On stability and periodic oscillations of an income-capital model with
time delay and spatial diffusion
Abstract
The paper aims to establish a realistic income-capital model and gives
economic conclusions by some mathematical analysis. Firstly, unlike
other known models which either neglect time delay or neglect spatial
diffusion, our model includes both time delay and spatial diffusion.
Secondly, taking the time delay as bifurcation parameter, the stability
of positive equilibrium and periodic oscillations are studied by
theoretical and numerical analysis under two different boundary
conditions. At last, the theoretical results yield the following
economic conclusions: 1) For the closed economy or the open economy,
there exists a critical threshold of time delay. If the time delay is
smaller than the critical threshold, then the economic system will keep
balanced at the present state; If the time delay is lager than the
critical threshold, the stability of present state will be destroyed,
and the periodic oscillations will emerge; 2) The biggest difference
between the critical threshold of open economy and that of closed
economy is that the former is related to diffusion coefficients, while
the latter is independent of diffusion coefficients; 3) The periodic
oscillations are spatially homogeneous for closed economy, but are
spatially inhomogeneous for open economy; 4) Regional income and capital
disparities are more likely to occur in open economies than in closed
economies; 5) Results reveal to some extent the causes of the gap
between the rich and the poor and also provide insight into why
developed economies are more likely to polarize than underdeveloped
ones. Our theoretical analysis is based on the center manifold theorem,
normal forms and Hopf bifurcation theory.