Expansion Of Solution Of A Competitive Population Model With Time-delay

Recently, population dynamics with time-delay has become a very challenging hot issue. To study the dynamic mechanics of an ecological system and give the dynamic prediction of it is one of the leading problems about ecology. The study of the dynamic behavior of populations is not only of great practical importance, but also a challenging mathematical problem. At present, the existence, stability, oscillation and bifurcation, etc, of the solution are the main aspects in the study of population dynamics with time-delay. In this paper, we shall give the local structure of the solution near its equilibrium and measure the rapidity of convergence of the solution from a quantitative viewpoint. Since the stability of a system depends totally upon the structure of the solution, we can consider the two problems at the same time. Besides, this quantitative analysis gives an explicit description of the solution and it is convenient for numerical simulation.According to the idea above, the present paper gives a quantitative analysis for a specific ecological problem—competitive population dynamics with two discrete time-delays. We obtain some good results concerning the structure and the numerical simulation of its dynamic solution.First, assume that there exists unique a positive equilibrium, then the linearization of the system near it yields to the corresponding linearized model. Applying semigroup theory of linear operators, we get the well-posed-ness of the linearized model. Then we provide the asymptotic expressions of the eigenvalues of the system operator and their corresponding eigenvectors with a detailed spectrum analysis. Although we prove that the eigenvector sequence fails to form a basis for the state space by estimating the norm of Riesz projection of the system operator, we still give the asymptotic ex-pansion of the solution according to the eigenvectors, on the basis of which we do the numerical simulation and error estimation. We point out that in short time, there may be errors we cannot neglect between the accurate solution and its simulation, but for long time situation, the simulation is of high precision, and the asymptotic expansion is exponentially convergent.In this paper, we take the competitive population model with time-delay as an example to illustrate our idea. However, the method we use here can be applied for other delay differential equations because of its generality.