中文摘要：

对于一类两种群互惠模型,如果扩散项为-u~pΔu和-v~qΔv,则在p,q和反应函数的系数满足一定条件时该模型的解在有限时间爆破,而在另外的一些条件下存在整体解。证明：如果扩散项为-Δu~m和-Δv~n,则解的动力学性态会完全不同。在适当的条件下,该时变问题存在唯一整体解,相应的平衡态问题存在正的最大和最小解。此外,这个时变解在一些初值条件下收敛到最大平衡解,而在另一些初值条件下收敛到最小平衡解。这种收敛性对反应函数的任意系数都成立。这意味着带不同退缩扩散项的互惠模型的动力学性态也可能不同。

英文摘要：

For a two-species mutualistic model, if the degenerate diffusion terms are -u~p Δu and -v~q Δv then for a certain set of reaction rates in the reaction function the solution of the model blows up in finite time, and for another set of reaction rates, a unique global solution exists. In this paper, it proves that if the diffusion terms are -Δu~m and -Δv~n then the dynamic behavior of the solution can be quite different. For this model, under appropriate conditions, the time-dependent problem has a unique bounded global solution, and the corresponding steady-state problem has a positive maximal solution and a positive minimal solution. Moreover, the time-dependent solution converges to the maximal solution for one class of initial functions, and to the minimal solution for another class of initial functions. The above convergence property holds true for any reaction rates in the reaction function. This means the dynamic behavior of a mutualistic model with different degenerate diffusion terms can be different.