This research introduces a novel two-parameter family of orthogonal polynomials that emerge as solutions to a doubly confluent Heun-type differential equation. We investigate these polynomials, examining their geometric properties and analyzing the behavior and distribution of their zeros under varying parameter conditions. Leveraging machine learning techniques, we successfully derive symbolic expressions for the corresponding weight functions associated with these orthogonal polynomials. Our numerical results demonstrate the efficacy of this approach, achieving a maximum absolute error of order ${10}^{-4}$ in weight function approximation. Furthermore, we present a comparison between our proposed model and conventional approximation methods, including cubic spline interpolation and Lagrange polynomial interpolation, highlighting the advantages of our methodology.

In this article, we derived conditions on the order $\nu$ of the classical Bessel functions $J_{\nu }$ that guarantee the inclusion of three distinct normalized forms of $J_{\nu }$ in a subclass of $\alpha$-spirallike functions. The primary goal was to determine the subintervals within $(-\frac{\pi }{2},\frac{\pi }{2})$ where these inclusion conditions are satisfied. A key component in establishing our results was the upper bound of the ratio $J_{\nu +1}(1)/J_{\nu }(1)$. The theoretical findings were validated through numerical experiments and accompanying graphical demonstrations.

This paper presents a nutritional state-structured model (NSM) to explore the dynamics of starvation and recovery in consumer-resource systems, focusing on full consumers ( $F$), hungry consumers ( $H$), and resources ( $R$). The model employs a system of differential equations to capture ecological processes such as reproduction, starvation, and resource regeneration. Through bifurcation analysis, we identified critical thresholds, notably the starvation rate ( $\sigma$) relative to the reproduction rate ( $\lambda$), that dictate system stability, transitioning between extinction and coexistence equilibria. Parameter sensitive and numerical simulations revealed how parameter variations influence population persistence and resource sustainability, with $\sigma >\lambda$ promoting balanced ecosystems and $\lambda >\sigma$ leading to potential overexploitation. The analogue of the basic reproduction number ( ${\mathcal{R}}_{\mathrm{e}}$) was derived using the next-generation matrix method, providing insights into the invasion dynamics and stability conditions of the system. This framework serves as a robust tool for analyzing eco-evolutionary interactions and assessing population persistence under resource-limited conditions. Finally, we demonstrated how higher fat reserves enhance competitive advantage, thereby driving the evolutionary trend toward larger body sizes as predicted by Cope's rule.