Numerical dynamics of insect–pathogen models with linearly implicit methods and neural networks

The application of linearly implicit methods in insect–pathogen models remains underexplored. This study investigates two main aspects: Firstly, we apply the linearly implicit methods to insect–pathogen models and find that it preserves positivity and boundedness of solutions. However, the numerical stability proves challenging for the model. To address this, we propose an improved linearly implicit method that maintains these desirable properties while ensuring local stability. Moreover, a transmission rate parameter β of the model controls population dynamics. Theoretical analysis and numerical experiments reveal that variations in β can drive the system to transition between steady states and periodic solutions. For periodic solutions analysis, we first employ neural network techniques to successfully capture limit cycles, verifying periodic characteristics under the improved linearly implicit method. Subsequently, we consider the Laplacian diffusion terms and develop a fully discrete scheme based on finite element methods and the improved linearly method, with numerical experiments again confirming of the model periodic behavior. This work not only extends the application of linearly implicit methods in ecological modeling but also provides a novel numerical method for dynamical analysis of insect–pathogen models.