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The purpose of this study is to explore the coopetition relationships between platform owners and complementors in complementary product markets. Drawing on the coopetition theory, the authors examined the evolutionary trends of the coopetition relationships between platform owners and complementors and explore the main influence factors.

The authors used Lotka–Volterra model to analyze the coopetition relationship between platform owners and complementors, including the evolutionary trends as well as the results. Considering the feasibility of sample data collection, simulation is used to verify the effects of different factors on the evolution of coopetition relationships.

The results show that there are four possible results of the competition in the complementary products market. That comprises “winner-take-all for platform owners,” “winner-take-all for complementors,” “stable competitive coexistence” and “unstable competitive coexistence,” where “stable competitive coexistence” is the optimal evolutionary state. Moreover, the results of competitive evolution are determined by innovation subjects’ interaction parameters. However, the natural growth rate, the initial market benefits of the two innovators and the overall benefits of the complementary product markets influence the time to reach a steady state.

The study provides new insights into the entry of platform owners into complementary markets, and the findings highlight the fact that in complementary product markets, platform owners and complementors should seek “competitive coexistence” rather than “winner-takes-all.” Moreover, the authors also enrich the coopetition theory by revealing the core factors that influence the evolution of coopetition relationships, which further enhance the analysis of the evolutionary process of coopetition relationships.

This paper aims to introduce a new numerical approach based on the local weak form and the Petrov–Galerkin idea to numerically simulation of a predator–prey system with two-species, two chemicals and an additional chemotactic influence.

In the first proceeding, the space derivatives are discretized by using the direct meshless local Petrov–Galerkin method. This generates a nonlinear algebraic system of equations. The mentioned system is solved by using the Broyden’s method which this technique is not related to compute the Jacobian matrix.

This current work tries to bring forward a trustworthy and flexible numerical algorithm to simulate the system of predator–prey on the nonrectangular geometries.

The proposed numerical results confirm that the numerical procedure has acceptable results for the system of partial differential equations.

This paper aims to develop a novel grey Bernoulli model with memory characteristics, which is designed to dynamically choose the optimal memory kernel function and the length of memory dependence period, ultimately enhancing the model's predictive accuracy.

This paper enhances the traditional grey Bernoulli model by introducing memory-dependent derivatives, resulting in a novel memory-dependent derivative grey model. Additionally, fractional-order accumulation is employed for preprocessing the original data. The length of the memory dependence period for memory-dependent derivatives is determined through grey correlation analysis. Furthermore, the whale optimization algorithm is utilized to optimize the cumulative order, power index and memory kernel function index of the model, enabling adaptability to diverse scenarios.

The selection of appropriate memory kernel functions and memory dependency lengths will improve model prediction performance. The model can adaptively select the memory kernel function and memory dependence length, and the performance of the model is better than other comparison models.

The model presented in this article has some limitations. The grey model is itself suitable for small sample data, and memory-dependent derivatives mainly consider the memory effect on a fixed length. Therefore, this model is mainly applicable to data prediction with short-term memory effect and has certain limitations on time series of long-term memory.

In practical systems, memory effects typically exhibit a decaying pattern, which is effectively characterized by the memory kernel function. The model in this study skillfully determines the appropriate kernel functions and memory dependency lengths to capture these memory effects, enhancing its alignment with real-world scenarios.

Based on the memory-dependent derivative method, a memory-dependent derivative grey Bernoulli model that more accurately reflects the actual memory effect is constructed and applied to power generation forecasting in China, South Korea and India.

Credit risk evaluation is a crucial task for banks and non-bank financial institutions to support decision-making on granting loans. Most of the current credit risk methods rely solely on expert knowledge or large amounts of data, which causes some problems like variable interactions hard to be identified, models lack interpretability, etc. To address these issues, the authors propose a new approach.

First, the authors improve interpretive structural model (ISM) to better capture and utilize expert knowledge, then combine expert knowledge with big data and the proposed fuzzy interpretive structural model (FISM) and K2 are used for expert knowledge acquisition and big data learning, respectively. The Bayesian network (BN) obtained is used for forward inference and backward inference. Data from Lending Club demonstrates the effectiveness of the proposed model.

Compared with the mainstream risk evaluation methods, the authors’ approach not only has higher accuracy and better presents the interaction between risk variables but also provide decision-makers with the best possible interventions in advance to avoid defaults in the financial field. The credit risk assessment framework based on the proposed method can serve as an effective tool for relevant policymakers.

The authors propose a novel credit risk evaluation approach, namely FISM-K2. It is a decision support method that can improve the ability of decision makers to predict risks and intervene in advance. As an attempt to combine expert knowledge and big data, the authors’ work enriches the research on financial risk.