Analysis Of The Dynamical Behavior Of Generalized Lotka-Volterra Systems

The Lotka-Volterra system was independently proposed in the early 20th centuryby the American mathematician A.J.Lotka and Italian mathematician V.Volterra,respectively.The Lotka-Volterra system has devolved into a landmark model in the study of ecosystems.It provides a powerful tool for understanding the evolution of complex system.However,many questions remain open for high-dimensional Lotka-Volterra model.For this reason,we investigate the global dynamics of n dimensional anti-symmetric Lotka-Volterra system,where parameter ci,aij∈R and aij=-aji.xi(t)represents the number of individuals for the i-th species at time t and ci∈R.In the context of predatorprey interactions,different species are labeled with i(or j),i,j=1,2,…,n,xi(t)represents the number of individuals for the i-th species at time t and ci accounts for its intrinsic growth rate.Parameter aij represents the effect of the i-th species upon the j-th species.The matrix A=(aij)n×n is called the interaction matrix.We find that the long-time behaviors of the solutions to system(0.0.3)is closely related with the linear inequalities Necessary and sufficient conditions for the existence of solutions to the linear inequalities(0.0.4)are established.We prove that when the solutions of the linear inequalities(0.0.4)exist,the solutions of(0.0.3)are uniformly bounded,and the persistence and extinction of any species can be classified by an index set I,I:={i∈S:yi>0,y≥0 solves(0.0.4)}.the i-th species will persist if i∈I and it will go extinct if i.(?)I.This is independent of the initial state of the system.When the solutions of(0.0.3)do not exist,the solutions of system(0.0.3)are unbounded.We prove that the index set I is completely determined by the interaction matrix A and intrinsic growth rate c=(c1,c2,…,cn)T.That is,the persistence or extinction of species in system(0.0.3)depends only on the interaction matrix and the intrinsic growth rate independent of the initial state.As an application of the main results,we specifically analyze the dynamics of system(0.0.3)with three species,and fully classified the persistence and extinction of species.The dynamic processes in ecological environments often have memory effects,and fractional calculus is very suitable for describing this phenomenon due to its non local characteristics.Therefore,introducing fractional-order calculus into the Lotka-Volterra model can better study the interactions of biological populations in ecosystem and facilitate a deeper understanding of the evolutionary patterns of biological communities.We investigate the global dynamics of n dimensional fractional-order anti-symmetric Lotka-Volterra system,It is proved that when the solution of the antisymmetric Lotka-Volterra system is uniformly bounded(unbounded),the solution of the corresponding fractional-order system is also uniformly bounded(unbounded).We discuss the stability of the equilibrium of the system(0.0.3)and corresponding fractional-order system,and characterize the effect of fractional-order derivatives on the asymptotic behavior of the antisymmetric Lotka-Volterra system.We conducted many numerical simulations for different values of fractional derivatives to further demonstrate the impact of fractional derivative.