Multiple periodic solutions in generalized Gause-type predator–prey systems with non-monotonic numerical responses

By using a continuation theorem based on coincidence degree theory, we establish easily verifiable criteria for the existence of multiple positive periodic solutions in periodic Gause-type predator–prey systems with non-monotonic numerical responses and time delays. As corollaries, some applications are listed. In particular, our results improve and supplement those obtained by Chen [Y. Chen, Multiple periodic solutions of delayed predator–prey systems with type IV functional responses, Nonlinear Anal. Real World Appl. 5 (2004) 45–53].     

Permanence and extinction of periodic predator–prey systems in a patchy environment with delay

This paper studies two species predator–prey Lotka–Volterra type dispersal systems with periodic coefficients and infinite delays, in which the prey species can disperse among n-patches, but the predator species is confined to one patch and cannot disperse. Sufficient and necessary conditions of integrable form for the permanence, extinction and the existence of positive periodic solutions are established, respectively. Some well-known results on the nondelayed periodic predator–prey Lotka–Volterra type dispersal systems are improved and extended to the delayed case.     

What generalizations do students achieve with respect to trigonometric functions in the transition from angles in degrees to real numbers?

The aim of this study is to examine prospective mathematics teachers’ generalizations of trigonometric functions from the unit circle to the Cartesian coordinate system. The researcher developed a test that aimed to reveal students’ generalizations, as well as the possible differences between them. The test was administered to 30 students who were near completion of their university degree program. The findings showed that the students were unable to establish the link between the unit circle and the Cartesian coordinate representation system; and therefore, they were not able to interpret the outputs of trigonometric functions with input of a real number that is not a multiple of π. The researcher also found that the students had developed certain misconceptions regarding the properties of trigonometric functions. To improve the teaching of trigonometric functions an instructional sequence and an alternative definition for trigonometric functions is proposed.     

How do inhibitors and modifiers of individual enzymes affect steady-state fluxes and concentrations in metabolic systems?

By using metabolic control theory we show how one can calculate the effect on steady-state metabolite concentrations and steady-state fluxes of small amounts of inhibitors or activators of individual enzymes in a metabolic system of arbitrary complexity. It turns out that only limited knowledge of the kinetics of the enzymes in the systems, summarized in their “Elasticity Coefficients,” is needed. An example of an actual calculation is given.     

How do path generation methods affect the accuracy of quasi-Monte Carlo methods for problems in finance?

Quasi-Monte Carlo (QMC) methods are important numerical tools in computational finance. Path generation methods (PGMs), such as Brownian bridge and principal component analysis, play a crucial role in QMC methods. Their effectiveness, however, is problem-dependent. This paper attempts to understand how a PGM interacts with the underlying function and affects the accuracy of QMC methods. To achieve this objective, we develop efficient methods to assess the impact of PGMs. The first method is to exploit a quadratic approximation of the underlying function and to analyze the effective dimension and dimension distribution (which can be done analytically). The second method is to carry out a QMC error analysis on the quadratic approximation, establishing an explicit relationship between the QMC error and the PGM. Equalities and bounds on the QMC errors are established, in which the effect of the PGM is separated from the effect of the point set (in a similar way to the Koksma–Hlawka inequality). New measures for quantifying the accuracy of QMC methods combining with PGMs are introduced. The usefulness of the proposed methods is demonstrated on two typical high-dimensional finance problems, namely, the pricing of mortgage-backed securities and Asian options (with zero strike price). It is found that the success or failure of PGMs that do not take into account the underlying functions (such as the standard method, Brownian bridge and principal component analysis) strongly depends on the problem and the model parameters. On the other hand, the PGMs that take into account the underlying function are robust and powerful. The investigation presents new insight on PGMs and provides constructive guidance on the implementation and the design of new PGMs and new QMC rules.