Nonlinear Analysis On The Dynamical Model Of Predator-prey With An External Interference

In this paper，we use Euler method to obtain a two-dimensional difference system，establish a nonlinear predator-prey fishery resource model in which predators are influencedby adding an external interference to research and discuss the fishery dynamical model. Herelocal dynamics properties and global dynamics of the mapping system are mainly analyzed.First，we would give some brief overview of the problems as well as some related researchsituation and background，and explain the relevant concepts and basic theoretical knowledgeabout this article in order to study some problem，and we attain a preliminary understandingabout our researching model.Then，on the one hand，according to establishing the nonlinear fishery resourcemodel，we quantitatively study the existence, stability of positive equilibrium andlocal bifurcations. Then we find that both Neimark and flip bifurcations probably occur.Finally we show the extents of sustainable use of fishery resources, that is the exhaustedconditions of resources, by the use of global analysis.On the other hand，because of influence of predator-prey fishery dynamical model inwhich one species are imposed by an external interference，the possibility of existence of abionomic equilibrium is also considered. By a computer-assisted study，we investigated someglobal bifurcations that change the domains of feasible trajectories (bounded discretetrajectories having an ecological sense). Through a specific example it is shown that thedomain boundaries of two-dimensional map can be generally obtained by the union of allrank preimages of two axes. The main results of this paper are given by the study of someglobal bifurcations that occur to study the global dynamics behavior of the mapping systemand explain the structures of attractor and structures of the basin，when the feasibledomain boundary contacts with the critical line and nondefinition set for one of the twoinverses of the map, i.e. this inverse has a vanishing denominator on this line.