Narrow Results

This work presents the numerical investigations for the deterministic prey–predator model. It is a reaction-diffusion mathematical model for the two-prey interacting with a single predator. The predator–prey model is an important part of the ecosystem, especially for recognizing how the population collaborates in their natural surroundings. The computational numerical scheme, namely, implicit finite difference scheme is developed. The numerical solution of the underlying model is successfully gained — the positivity of the reaction-diffusion model is established. To show the feasibility of the scheme, consistency, positivity, and stability are established by applying mathematical techniques. The numerical scheme yields stable and consistent approximations that remain positive throughout the domain. Moreover, the non-negative initial conditions and homogeneous Neumann conditions are used. For various parameter values, 3D, 2D, and parametric graphs are produced. In addition, the mathematical results are verified through numerical calculations.

In this paper, we investigate the spatiotemporal patterns of solutions to diffusive nonlocal Nicholson’s blowflies equations, wherein a natural death rate of the immature population is included in the distribution function. We first prove the positivity and boundedness of positive solutions in the model by using the minimum principle and the method of lower and upper solutions. Subsequently, we conduct a detailed bifurcation and stability analysis to obtain conditions on all the diffusion coefficients and the death rate coefficient of the immature population required for the emergence of spatiotemporal patterns, including spatially nonhomogeneous time periodic orbits. Our results indicate that the model can undergo Hopf bifurcation when the diffusion rate of the mature population passes through a sequence of critical values. Additionally, we examine the dependence of Hopf bifurcation points and bifurcated oscillations on model parameters, including the diffusion rate and death rate of the immature population. Finally, we report numerical simulations based on the bifurcation analysis to demonstrate the theoretical results, and it will help us better understand the ecological characteristics and behavioral patterns of the blowfly population.

In this short note we discuss some properties of multiplier ideal sheaves on singular varieties, for example, in how far they are compatible with fibre products.

In the present manuscript, we introduce a finite-difference scheme to approximate solutions of the two-dimensional version of Fisher's equation from population dynamics, which is a model for which the existence of traveling-wave fronts bounded within (0,1) is a well-known fact. The method presented here is a nonstandard technique which, in the linear regime, approximates the solutions of the original model with a consistency of second order in space and first order in time. The theory of M-matrices is employed here in order to elucidate conditions under which the method is able to preserve the positivity and the boundedness of solutions. In fact, our main result establishes relatively flexible conditions under which the preservation of the positivity and the boundedness of new approximations is guaranteed. Some simulations of the propagation of a traveling-wave solution confirm the analytical results derived in this work; moreover, the experiments evince a good agreement between the numerical result and the analytical solutions.

During this analysis, as per natural control approach in pest management, a plant-pest dynamics with biological control is proposed, here assuming that the pest and natural enemy are having different levels of gestation delay and harvesting rate of pests by natural enemy follows Holling type-III response function. Boundedness and positivity of the system are studied. Equilibria and stability analysis is carried out for possible equilibrium points. The existence of Hopf bifurcation at interior equilibrium is presented. The sensitivity analysis of the system at interior equilibrium point for model parameters has been explored. Numerical simulations are performed to support our analytic findings.

This paper introduces a one-dimensional NPZD-model developed to simulate biological activity in a turbulent ocean water column. The model consists of a system of coupled semilinear parabolic equations. An initial-boundary value problem is formulated and the existence of a unique positive weak solution to it is proved. The existence result is derived using a variational formulation, an approximate model and a fixed-point method. It is shown that the qualitative analysis performed still applies if different parametrizations of several biological processes found in the biogeochemical modeling literature are used.

All the variables of the biological models are positive. We examine the constraints that one has to put on the model to verify such a property. For a linear (differential) model, it implies that the equilibrium is stable. For the n-dimensional Lotka-Volterra models, the positive orthant is invariant, but we impose stronger constraints to prevent the solutions to go upon or below some given thresholds, that define the space scale of the model. Then we show that, under reasonable hypothesis, this implies the global convergence towards the equilibrium.

