Purpose

This paper aims to introduce a new numerical approach based on the local weak form and the Petrov–Galerkin idea to numerically simulation of a predator–prey system with two-species, two chemicals and an additional chemotactic influence.

Design/methodology/approach

In the first proceeding, the space derivatives are discretized by using the direct meshless local Petrov–Galerkin method. This generates a nonlinear algebraic system of equations. The mentioned system is solved by using the Broyden’s method which this technique is not related to compute the Jacobian matrix.

Findings

This current work tries to bring forward a trustworthy and flexible numerical algorithm to simulate the system of predator–prey on the nonrectangular geometries.

Originality/value

The proposed numerical results confirm that the numerical procedure has acceptable results for the system of partial differential equations.

中文翻译：

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通过直接无网格局部 Petrov-Galerkin 方法模拟具有两种物种、两种化学物质和附加趋化影响的捕食者-被捕食者系统

目的

本文旨在引入一种基于局部弱形式和 Petrov-Galerkin 思想的新数值方法，对具有两种物种、两种化学物质和附加趋化影响的捕食者-被捕食者系统进行数值模拟。

设计/方法论/途径

在第一步中，使用直接无网格局部 Petrov-Galerkin 方法对空间导数进行离散化。这生成了非线性代数方程组。上述系统是通过使用布罗伊登方法来求解的，该技术与计算雅可比矩阵无关。

发现

目前的工作试图提出一种可靠且灵活的数值算法来模拟非矩形几何形状上的捕食者-被捕食者系统。

原创性/价值

所提出的数值结果证实了数值程序对于偏微分方程组具有可接受的结果。