Bifurcation Analysis In Bioecological Models

This work investigates bifurcation dynamics in two kinds of nonlinear dynamical systems: the population model with fear effects and the cancer growth model.The existence,stability,and types of equilibrium points are discussed by using dynamical system stability and qualitative theory.Additionally,the existence of codimension-one bifurcations is analyzed through the central manifold theorem and local bifurcation theory,and the codimension-two bifurcations are analyzed through normal form theory and local bifurcation theory.A multitude of numerical simulations portray this investigation,including local bifurcation diagrams for both equilibrium points and periodic orbits,bifurcation curves diagrams,phase diagrams,and time series diagrams.Other complex dynamics in the systems are also identified.This paper comprises of five chapters.A brief introduction to the research background of nonlinear dynamical system,development of two kinds of models,and codimension-one and codimension-two bifurcation theory is presented in Chapter one.The codimension-one bifurcations of a population model with fear bifurcation parameter are investigated in Chapter two.The existence of transcritical bifurcation,saddle-node bifurcation and supercritical Hopf bifurcations are proved.Numerical simulations virified the theoretical analysis,and the biological explanations are presented based on ecosystem significance.The codimension-two bifurcations in the population model with fear effects are considered in Chapter three.Using codimension-two bifurcation theory,two Zero-Hopf bifurcations and a generalized Hopf bifurcation are proved to exist.The normalized forms and the bifurcation curves of codimension-two bifurcations are derived by normal form theory and local bifurcation theory.The distribution of solutions in the plane divided by the bifurcation curves is analyzed.Complex dynamics of the system are simulated.A codimension-two bifurcation in a cancer growth model is discussed in Chapter four.The existence of a Cusp bifurcation is proved using codimension-two bifurcation theory.The normalized form and bifurcation curves of the Cusp bifurcation are derived and numerical simulation results are presented.Chapter five provides a summary of the entire article and suggests future research directions.Overall,the paper provides valuable insights into the dynamics of these two nonlinear systems and identifies potential avenues for further research.