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 controlsx˙i(t)=xi(t)ri(t)-∑j=1Naij(t)xjαij(t)-∑j=1Nbij(t)xjβij(t-τij(t))-di(t)ui(t)-∑j=1,j≠iNcij(t)xiαii(t)xjαij(t)-∑j=1N∫-σij0gij(t,s)xjγij(t+s)ds,u˙i(t)=hi(t)-ei(t)ui(t)+fi(t)xiαii(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] and [39]. Finally, an example together with its numeric simulations show the feasibility and effectiveness of our main results.

Research highlights
► The problem considered in this paper is in many aspects more general and incorporates as special cases various problems which have been extensively studied in the literature.
► Applying the our main results to some special cases, we derive some new criteria which generalize and greatly improve some well known results.
► The coeffcients of the nonlinear N-species competitive model with feedback controls are almost periodic, not periodic. It is more diffcult to be analyzed.
► The method is based on comparison theorem and constructing suitable Lyapunov functional.