Institut für Angewandte Mathematik, Rheinische Friedrich-Wilhelms-Universität, Endenicher Allee 60, 53129, Bonn, Germany.
J Math Biol. 2022 Apr 28;84(6):42. doi: 10.1007/s00285-022-01753-z.
We consider a reaction-diffusion system of densities of two types of particles, introduced by Hannezo et al. (Cell 171(1):242-255.e27, 2017). It is a simple model for a growth process: active, branching particles form the growing boundary layer of an otherwise static tissue, represented by inactive particles. The active particles diffuse, branch and become irreversibly inactive upon collision with a particle of arbitrary type. In absence of active particles, this system is in a steady state, without any a priori restriction on the amount of remaining inactive particles. Thus, while related to the well-studied FKPP-equation, this system features a game-changing continuum of steady state solutions, where each corresponds to a possible outcome of the growth process. However, simulations indicate that this system self-organizes: traveling fronts with fixed shape arise under a wide range of initial data. In the present work, we describe all positive and bounded traveling wave solutions, and obtain necessary and sufficient conditions for their existence. We find a surprisingly simple symmetry in the pairs of steady states which are joined via heteroclinic wave orbits. Our approach is constructive: we first prove the existence of almost constant solutions and then extend our results via a continuity argument along the continuum of limiting points.
我们考虑了 Hannezo 等人引入的两种粒子密度的反应扩散系统（Cell 171(1):242-255.e27, 2017）。这是一个简单的生长过程模型：活跃的、分支的粒子形成静态组织的生长边界层，而不活跃的粒子则代表组织的其余部分。活跃的粒子在与任意类型的粒子碰撞时会扩散、分支并不可逆地变为不活跃。在没有活跃粒子的情况下，该系统处于稳态，对剩余不活跃粒子的数量没有任何先验限制。因此，尽管与研究得很好的 FKPP 方程有关，但该系统具有一系列改变游戏规则的稳态解，每个解对应于生长过程的可能结果。然而，模拟表明该系统会自我组织：在广泛的初始数据下，会出现具有固定形状的传播前沿。在本工作中，我们描述了所有正的有界传播波解，并得到了它们存在的充分必要条件。我们发现，通过异宿波轨连接的稳态对之间存在一种令人惊讶的简单对称性。我们的方法是建设性的：我们首先证明了几乎常数解的存在，然后通过沿极限点连续统的连续性论证扩展了我们的结果。
