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Our main results are presented and proved in Sections 3 and 4. In Section 3, we obtain some sufficient conditions for nonexistence, existence and uniqueness of positive equilibriums to (1.2) by employing the method of super-sub solutions.
By employing the method of super-sub solutions, combined with the careful analysis for (rho eta,x,y)), they proved nonexistence, existence and uniqueness of positive equilibriums to (1.5).
2, we show the existence and the asymptotic behaviors of traveling front solutions of (1.1) and give some preparations that will be used to construct the sup-sub solutions.
Later, Guo and Morita in [12] extended the results in [8] and [10] to more general cases including a discrete KPP equation by the super-sub solutions method and the comparison theorem.
At the same time, Chen and Guo in [13] proved the existence of entire solutions by showing a different technique which used only one function to construct a pair of deterministic super-sub solutions.
In particular, developed is a technique that combines the method of super-sub solutions, the variation-of-constants formula for the delay differential equation and the estimation of integral kernels, which enables us to obtain some sufficient conditions for the global attractiveness of the unique positive steady state for (1.1).
In Section 4, developed is a technique that combines the method of super-sub solutions, the variation-of-constants formula for the delay differential equation and the estimation of integral kernels, which enables us to obtain some sufficient conditions for the global attractivity of the unique positive equilibrium.
With the aid of the comparison theorem and the sup-sub solutions method, we construct some entire solutions that behave as two opposite traveling front solutions with critical speeds moving towards each other from both sides of x-axis and then annihilating.
In particular, developed is a technique that combines the method of super-sub solutions, the variation-of-constants formula for the delay differential equation and the estimation of integral kernels, which enables us to obtain some sufficient conditions for the global attractiveness of the unique positive equilibrium.
