The Global Behavior Analysis Of Two Competitive Models In Higher Dimensional Systems

In this paper, the global behaviors of an innovation di?usion model for many com-petitive products and a Gompertz model of three competing populations are studied onthe basis of the theory of competitive systems.In chapter 2, motivated by the ideas from the population dynamics and the epidemicsdynamics, the innovation di?usion model is established to describe the dynamics of manycompeting products in a market. The global behavior is analyzed in detailed. First,an abstract function about acceptance of products is introduced into innovation modelwith the three competing products. The conditions for coexistence of three products areobtained by showing system without periodic orbits via Stokes theory, the the nonlinearODE theory, Hurwitz theory and the generalized Poincar′e-Bendixson theorem. Then,the results of previous innovation models of three products with bilinear acceptance areextended to model with n products by using a clever Liapunov function to show thatthe unique positive equilibrium is globally asymptotically stable. From the analysis ofthe e?ect on the advertisement, the conditions that either the nonnegative equilibrium orthe positive equilibrium is globally asymptotically stable are established. The thresholdbetween the extinction and existence of the product without advertisement in the marketis obtained and the decision about competition is obtained according to the analysis.The main task of mathematical biology is to consider the population dynamics. Dur-ing last seventy years, the population models are mostly developed from Logistic growth.Recently, many data have shown that per capita growth functions of populations are wellfitted by Gompertz growth. Thus the Gompertz model for three competing populationis proposed on the basis of the Gompertz growth of each population in the absence ofothers. The dynamical behavior have been researched in four chapters according to therelation of the extent between the interspecific and the intro-specific competitions.The conditions for Gompertz system for three competing populations without peri-odic orbits are obtained via the methods provided by Busenberg and van den Driesscheif all of the interspecific competitions either are strong or are weak, and if only one of...