Dynamic Behaviors Of Three Kinds Of Non-autonomous Food-chain Systems

Studying the dynamics of food-chain system is of great significance for the continued survival and extinction of biological populations.Firstly,we consider a three species food-chain system with Hollingâ…£functional response.By using comparison theorem of differential equation,a set of sufficient conditions which guarantee the permanence of the system are obtained.The conditions approximately indicate that the intrinsic growth rate r₁(t) of the prey X₁ is comparatively large,the death rates r_i(t)(i=2,3) of the predator X_i(i=2,3) are less and their respective digestion d₂(t),d₄(t) are relatively large,then the system is permanent. By constructing a suitable Lyapunov equation,we present a set of sufficient conditions which ensure the global attractivity of the system.The conditions roughly show that the density-dependent a_i(t)(i=1,2,3) of X_i(i=1,2,3) are relatively large,then the system is globally attractive.Example together with its numerical simulation of the permanence for the system shows that our main results are feasible.Secondly,we study a nonlinear Lotka-Volterra three species food-chain system.By applying a continuation theorem based on coincidence degree theory,a set of criteria which ensure the existence of at least one positive periodic solution of the system is established.The criteria roughly indicate that the average density-dependent(?)₁ of the prey X₁ is comparatively large,the average digestion(?)₂₁ of the predator X₂ is less,the average density-dependent(?)₃ and the average interference(?)₃₂ of the predator X₃ are both less,then the system has at least oneωperiodic solution.Finally,we study a ratio-dependent three species food-chain system with diffusion and time delay.With the help of differential inequality theory,sufficient conditions which guarantee the permanence of the system and the extinction of the predators are obtained.The conditions approximately show that the density-dependent b₁(t) of the prey X₁ in patch 1 is relatively large,the digestion k_i(t)(i=1,2) of the predator Y_i(i=1,2) are larger and their death rates r_i(t)(i=1,2) are less,then the system is permanent.The conditions also indicate that the average digestion(?)_i(i=1,2) of the predator Y_i(i=1,2) are separately less than their death rates(?)_i(i=1,2),then the predator Y_i(i=1,2) will tend to extinct.Examples together with their respective numerical simulations show that our main results are verifiable.The results have a certain reference value for the biological control in the theory, they also practically have a certain guiding significance for the pest control and the accumulation of heavy metal through the food-chain.