All the time, differential dynamical models with delay has witness a tremendous application value in characterizing the internal law among diverse biological populations in biology. In the current article, on the basis of the previous publications, we formulate a new Lotka–Volterra commensal symbiosis system accompanying delay. Utilizing fixed point theorem, inequality tactics and an appropriate function, we gain the sufficient criteria on existence and uniqueness, non-negativeness and boundedness of the solution to the formulated delayed Lotka–Volterra commensal symbiosis system. Making use of stability and bifurcation theory of delayed differential equation, we focus on the emergence of bifurcation behavior and stability nature of the formulated delayed Lotka–Volterra commensal symbiosis system. A new delay-independent stability and bifurcation conditions on the model are presented. By constructing a positive definite function, we explore the global stability. By constructing two diverse hybrid delayed feedback controllers, we can adjusted the domain of stability and time of appearance of Hopf bifurcation of the delayed Lotka–Volterra commensal symbiosis system. The effect of time delay on the domain of stability and time of appearance of Hopf bifurcation of the model is given. Matlab experiment diagrams are provided to sustain the acquired key outcomes.

With economic and social development, the transport sector is growing rapidly, leading to a surge in energy consumption and environmental degradation. New Energy Vehicles (NEVs) are regarded as an important tool to alleviate energy and environmental issues, but the market share of NEVs in China is low, and how to increase the market share of NEVs is an urgent issue. This study constructs an evolutionary game model between local governments and automakers to explore the evolutionary stable strategy (ESS) of both under different combinations of subsidy and carbon tax, aiming to find rational policies to promote the development of the NEVs. Considering the possible delay effect of the game process, a time-delayed differential equation is established to analyze the effects of random time delay on the stability of the system. Finally, a numerical simulation is carried out to obtain the following conclusions: constant subsidy and carbon tax policy is not conducive to the proliferation of NEVs, variable subsidy and carbon tax policy is effective in the short term, and variable subsidy and constant carbon tax policy is effective in the long term; strengthening the entry of carbon tax can help to develop the NEVs industry; the initial value will affect the evolution of system under constant policy; excessive time delay can cause the system to lose its original stability.

Establishing dynamical models to characterize the relation of different chemical compositions is an important topic in chemistry and mathematics. However, a lot of dynamical models are merely concerned with the integer-order dynamical models. The report on fractional-order chemical dynamical systems is quite few. In this current article, based on the earlier publications, we establish a new fractional-order coupled Oregonator model incorporating time delay. A set of sufficient conditions which ensure the stability and the onset of Hopf bifurcation of fractional-order coupled Oregonator model incorporating time delay are derived by regarding the time delay as bifurcation parameter. The exploration manifests that time delay has a vital influence on stabilizing system and controlling bifurcation of the investigated fractional-order coupled Oregonator model. At last, Matlab simulation results are adequately displayed to corroborate the derived theoretical achievements.