In this paper, we propose a diffusion-type predator–prey interaction model with harvest and disease in prey, and conduct stability analysis and pattern formation analysis on the model. For the temporal model, the asymptotic stability of each equilibrium is analyzed using the linear stability method, and the conditions for Hopf bifurcation to occur near the positive equilibrium are investigated. The simulation results indicate that an increase in infection force might disrupt the stability of the model, while an increase in harvesting intensity would make the model stable. For the spatiotemporal model, a priori estimate for the positive steady state is obtained for the non-existence of the non-constant positive solution using maximum principle and Harnack inequality. The Leray–Schauder degree theory is used to study the sufficient conditions for the existence of non-constant positive steady states of the model, and pattern formation are achieved through numerical simulations. This indicates that the movement of prey and predators plays an important role in pattern formation, and different diffusions of these species may play essentially different effects.

Air pollution is one of the most important environmental problems in the world, it does harm to human health as well as hindering the sustainable development of the economy. The grim situation deserves wide attention and active participation in improvement. Thus, in this paper, considering the pollution control and sustainable development of the economy, we develop and investigate the economic-environment model with unilateral and bilateral control. To begin with, we briefly describe the dynamic behavior of the free system. It then follows the analysis of the system with unilateral and bilateral control. The existence of the order-1 and order-2 periodic solutions are investigated under different conditions. In addition, their stability is proven. The results show that the bilateral control strategy is beneficial to the sustainable development of economy.

In this paper, we give another definition of Ruston elements and almost Ruston elements, which is equivalent to the definitions given by Mouton and Raubenheimer in the case that the homomorphism has a closed range and Riesz property. For two homomorphisms, we consider the preserver problems of Fredholm theory and Fredholm spectrum theory. In addition, we study the spectral mapping theorems of Fredholm (Weyl, Browder, Ruston, and almost Ruston) elements relative to a homomorphism. Last but not least, the dependence of Fredholm theory on three homomorphisms is considered, and meanwhile, the transitivity of Fredholm theory relative to three homomorphisms is illustrated. Furthermore, we consider the Fredholm theory relative to more homomorphisms.