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We propose a nonsmooth Filippov refuge ecosystem with a piecewise saturating response function and analyze its dynamics. We first investigate some key elements to our model which include the sliding segment, the sliding mode dynamics and the existence of equilibria which are classified into regular/virtual equilibrium, pseudo-equilibrium, boundary equilibrium and tangent point. In particular, we consider how the existence of the regular equilibrium and the pseudo-equilibrium are related. Then we study the stability of the standard periodic solution (limit cycle), the sliding periodic solutions (grazing or touching cycle) and the dynamics of the pseudo equilibrium, using quantitative analysis techniques related to nonsmooth Filippov systems. Furthermore, as the threshold value is varied, the model exhibits several complex bifurcations which are classified into equilibria, sliding mode, local sliding (boundary node and focus) and global bifurcations (grazing or touching). In conclusion, we discuss the importance of the refuge strategy in a biological setting.

AutoTutor is an intelligent tutoring system that holds conversations with learners in natural language. AutoTutor uses Latent Semantic Analysis (LSA) to match student answers to a set of expected answers that would appear in a complete and correct response or which reflect common but incorrect understandings of the material. The correctness of student contributions is decided using a threshold value of the LSA cosine between the student answer and the expectations. In previous work LSA has shown to be effective in detecting good answers of students. The results indicate that the best agreement between LSA matches and the evaluations of subject matter experts can be obtained if the cosine threshold is allowed to be a function of the lengths of both student answer and the expectation being considered. Based on some of our experiences with LSA and AutoTutor, we are developing a new mathematical model to improve the precision of AutoTutor's natural language understanding and discriminative ability.

The COVID-19 pandemic started, a global effort to develop vaccines and make them available to the public, has prompted a turning point in the history of vaccine development. In this study, we formulate a stochastic COVID-19 epidemic mathematical model with a vaccination effect. First, we present the model equilibria and basic reproduction number. To indicate that our stochastic model is well-posed, we prove the existence and uniqueness of a positive solution at the beginning. The sufficient conditions of the extinction and the existence of a stationary probability measure for the disease are established. For controlling the transmission of the disease by the application of external sources, the theory of stochastic optimality is established. The nonlinear least-squares procedure is utilized to parametrize the model from actual cases reported in Pakistan. The numerical simulations are carried out to demonstrate the analytical results.

In this study, the effects of threshold variation in image segmentation of micro CT images of cancellous bone in the determination of the architectural parameters and stiffness were investigated. A total of 42 samples of 6 × 6 × 6 mm³ cubes with threshold values set between 500–1100 greyscale in increment of 100 of CT images of six human C5 vertebral bodies were analyzed. Threshold value of 800, based on Otsu's method, was set for the control group. From various threshold values, the respective architectural parameters, and the corresponding stiffness in three orthotropic directions (Exx, Eyy, Ezz) of each cube were computed from the voxel-based micro-finite element models under compressive simulation. The results showed that 1% variation of threshold value resulted in a 3.4% variation in BV/TV, 2% in Tb.N, 3.1% in Tb.Th, 2.9% in BS/BV, 1.8% in Tb.Sp, 29.2% in Exx, 28.7% Eyy and 27.7% in Ezz. Statistical analysis showed that 2.9% threshold variation caused significant change in BV/TV, Tb.Th, Exx, Eyy and Ezz values. The study shows that with threshold variation of more than 2.9%, significant differences in the architectural parameters and stiffness compared to those based on Otsu's method.

Metal castings with the presence of shrinkage porosity are often recycled or rejected and in turn reduce productivity of the process as well as increase energy cost involved in the process. This can be overcome by prevention of its occurrence using a suitable prediction technique. A detailed study of literature reveals that several Criterion Functions (an empirical model that connects solidification phenomena with formation of shrinkage porosity) have been employed to predict the location of shrinkage porosity in castings manufactured using particular process-alloy combination by values of process parameters (mostly thermal gradient, cooling rate, and velocity of molten metal). However, criterion function considering the effect of geometric variation in stainless steel castings on an extent of shrinkage porosity need to be established. In the present work, a benchmark casting, a combination of three T junctions, has been cast and used for the development of geometry-driven criterion function for stainless steel castings. Real experimental results with the presence of shrinkage porosity were used for superimposing on virtual experimental results (simulated results) for establishment of local simulation conditions. These conditions are further used in extrapolating results using casting simulation. The developed geometry-driven criterion function was further validated and found to be effective in prediction of shrinkage porosity.

For the non-stationary signal denoising, an effective method for dropping ambient noise is based on discrete wavelet transform. Also, in order to minimize the loss of useful signal and get high SNR in the wavelet denoising, it is very important that the thresholding is suitable for the characteristics of signal. In this paper, we propose new thresholding method to reduce an ambient noise and to detect effectively the useful signal. First, we analyze four kinds of previous wavelet threshold functions (Hard, Soft, Garrote and Hyperbola) and propose new wavelet threshold function compromised between Garrote and Hyperbola threshold functions. Next, a threshold value is selected by value to reduce exponentially according to the wavelet decomposition level. We also analyze a continuity and monotonicity, and prove the logic of new threshold function. The results of theoretical analysis show that new threshold function solves the problems of constant error and discontinuity of previous threshold functions, and minimizes the information loss of useful signal. The results of experiment show that SNR of new thresholding method is highest and RMSE and Entropy are smallest. The results of theoretical analysis and experiment show that new thresholding method is more appropriate to wavelet denoising for dropping ambient noise than previous methods.

Based on the facts of releasing natural enemies and spraying pesticides at different time points, we propose a generalized predator-prey model with impulsive interventions. The threshold values for the existence and stability of pest eradication periodic solution are provided under the assumptions of releasing natural enemies either more or less frequent than spray. In order to address how the different pulse time points, control tactics affect the pest control (i.e. the threshold value), the Holling Type II Lotka-Volterra predator-prey system, as an example, with impulsive intervention at different time points are investigated carefully. The numerical results show how the threshold values are affected by the factors including instantaneous killing rates of pesticides on pests and natural enemies, the release rate of natural enemies and release constant, timing of pesticide application and timing of release period. Furthermore, it is confirmed that the system has the coexistences of pests and natural enemies for a wide range of parameters and with quite different pest amplitudes.

The coronavirus disease (COVID-19) is a dangerous pandemic and it spreads to many people in most of the world. In this paper, we propose a COVID-19 model with the assumption that it is affected by randomness. For positivity, we prove the global existence of positive solution and the system exhibits extinction under certain parametric restrictions. Moreover, we establish the stability region for the stochastic model under the behavior of stationary distribution. The stationary distribution gives the guarantee of the appearance of infection in the population. Besides that, we find the reproduction ratio $R_0^S$ for prevail and disappear of infection within the human population. From the graphical representation, we have validated the threshold conditions that define in our theoretical findings.