Local stability of a co-opetition model between two transportation modes with delays

In this paper, a co-opetition model with discrete delays which reflects the relation between two transportation modes is established based on the Lotka-Volterra model. The local stability of the positive equilibrium is investigated through constructing a proper Lyapunov functional. A detailed and explicit procedure of obtaining the sufficient conditions for local stability of the positive equilibrium of the system along with an estimate about the size of allowable delay is provided. Then the inherent dynamics relation and evolution mechanism of the harmonious developments between two different transportation modes are discussed, the theoretic condition in which the two different transportation modes can coexist harmoniously is given.