Traveling waves for a generalized Holling Tanner predator-prey model
Traveling waves for a generalized Holling Tanner predator-prey model
复制标题
广义 Holling-Tanner 捕食者-被捕食者模型的行波
DOI：
10.1016/j.jde.2017.08.021
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发表时间：
2017
影响因子：
2.4
通讯作者：
Rui Peng
中科院分区：
文献类型：
--
作者：
Shangbing Ai;Yihong Du;Rui Peng
我们研究了霍林-坦纳型捕食者-被捕食者模型的行波解，其中捕食者方程在被捕食者种群为零时具有奇点。这里的行波解将仅猎物平衡 (1, 0) 与唯一的常数共存平衡 (u⁎, v⁎) 连接起来。首先，我们针对相当一般的捕食者-被捕食者系统的弱行波解给出了尖锐的存在性结果，并明确确定了最小速度。这样的弱行波 (u (xi), v (xi)) 在 xi=−∞ 处连接 (1, 0)，但不需要在 xi=∞ 处连接 (u⁎, v⁎)。接下来，我们修改Holling-Tanner模型以消除其奇异性，并应用一般结果来获得修改模型的弱行波解，并表明该弱行波解中的猎物分量具有正的下界，因此是原始模型的弱行波解。弱行波解的这些结果在相当一般的条件下成立。然后我们用挤压法和李亚普诺夫函数法两种方法证明，在附加条件下，弱行波解实际上是行波解，即它们收敛到共存平衡点 ψ → ∞。
We study traveling wave solutions for Holling–Tanner type predator–prey models, where the predator equation has a singularity at zero prey population. The traveling wave solutions here connect the prey only equilibrium (1, 0) with the unique constant coexistence equilibrium (u⁎, v⁎). First, we give a sharp existence result on weak traveling wave solutions for a rather general class of predator–prey systems, with minimal speed explicitly determined. Such a weak traveling wave (u (ξ), v (ξ)) connects (1, 0) at ξ=−∞ but needs not connect (u⁎, v⁎) at ξ=∞. Next we modify the Holling–Tanner model to remove its singularity and apply the general result to obtain a weak traveling wave solution for the modified model, and show that the prey component in this weak traveling wave solution has a positive lower bound, and thus is a weak traveling wave solution of the original model. These results for weak traveling wave solutions hold under rather general conditions. Then we use two methods, a squeeze method and a Lyapunov function method, to prove that, under additional conditions, the weak traveling wave solutions are actually traveling wave solutions, namely they converge to the coexistence equilibrium as ξ→∞.