Multiple spatiotemporal coexistence states and turing-hopf bifurcation in a lotka-volterra competition system with nonlocal delays

We consider a two-species Lotka-Volterra competition system with both local and nonlocal intraspecific and interspecific competitions under the homogeneous Neumann condition. Firstly, we obtain conditions for the exis-tence of Hopf, Turing, Turing-Hopf bifurcations and the necessary and suficient condition that Turing instability occurs in the weak competition case, and find that the strength of nonlocal intraspecific competitions is the key factor for the stability of coexistence equilibrium. Secondly, we derive explicit formulas of normal forms up to order 3 by applying center manifold theory and normal form method, in which we show the difierence compared with system with-out nonlocal terms in calculating coeficients of normal forms. Thirdly, the existence of complex spatiotemporal phenomena, such as the spatial homoge-neous periodic orbit, a pair of stable spatial inhomogeneous steady states and a pair of stable spatial inhomogeneous periodic orbits, is rigorously proved by analyzing the amplitude equations. It is shown that suitably strong nonlocal intraspecific competitions and nonlocal delays can result in various coexistence states for the competition system in the weak competition case. Lastly, these complex spatiotemporal patterns are presented in the numerical results.