A new model for the growth of a single-species with a two-stage structure is developed employing a state-dependent age of maturity. The paper discusses some remarks in a previous model where the time to maturity is state dependent. Then, we present a new model which is mainly based on making the maturation period of juveniles depend on the total population size not at the present time t but at an earlier time stage, namely, the time when they were born.

The paper considers both biological and mathematical aspects. Emphasis is given to the basic theory of the solutions, such as local and global properties of existence, uniqueness, positivity and boundedness.

Fractal interpolation is an advance technique for visualization of scientific shaped data. In this paper, we present a new family of partially blended rational cubic trigonometric fractal interpolation surfaces (RCTFISs) with a combination of blending functions and univariate rational trigonometric fractal interpolation functions (FIFs) along the grid lines of the interpolation domain. The developed FIFs use rational trigonometric functions $\frac{p_{i,j}(\theta )}{q_{i,j}(\theta )}$, where $p_{i,j}(\theta )$ and $q_{i,j}(\theta )$ are cubic trigonometric polynomials with four shape parameters. The convergence analysis of partially blended RCTFIS with the original surface data generating function is discussed. We derive sufficient data-dependent conditions on the scaling factors and shape parameters such that the fractal grid line functions lie above the grid lines of a plane $\mathrm{\Pi }$, and consequently the proposed partially blended RCTFIS lies above the plane $\mathrm{\Pi }$. Positivity preserving partially blended RCTFIS is a special case of the constrained partially blended RCTFIS. Numerical examples are provided to support the proposed theoretical results.

The appearance of fractal interpolation function represents a revival of experimental mathematics, raised by computers and intensified by powerful evidence of its applications. This paper is devoted to establish a method to construct $\alpha$-fractal rational quartic spline, which eventually provides a unified approach for the generalization of various traditional nonrecursive rational splines involving shape parameters. We deduce the uniform error bound for the $\alpha$-fractal rational quartic spline when the original function is in ${{𝒞}}^4(I)$. By solving a system of linear equations, appropriate values of the derivative parameters are determined so as to enhance the continuity of the $\alpha$-fractal rational quartic spline to ${{𝒞}}^2$. The elements of the iterated function system are identified befittingly so that the class of $\alpha$-fractal function $Q^{\alpha }$ incorporates the geometric features such as positivity, monotonicity and convexity in addition to the regularity inherent in the germ $Q$. This general theory in conjunction with shape preserving aspects of the traditional splines provides algorithms for the construction of shape preserving fractal interpolation functions.

In this paper, we study shape preserving aspects of bivariate $\alpha$-fractal functions. Its specific aims are: (i) to solve the range restricted problem for bivariate fractal approximation (ii) to establish the fractal analogue of lionized Weierstrass theorem of bivariate functions (iii) to study the constrained approximation by ${{𝒞}}^r$-bivariate $\alpha$-fractal functions (v) to investigate the conditions on the parameters of the iterated function system in order that the bivariate $\alpha$-fractal function $f^{\alpha }$ preserves fundamental shapes, namely, positivity and convexity (concavity) in addition to the smoothness of $f$ over a rectangle (vi) to establish fractal versions of some elementary theorems in the shape preserving approximation of bivariate functions.

Measles is a highly transmissible disease in children around the world. According to the World Health Organization (WHO), 73% of deaths of children were due to measles in 2018. This study describes the physical solution of the SIQR model for measles spread under the effect of natural delay amongst different compartments. By three different numerical techniques, the efficacies of solutions of the underlying system have been compared and a clear preference of nonstandard finite-difference (NSFD) scheme over the rest has been established. It has also been observed, on principle, that the NSFD formulation recovers all the essential traits of a continuous model namely the boundedness, positivity and stability of equilibriums of populations. The numerical results have also been supported by a very strong classical analysis of the model where the existence of a solution vector in explicit subsets of the function spaces has been guaranteed which leads to optimization of fixed-point methods.

