Internal boundary value problems with displacement for the mixed-wave equation

В работе исследованы краевые задачи с внутреннекраевым смещением для модельного смешанно-волнового уравнения, которые являются обобщениями задачи Гурса и задач с данными на противоположных характеристиках. Показано, что при определенных условиях на заданные функции решение исследуемых задач существует, единственно и выписывается в явном виде. The paper investigates boundary value problems with an internal boundary displacement for a model mixed-wave equation, which are generalizations of the Goursat problem and problems with data on opposite characteristics. It is shown that, under certain conditions for given functions, the solution to the problems under study exists, is unique, and is written out in an explicit form.

Estimating Invasion Dynamics with Geopolitical Unit-Level Records: The Optimal Method Depends on Irregularity and Stochasticity of Spread

Biological invasions are an ongoing threat for sustainability of ecosystems, and estimating the spread of invasive species is critical for making management decisions. Geopolitical unit-level data (GULD) are often used to estimate invasions due to their wide availability, and researchers had evaluated the abilities of multiple methods to estimate invasion with GULD. However, earlier studies were case based and only addressed limited information on the spread, thus making it inadequate to determine which method to choose to estimate invasions with GULD under various spread scenarios. Here, we conducted a simulation study to (1) evaluate performances of eight methods on estimating expansion patterns, spread rates, and spread dynamics of invasive species with GULD; (2) assess the impact of size and homogeneity of size of geopolitical unit on the estimations by studied methods; (3) evaluate the similarities of all studied methods. Additionally, we presented a concave hull boundary displacement method (Ctd_BD) and an area-based regression method (SqrtNA_R) to estimate spread with GULD. Three regions with varying sizes of counties in the United States (U.S.) were selected to conduct the simulations, and three spread scenarios and three expansion patterns were simulated. AIC, and R2 and root mean square error (RMSE) were used to evaluate the accuracy of methods on estimating expansion pattern, and overall spread rate and spread dynamics, respectively. Correlation coefficient and RMSE were used to assess the similarity of eight methods. We found Ctd_BD and area-based regression methods consistently estimated the right expansion patterns. Boundary displacement and area-based regression methods estimated highly correlated spread rates and dynamics. Distance-based regression methods provided a high accuracy on estimating overall spread rate without long-distance jump dispersal but performed poorly on estimating the spread dynamics. We recommend boundary displacement method, especially Ctd_BD, for estimating spread with GULD, whereas for spread without clear infestation boundaries, distance-based regression can be used to estimate overall spread rate and area-based regression can be used to estimate spread dynamics.