Huang et al. [10] developed a hybrid continuous/discrete-time model to describe the persistence and invasion dynamics of Zebra mussels in rivers. They used a net reproductive rate $ R_0 $ to determine population persistence in a bounded domain and estimated spreading speeds by applying the linear determinacy conjecture and using the formula in [16]. Since the associated solution operator is non-monotonic and non-compact, it is nontrivial to rigorously establish these quantities. In this paper, we analyze the spatial dynamics of this model mathematically. We first solve the parabolic equation and rewrite the model into a fully discrete-time model. In a bounded domain, we show that the spectral radius $ \hat{r} $ of the linearized operator can be used to determine population persistence and that the sign of $ \hat{r}-1 $ is the same as that of $ R_0-1 $, which confirms that $ R_0 $ defined in [10] can be used to determine population persistence. In an unbounded domain, we construct two monotonic operators to control the model operator from above and from below and obtain upper and lower bounds of the spreading speeds of the model.

中文翻译：

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河流环境中斑马贻贝模型的空间动力学

黄等。[10]开发了一个混合的连续/离散时间模型来描述斑马贻贝在河流中的持久性和入侵动力学。他们使用净繁殖率$ R_0 $来确定有界域中的种群持久性，并通过应用线性确定性猜想并使用[16]。由于关联的解决方案算子是非单调且非紧凑的，因此严格确定这些数量并不容易。在本文中，我们用数学方法分析了该模型的空间动力学。我们首先求解抛物线方程，然后将模型重写为完全离散时间的模型。在有界域中，我们表明线性化算子的谱半径$ \ hat {r} $可用于确定总体持久性，并且$ \ hat {r} -1 $的符号与$的符号相同R_0-1 $，以确认[10可以用来确定人口的持久性。在一个无界域中，我们构造了两个单调算子以从上方和下方控制模型算子，并获得模型扩展速度的上限和下限。