The application of mathematics to fields such as biology, medicine, and biotechnology has resulted in quantitative relations, and mathematical biology, an interdisciplinary branch, has emerged. The considered models of cell/tissue growth are determined by cross-reaction-diffusion equations with possible transitions between malignant cells with different growth rates or healthy cells. For the considered models, the exact solutions in the explicit form were not obtained in the existing literature to our knowledge. Our main aim is to fill this gap by revealing their exact solutions. The obtained results have a major role in the literature so that the considered models are seen in a large scale of applications.

中文翻译：

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组织生长的反应-交叉扩散模型

数学在生物学、医学、生物技术等领域的应用，产生了数量关系，数学生物学这一跨学科的分支应运而生。所考虑的细胞/组织生长模型由交叉反应扩散方程确定，其中具有不同生长速率的恶性细胞或健康细胞之间可能发生转变。对于所考虑的模型，据我们所知，在现有文献中没有获得显式形式的精确解。我们的主要目标是通过揭示他们的确切解决方案来填补这一空白。获得的结果在文献中具有重要作用，因此所考虑的模型可以在大规模应用中看到。