Monotone Method In The Ecological Model

The whole paper contains four parts and mainly study the convergent tendency of differential systems. The monotonicity principles are utilized to study the global asymptotical stability and the existence of the periodic solutions of the finite dimensional mathematical models in ecology in the first three parts. The last part discusses the asymptotic behavior of solutions for a class of nonautonomous delay differential systems.In the first part, the global stability of two-dimensional and three-dimensioal irreducible systems of cooperation or competition are discussed by non-positive divergence and the monotonicity principles and the given concrete examples interpret the results. The theorems obtained improve some of well-known results in the literature.The second part is mainly concerned with the sufficient condition for the global asymptotical stability of three-dimensional cooperative systemby the coordinate transformation, projection, non-positive divergence and the principles of the limit set of monotone dynamical systems. We demonstrate the truth of the guess given by Hirsch and obtain the conditions for global asymptotical stability for the equilibrium p with some of eigenvalues of DF(p) having Reλ= 0,giving conditions for concrete mathematical models in ecology to ideal state of balance. Using projection and the method of monotone, the third part discusses the non-existence of periodic solution of the three-dimensional Lotka-Volterra competitive systemwith coefficients meeting certain conditions and at the same time we gain the conditions for the global asymptotical stability of this system.Utilizing the specific property of limit set, we study the asymptotic behavior of solutions for a class of nonautonomous delay differential systemsand get the conclusion that each bounded solutions of such systems tend to an equilibrium.