We study the collective dynamics of mobile species under cyclic competition by breaking the symmetry in the initial populations and examining the basins of the two distinct asymptotic states: extinction and coexistence, the latter maintaining biodiversity. We find a rich dependence of dynamical properties on initial conditions. In particular, for high mobility, only extinction basins exist and they are spirally entangled, but a basin of coexistence emerges when the mobility parameter is decreased through a critical value, whose area increases monotonically as the parameter is further decreased. The structure of extinction basins for high mobility can be predicted by a mean-field theory. These results provide a more comprehensive picture for the fundamental issue of species coexistence than previously achieved.
Based on some important experimental dates, in this paper we shall introduce time delays into Mehrs's non-linear differential system model which is used to describe proliferation, differentiation and death of T cells in the thymus (see, for example, [3], [6], [7] and [9]) and give a revised nonlinear differential system model with time delays. By using some classical analysis techniques of functional differential equations, we also consider local and global asymptotic stability of the equilibrium and the permanence of the model.
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