Bridging the gap between models based on ordinary, delayed, and fractional differentials equations through integral kernels.

Evolution equations with convolution-type integral operators have a history of study, yet a gap exists in the literature regarding the link between certain convolution kernels and new models, including delayed and fractional differential equations. We demonstrate, starting from the logistic model structure, that classical, delayed, and fractional models are special cases of a framework using a gamma Mittag-Leffler memory kernel. We discuss and classify different types of this general kernel, analyze the asymptotic behavior of the general model, and provide numerical simulations. A detailed classification of the memory kernels is presented through parameter analysis. The fractional models we constructed possess distinctive features as they maintain dimensional balance and explicitly relate fractional orders to past data points. Additionally, we illustrate how our models can reproduce the dynamics of COVID-19 infections in Australia, Brazil, and Peru. Our research expands mathematical modeling by presenting a unified framework that facilitates the incorporation of historical data through the utilization of integro-differential equations, fractional or delayed differential equations, as well as classical systems of ordinary differential equations.

Lotka-Volterra model with Allee effect: equilibria, coexistence and size scaling of maximum and minimum abundance.

The Lotka-Volterra competition model (LVCM) is a fundamental tool for ecology, widely used to represent complex communities. The Allee effect (AE) is a phenomenon in which there is a positive correlation between population density and fitness, at low population densities. However, the interplay between the LVCM and AE has been seldom analyzed in multispecies models. Here, we analyze the mathematical properties of the LVCM [Formula: see text] AE, investigating the coexistence of species interacting through neutral diffuse competition, their equilibria and stable points. Minimum viable population density arises as the threshold below which species go extinct, characteristic of strong Allee effects. Then, by imposing relationships of main parameters to body size, i.e. allometric scaling, we derive a general solution to the size-scaling maximum and minimum expected density under plausible scenarios. The scaling of maximum population density is consistent with the literature, but we also provide novel predictions on the scaling of the lower limit to population density, a critical value for conservation science. The resulting framework is general and yields results that increase our current understanding of how complex demographic processes can be linked to ubiquitous ecological patterns.

Limitations and applications in a fractional Barbalat's Lemma.

Barbalat's Lemma is a mathematical result that can lead to the solution of many asymptotic stability problems. On the other hand, Fractional Calculus has been widely used in mathematical modeling, mainly due to its potential to make explicit the dependence of previous stages through nonlocal operators. In this work, we present a fractional Barbalat's Lemma and its proof, as proposed in [31]. The proof is analyzed in order to show an imprecision. In fact, for orders 0 < α < 1 , we are not able to get the supreme limit of the integrand. Then, a counterexample and a corrected version of the lemma are presented, according to [9]. The objective of this work is to draw attention to the potential and limitations of a fractional Barbalat's Lemma, given its wide use in recent articles. In a fractional SIR model, we exhibit the constraint of the result by introducing a non-periodic relapse. So, the supreme limit could not be verified. Also in this context, we provide a general discussion of the classical Calculus' properties that are not inherited if we change the integer orders to fractional ones.

Antihypertensive Activity of the Alkaloid Aspidocarpine in Normotensive Wistar Rats.

The alkaloid Aspidocarpine was isolated from the bark of Aspidosperma desmanthum. Its structure was elucidated by the spectral data of 1H and 13C-NMR (1D and 2D) and high-resolution mass spectrometry (HRESIMS). The antihypertensive activity was investigated by intravenous infusion in Wistar rats. This alkaloid significantly reduced (p < 0.05) the systolic, median, and diastolic blood pressures of rodents, without causing motor incoordination and imbalance in the rotarod test. The results indicate that the alkaloid Aspidocarpine exerts its antihypertensive activity without causing sedation or the impairment of motor functions.

New hydroperoxycycloartane-complete chemical shifts assignment of 13C and 1H-and cytotoxicity evaluation of cycloartanes isolated from Trichilia casaretti.

New cycloartane, 22-hydroxy-25-hydroperoxycycloart-23E-en-3-one (1), along with six known analogues (2-7) and three steroids (8-10), were isolated from the leaves of Trichilia casaretti. Structures were elucidated mainly on the basis of the analysis of 1D and 2D NMR (1H and 13C) and HRESIMS spectroscopic data, involving comparison with data of the literature. The cytotoxic activities of 1-7 and 10 isolated compounds were also evaluated against human leukemia cell line Molt-4 (acute lymphoblastic) and exhibited good cytotoxic activity with IC50 values ranging from 10.62 to 21.14 µM.

Chemical Compounds and Biologic Activities: A Review of Cedrela Genus.

The genus Cedrela P. Browne, which belongs to the Meliaceae family, has eighteen species. Trees of this genus are of economic interest due to wood quality, as well as being the focus of studies because of relevant biologic activities as in other Meliaceae species. These activities are mainly related to limonoids, a characteristic class of compounds in this family. Therefore, the aim of this review is to perform a survey of the citations in the literature on the Cedrela genus species. Articles were found on quantitative and qualitative phytochemical studies of the Cedrela species, revealing the chemical compounds identified, such as aliphatics acid and alcohol, flavonoids, tocopherol, monoterpenes, sesquiterpenes, triterpenes, cycloartanes, steroids, and limonoids. Although some activities were tested, the majority of studies focused on the insecticidal, antifeedant, or insect growth inhibitor activities of this genus. Nonetheless, the most promising activities were related to their antimalarial and antitripanocidal effects, although further investigations are still needed.