Mathematical Methods in the Applied Sciences最新文献

Super-resolution (SR) remains a critical challenge in imaging science, particularly in applications demanding fine-texture reconstruction and noise suppression. In this work, we introduce a novel nonvariational anisotropic diffusion model, formulated as a high-order nonlinear reaction-diffusion system, designed to enhance SR capabilities. Our approach utilizes a decomposition strategy based on the $��{�H�}^{�-�1�}�$ norm, which effectively captures and preserves fine details and intricate textures in images. Theoretical analysis establishes the well-posedness of the model via the Schauder fixed-point theorem, ensuring mathematical rigor. To validate its performance, we implement a finite difference numerical scheme, testing the model against state-of-the-art SR techniques on challenging image datasets. Results demonstrate significant improvements in texture restoration, edge sharpness, and robustness to noise and degradation, surpassing competitive methods. This research lays the foundation for advanced PDE-based SR frameworks, offering significant potential for applications in biomedical imaging, satellite vision, and high-fidelity computational photography.

Effective integrated pest management should control pests while safeguarding the environment and human health. This study proposes a novel pest management system incorporating cooperative predation, fear effect, impulsive density-dependent nonlinear prey (pest) trapping and predator (natural enemy) release. The conditions for global asymptotic stability of the prey extinction periodic solution and system permanence are established. A supercritical bifurcation occurs when the impulsive control period serves as a bifurcation parameter. This implies that a transition from a prey extinction periodic solution to a positive periodic solution of prey–predator coexistence when the impulsive period exceeds a critical value. Meanwhile, simulations are conducted to verify the theoretical results, determine the key factors influencing pest extinction threshold condition, and explore the complex dynamics of the system. This analysis will aid future research and the development of effective pest control strategies.

In this paper, we study a class of fractional Kirchhoff-Choquard equations with fixed mass constraints in $��{�ℝ�}^{�3�}�$. Based on the Pohozaev manifold, by establishing the compactness lemma of the Palais–Smale sequence and using Ekeland variational principle, the existence of solutions is obtained and their asymptotic behavior is analyzed for $��s�\in �\left(�\frac{�3�}{�4�}�,�1�\right)�$. Through variable substitution, the low-order fractional Kirchhoff equation is transformed into an equivalent system. Combining fractional inequalities and variational methods, the existence of solutions is obtained for $��s�\in �\left(�0�,�\frac{�3�}{�4�}�\right)�$.

The novel case of nonstationary flows of helical type governed by gravity-driven viscous-plastic flow dynamics derived for incompressible non-Newtonian fluid is investigated here analytically in case of given (constant) value of Bernoulli-function. In such types of flow, the velocity is linearly and spatially dependent on curl field. Conditions for the existence of the exact solution for the aforementioned type of flows are obtained. Equations of momentum, continuity, and that one for the components of stress tensor are studied. The spatial and time-dependent structure of pressure field of the fluid flow should be determined via constant Bernoulli-function if components of the velocity of the flow are already obtained. It is remarkable that the case of non-stationary helical or Beltrami flow for incompressible gravity-driven viscous-plastic flow governed by three-dimensional equations has been investigated for the first time to the best of our knowledge.

MSC Classification: 35Q35, 76D17

We study the two-dimensional Cauchy–Dirichlet problems for the quasilinear pseudoparabolic integro-differential Volterra equation and for the quasilinear pseudoparabolic integro-differential Fredholm equation. In each of these two problems, the integral term depends on a positive integer parameter $��n�$ and, as $��n�\to �+�\infty �$, converges weakly$��{}^{�\star �}�$ to the expression incorporating the Dirac delta-function, which models an instantaneous impulsive impact. For fixed $��n�\in �\mathbb{N}�$ we prove the well-posedness of each of the two problems in the class of regular weak solutions. Further, for each of the two problems, we establish that the initial shock layer, associated with the Dirac delta-function, is formed as $��n�\to �+�\infty �$, and that the family of regular weak solutions of the original problem converges to the strong solution of a limit two-scale microscopic–macroscopic model.

In this paper, we put forward and study a modified Leslie-Gower predator-prey model with Crowley-Martin functional response and Allee effect in the growth rate of a predator population. Crowley–Martin-type functional response is considered to describe mutual interference between predators. Our analysis shows that the intensity of Allee effect affect the stability of all equilibria, and by plotting the two-parameter bifurcation diagram of the degree of mutual interference between predators and the intensity of Allee effect, it is found that under their influence, the system will appear Bi-stable phenomenon. In addition, by considering the intensity of Allee effect as a bifurcation parameter, the one-parameter bifurcation diagram of the predator appears saddle-node bifurcation and Hopf bifurcation. Finally, Turing instability occurs in the reaction-diffusion system. The study reveals that as the intensity of the Allee effect increases, predators face greater survival pressure, their population growth becomes limited, predation patterns changes, the distribution of the prey population changes, and low-density areas of the prey decrease. As mutual interference between predators increases, the distribution patterns of both predators and prey shift significantly. High interference reduces high-density prey clusters, while predator distribution, though fluctuating, becomes smoother overall. This study highlights the complex interaction between Allee effect and predator-predator interaction in predator-prey dynamics. The results may provide biological insights and ecological management suggestions for predator-prey interaction.

Anthropisation and excessive hunting in tropical forests threaten biodiversity, ecosystem maintenance, and human food security. In this article, we focus on the issue of coexistence between humans and wildlife in an anthropised environment. Assuming that humans move daily between their residential area and the surrounding forest to hunt, we study a resource-consumer model with consumer migration. A comprehensive analysis of the system is carried out using classical theory and monotone systems theory. We show that the possibilities for long-term coexistence between human populations and wildlife populations are determined by hunting rate thresholds. Depending on the level of anthropisation and the hunting rate, the system may converge towards a limit cycle or a co-existence equilibrium. However, the conditions for coexistence become more difficult as anthropisation increases. Numerical simulations are provided to illustrate the theoretical results.