Department of Mathematics, Bar Ilan University, Ramat Gan, Israel.
Math Biosci Eng. 2013 Jun;10(3):523-50. doi: 10.3934/mbe.2013.10.523.
We here study spatially extended catalyst induced growth processes. This type of process exists in multiple domains of biology, ranging from ecology (nutrients and growth), through immunology (antigens and lymphocytes) to molecular biology (signaling molecules initiating signaling cascades). Such systems often exhibit an extinction-proliferation transition, where varying some parameters can lead to either extinction or survival of the reactants. When the stochasticity of the reactions, the presence of discrete reactants and their spatial distribution is incorporated into the analysis, a non-uniform reactant distribution emerges, even when all parameters are uniform in space. Using a combination of Monte Carlo simulation and percolation theory based estimations; the asymptotic behavior of such systems is studied. In all studied cases, it turns out that the overall survival of the reactant population in the long run is based on the size and shape of the reactant aggregates, their distribution in space and the reactant diffusion rate. We here show that for a large class of models, the reactant density is maximal at intermediate diffusion rates and low or zero at either very high or very low diffusion rates. We give multiple examples of such system and provide a generic explanation for this behavior. The set of models presented here provides a new insight on the population dynamics in chemical, biological and ecological systems.
我们在这里研究空间扩展的催化剂诱导生长过程。这种类型的过程存在于多个生物学领域，从生态学（营养物质和生长）到免疫学（抗原和淋巴细胞）再到分子生物学（启动信号级联的信号分子）。这些系统通常表现出灭绝-增殖转变，改变某些参数可以导致反应物的灭绝或存活。当反应的随机性、离散反应物的存在及其空间分布被纳入分析中时，即使所有参数在空间上都是均匀的，也会出现不均匀的反应物分布。通过结合蒙特卡罗模拟和基于渗流理论的估计；研究了这种系统的渐近行为。在所有研究的情况下，事实证明，反应物种群在长期内的总体生存基于反应物聚集体的大小和形状、它们在空间中的分布和反应物的扩散速率。我们在这里表明，对于一大类模型，在中等扩散速率下反应物密度最大，而在非常高或非常低的扩散速率下则很低或为零。我们提供了此类系统的多个示例，并为这种行为提供了一个通用的解释。这里提出的模型集为化学、生物和生态系统中的种群动态提供了新的见解。
