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We explore local dynamics at fixed points, existence of periodic points and bifurcation analysis of a three species discrete predator-prey model. More precisely, it is explored that model has trivial fixed point, five boundary fixed points and the interior fixed point under definite parametric conditions. We have also constructed the linearized system for the model under consideration. Then we have investigated local stability analysis with topological classifications at each fixed points by linear stability theory. Further existence of periodic points and bifurcation analysis at each fixed points are also demonstrated. The extensive numerical simulations are provided to demonstrate theoretical results.

Humans have observed complex behaviors presented by nature. They have observed behaviors resembling power-law, to model these problems; they have used fractional differential operators with a power-law kernel. They have observed behaviors resembling a declining process, to model these problems; they used fractional differential operators based on exponential decay kernel. They have observed crossover behaviors from fading process to power-law, to model these problems; they used fractional derivatives based on the generalized Mittag-Leffler function. However, they have also observed a passage from randomness to power-law or vice-versa, or even from deterministic to randomness, these behaviors could not be depicted using above mentioned operators. To overcome these problems, the concepts of piece-wise differential and integral calculus have been introduced. In this paper, we adopt such a concept to model the dynamical interaction of predator–prey, several scenarios are considered and simulations are provided for each case. According to the results, it seems that the idea of using piece-wise operators seems provides a better understanding of the behavior of dynamic systems.

The Ambartsumian equation, based on the modified Riemann–Liouville fractional derivative, is analyzed in this paper. The solution is expressed as a power series of arbitrary powers and its convergence has been proven. In addition, we show that the present solution reduces to the results in the literature when the fractional derivative tends to 1. Moreover, the behavior of the obtained solution is discussed through figures.

Chemical investigation of Carthamus tinctorius L. flowers resulted in isolation of seven metabolites that were identified as; p-Hydroxybenzoic acid (1), trans hydroxy cinnamic acid (2), kaempferol-6-C-glucoside (3), astragalin (4), cartormin (5), kaempferol-3-O-rutinoside (6), and kaempferol–3-O-sophoroside (7). Virtual screening of the isolated compounds against human intestinal α-glucosidase, acetylcholinesterase, and butyrylcholinesterase was carried out. Additionally, the antioxidant activity of the bioactive compounds was assessed. Compounds 1 and 5 exhibited moderate binding affinities to acetylcholinesterase (binding energy −5.33 and −4.18 kcal/mol, respectively), compared to donepezil (-83.33kcal/mol). Compounds 1–7 demonstrated weak affinity to butyrylcholinesterase. Compounds 2 and 4 displayed moderate binding affinity to human intestinal α-glucosidase,compared to Acarbose (reference compound), meanwhile compound 2 exhibited lower affinity. Molecular dynamic studies revealed that compound 4 formed a stable complex with the binding site throughout a 100 ns simulation period. The in-vitro results were consistent with the virtual experimental results, as compounds 1 and 5 showed mild inhibitory effects on acetylcholinesterase (IC50s 150.6 and 168.7 µM, respectively). Compound 4 exhibited moderate α-glucosidase inhibition with an IC50 of 93.71 µM. The bioactive compounds also demonstrated notable antioxidant activity in ABTS [2,2′-azino-bis(3-ethylbenzothiazoline-6-sulfonic acid)], ORAC (oxygen radical-absorbance capacity), and metal chelation assays, suggesting their potential in improving dementia in Alzheimer’s disease (AD) and mitigating hyperglycemia.