The evolution of calcium fertilisation waves on the cortex of amphibian eggs can be described by a nonlinear reaction-diffusion process on the surface of a sphere. Here, we use the nonclassical symmetry technique to find an exact analytic solution that describes the evolution of the calcium concentration. The solutions presented compare well with published experimental results. The analytic solution can be used to give insight into the processes governing the fertilisation wave, such as the flow of calcium ions from the sperm entry point. By finding a spiral solution to an approximate equation linearised near saturation, we also demonstrate how solutions with other properties may be constructed using this technique.

中文翻译：

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卵表面钙离子施肥波的解析溶液。

两栖动物卵皮层上钙肥化波的演化可以通过球体表面的非线性反应扩散过程来描述。在这里，我们使用非经典对称技术找到描述钙浓度变化的精确解析溶液。提出的解决方案与已发布的实验结果很好地比较。该分析解决方案可用于深入了解控制受精波的过程，例如来自精子进入点的钙离子流。通过找到近似饱和度线性化近似方程的螺旋解，我们还演示了如何使用该技术构造具有其他性质的解。