of the respective mean matrices. The adjustment and validation of the models require the usage of the estimated structure vectors. The information enclosed in these matrices can be condensed into the pair consisting of the estimated structure vectors and the sum of squares of residuals. The results obtained allow for cross-sectional and longitudinal inference. For models of degree greater than one, it is also considered the possibility of truncating the model when eigenvalues are much higher than the others. A direct consequence of the adopted methodology is the application of the degree-one models to cross-product matrices and Hilbert-Schmidt scalar product matrices. In addition to these models, structured family models were also considered. The models of these families are associated with the treatments of a base design. The action of the factors considered in the base design on the structure vectors is also analyzed. In structured families with orthogonal base design, the designs are associated to partitions, and the hypotheses formulated are associated with the spaces of these partitions. We carry out ANOVA-like analysis for the action of the factors in the base design, on homolog components on estimated structure vectors, considering that the estimator’s structure vectors have, approximately, the same covariance matrix. To apply our results, we assume the factors in the base design to have fixed effects and that the base design has orthogonal structure. The action of factors in the base design is studied. An application is given, using a data set from a breeding program of durum wheat ( Triticum turgidum L., Durum Group) conducted in Portugal. The results show that our methodology is fully applicable to complete and incomplete data sets, often observed in multi-environmental trials.