研究一类基于个体尺度分布的竞争种群系统平衡态的稳定性. 利用种群再生数获得了平衡态的存在性条件,借助特征根的分布给出了平衡态的稳定性判据,运用离散化程序展示了两个稳定性实例.

This paper researches the stability of the steady states of a competitive population system, in which the individuals are classified by their body size. Firstly, we establish the existence of equilibria by defining the basic reproductive numbers of the involved populations, and show that the system owns a unique positive steady state if the reproductive numbers of the two populations are greater than one. On the other hand, the trivial solution is always an equilibrium. Secondly, we investigate the stability of the equilibria via linearization method, and carefully derive the corresponding characteristic equation. Then we obtain the criterion by a standard argument; that is, a steady state will be asymptotically stable if all the eigenvalues have negative real parts, while unstable if one of the eigenvalues has positive real part. Finally, we explore the stability by numerical simulations since the high complexity in the characteristic equation makes a clear and extensive theoretical analysis impossible. Introducing an appropriate difference format and choosing two examples, we present some intuitive demonstrations for the stability of the trivial equilibrium, which agree with the theoretical predictions.\ \indent In a word, size-structured population models are more realistic for some species, but they bring us more challenges to mathematical analysis. This work offers a helpful effort.