The aim is to study the dynamics of Coronavirus model using stochastic methods. Threshold parameter $R_0$ is obtained for the model. Afterwards, both the disease-free equilibrium (DFE) and endemic equilibrium (EE) points are acquired and the stability of the model is discussed. Both the equilibrium points are locally asymptotically stable. Euler–Maruyama, stochastic Euler scheme (SES), stochastic fourth-order Runge–Kutta scheme (SRKS) and stochastic non-standard finite difference technique (SNFDT) are applied to solve the model equations. Euler–Maruyama, SES, SRKS fail for large time step size, while, SNFDT preserves the dynamics of the proposed model for any step size. Numerical comparison of applied methods is provided using different step sizes.

We study the positivity of the first Chern class of a rank $r$ Ulrich vector bundle $𝜀$ on a smooth $n$-dimensional variety $X\subseteq {\mathbb{P}}^N$. We prove that $c_1(𝜀)$ is very positive on every subvariety not contained in the union of lines in $X$. In particular, if $X$ is not covered by lines we have that $𝜀$ is big and $c_1{(𝜀)}^n\ge r^n$. Moreover we classify rank $r$ Ulrich vector bundles $𝜀$ with $c_1{(𝜀)}^2=0$ on surfaces and with $c_1{(𝜀)}^2=0$ or $c_1{(𝜀)}^3=0$ on threefolds (with some exceptions).

Motivated by the finiteness of the set of automorphisms Aut($X$) of a projective manifold of general type $X$, and by Kobayashi–Ochiai’s conjecture that a projective manifold dim($X$)-analytically hyperbolic (also known as strongly measure hyperbolic) should be of general type, we investigate the finiteness properties of Aut($X$) for a complex manifold satisfying a (pseudo-) intermediate hyperbolicity property. We first show that a complex manifold $X$ which is (dim($X$)-1)-analytically hyperbolic has indeed finite automorphisms group. We then obtain a similar statement for a pseudo-(dim($X$)-1)-analytically hyperbolic, strongly measure hyperbolic projective manifold $X$, under an additional hypothesis on the size of the degeneracy set. Some of the properties used during the proofs lead us to introduce a notion of intermediate Picard hyperbolicity, which we last discuss.

In this note we are concerned with the solution of Forward–Backward Stochastic Differential Equations (FBSDE) with drivers that grow quadratically in the control component (quadratic growth FBSDE or qgFBSDE). The main theorem is a comparison result that allows comparing componentwise the signs of the control processes of two different qgFBSDE. As a by-product one obtains conditions that allow establishing the positivity of the control process.

Let Q be an acyclic quiver and let be the corresponding cluster algebra. Let H be the path algebra of Q over an algebraically closed field and let M be an indecomposable regular H-module. We prove the positivity of the cluster characters associated to M expressed in the initial seed of when either H is tame and M is any regular H-module, or H is wild and M is a regular Schur module which is not quasi-simple.

We show that the ordered rings naturally associated to compact convex polyhedra with interior satisfy a positivity property known as order unit cancellation, and obtain other general positivity results as well.

We consider Volterra integral equations and arbitrary order integro-differential equations. We establish positivity conditions and two-sided estimates for Green's functions. These results are then applied to obtain stability and positivity conditions for equations with nonlinear causal mappings (operators) and linear integro-differential parts. Such equations include differential, difference, differential-delay, integro-differential and other traditional equations.

Gaussian states — or, more generally, Gaussian operators — play an important role in Quantum Optics and Quantum Information Science, both in discussions about conceptual issues and in practical applications. We describe, in a tutorial manner, a systematic operator method for first characterizing such states and then investigating their properties. The central numerical quantities are the covariance matrix that specifies the characteristic function of the state, and the closely related matrices associated with Wigner's and Glauber's phase space functions. For pedagogical reasons, we restrict the discussion to one-dimensional and two-dimensional Gaussian states, for which we provide illustrating and instructive examples.