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We derive an alternative expression for a delayed logistic equation in which the rate of change in the population involves a growth rate that depends on the population density during an earlier time period. In our formulation, the delay in the growth term is consistent with the rate of instantaneous decline in the population given by the model. Our formulation is a modification of Arino et al. (J Theor Biol 241(1):109-119, 2006) by taking the intraspecific competition between the adults and juveniles into account. We provide a complete global analysis showing that no sustained oscillations are possible. A threshold giving the interface between extinction and survival is determined in terms of the parameters in the model. The theory of chain transitive sets and the comparison theorem for cooperative delay differential equations are used to determine the global dynamics of the model. We extend our delayed logistic equation to a system modeling the competition between two species. For the competition model, we provide results on local stability, bifurcation diagrams, and adaptive dynamics. Assuming that the species with shorter delay produces fewer offspring at a time than the species with longer delay, we show that there is a critical value, [Formula: see text], such that the evolutionary trend is for the delay to approach [Formula: see text]., (© 2022. The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.)

In this study, we focus on lateral size effects of graphene nanosheets as conductive additives for LiNi 0.5 Co 0.2 Mn 0.3 O 2 (NCM) cathode materials for Li-ion batteries. We used two different lateral sizes of graphene, 13 (GN-13) and 28 µm (GN-28). It can be found that the larger sheet sizes of graphene nanosheets give a poorer rate capability. The electrochemical measurements indicate that GN-13 delivers an average capacity of 189.8 mAh/g at 0.1 C and 114.2 mAh/g at 2 C and GN-28 exhibits an average capacity of 179.4 mAh/g at 0.1 C and only 6 mAh/g at 2 C. Moreover, according to the results of alternating current (AC) impedance, it can be found that the GN-28 sample has much higher resistance than that of GN-13. The reason might be attributed to that GN-28 has a longer diffusion distance of ion transfer and the mismatch of particle size between NCM and GN-28. The corresponding characterization might provide important reference for Li-ion battery applications.

A system of impulsive differential equations with state-dependent impulses is used to model the growth of a single population on two limiting essential resources in a self-cycling fermentor. Potential applications include water purification and biological waste remediation. The self-cycling fermentation process is a semi-batch process and the model is an example of a hybrid system. In this case, a well-stirred tank is partially drained, and subsequently refilled using fresh medium when the concentration of both resources (assumed to be pollutants) falls below some acceptable threshold. We consider the process successful if the threshold for emptying/refilling the reactor can be reached indefinitely without the time between successive emptying/refillings becoming unbounded and without interference by the operator. We prove that whenever the process is successful, the model predicts that the concentrations of the population and the resources converge to a positive periodic solution. We derive conditions for the successful operation of the process that are shown to be initial condition dependent and prove that if these conditions are not satisfied, then the reactor fails. We show numerically that there is an optimal fraction of the medium drained from the tank at each impulse that maximizes the output of the process.