A study of periodic solutions of several types of nonlinear models in biomathematics

Biomathematics is a cross-discipline formed by the interpenetration of mathematics with life sciences, biology, and other disciplines, and biomathematical models provide an effective tool for solving problems in the above application areas. Our aim in this paper is to combine mathematical analytical tools and numerical simulation methods to investigate the existence and steady state of periodic solutions in different nonlinear models. Time lags with both discrete and distributed characteristics are introduced into the Lotka-Volterra predator-feeder system, and based on the discussion of the central manifold theorem and canonical type theory, it is proved that the branching periodic solution exists when the discrete time lag parameter τ > τ₀. In the SEIRS infectious disease model with nonlinear incidence term and vertical transmission, the global stability of the disease-free equilibrium point and the local asymptotic stability of the endemic equilibrium point are analyzed through the computation and discussion of the fundamental regeneration number R₀ (p, q). A class of convergence-growth models with nonlinear sensitivity functions is studied, and the global boundedness of classical solutions and their conditions are demonstrated based on global dynamics. A mathematical generalization of the muscular vascular model is made by introducing a centralized parameter, the relationship between periodic solutions and chaotic phenomena is explored utilizing a systematic equivalence transformation, and the equation of the homoscedastic orbitals is deduced to be z2=x2(A-12x2) {z^2} = {x^2}\left( {A - {1 \over 2}{x^2}} \right) .