Almost periodic solutions of a nonlinear ecological model

By using comparison theorem and constructing suitable Lyapunov functional, we study the following almost periodic nonlinear N-species competitive model with feedback controls x ˙ i ( t ) = x i ( t ) r i ( t ) - ∑ j = 1 N a ij ( t ) x j α ij ( t ) - ∑ j = 1 N b ij ( t ) x j β ij ( t - τ ij ( t ) ) - d i ( t ) u i ( t ) - ∑ j = 1 , j ≠ i N c ij ( t ) x i α ii ( t ) x j α ij ( t ) - ∑ j = 1 N ∫ - σ ij 0 g ij ( t , s ) x j γ ij ( t + s ) ds , u ˙ i ( t ) = h i ( t ) - e i ( t ) u i ( t ) + f i ( t ) x i α ii ( t ) . A set of sufficient conditions are obtained for the existence and global asymptotic stability of a unique positive almost periodic solution of the above model. In addition, we will apply our main results to some important competition models with or without feedback controls which have been well studied by many authors. As you will see, our results improve and generalize many previous known results in [6,9,10,16,18,30,31,35,38,39]. Finally, an example together with its numeric simulations show the feasibility and effectiveness of our main results.