Neumann Gunter, Schuster Stefan
J Math Biol. 2007 Jun;54(6):815-46. doi: 10.1007/s00285-006-0065-3. Epub 2007 Apr 25.
In this work, important aspects of bacteriocin producing bacteria and their interplay are elucidated. Various attempts to model the resistant, producer and sensitive Escherichia coli strains in the so-called rock-scissors-paper (RSP) game had been made in the literature. The question arose whether there is a continuous model with a cyclic structure and admitting an oscillatory dynamics as observed in various experiments. The May-Leonard system admits a Hopf bifurcation, which is, however, degenerate and hence inadequate. The traditional differential equation model of the RSP-game cannot be applied either to the bacteriocin system because it involves positive interaction terms. In this paper, a plausible competitive Lotka-Volterra system model of the RSP game is presented and the dynamics generated by that model is analyzed. For the first time, a continuous, spatially homogeneous model that describes the competitive interaction between bacteriocin-producing, resistant and sensitive bacteria is established. The interaction terms have negative coefficients. In some experiments, for example, in mice cultures, migration seemed to be essential for the reinfection in the RSP cycle. Often statistical and spatial effects such as migration and mutation are regarded to be essential for periodicity. Our model gives rise to oscillatory dynamics in the RSP game without such effects. Here, a normal form description of the limit cycle and conditions for its stability are derived. The toxicity of the bacteriocin is used as a bifurcation parameter. Exact parameter ranges are obtained for which a stable (robust) limit cycle and a stable heteroclinic cycle exist in the three-species game. These parameters are in good accordance with the observed relations for the E. coli strains. The roles of growth rate and growth yield of the three strains are discussed. Numerical calculations show that the sensitive, which might be regarded as the weakest, can have the longest sojourn times.
在这项工作中，阐明了产细菌素细菌的重要方面及其相互作用。文献中已进行了各种尝试，以在所谓的“石头 - 剪刀 - 布”（RSP）博弈中对耐药、产细菌素和敏感的大肠杆菌菌株进行建模。问题在于是否存在一个具有循环结构且能呈现如各种实验中所观察到的振荡动力学的连续模型。梅 - 伦纳德系统存在霍普夫分岔，然而，该分岔是退化的，因此并不适用。RSP博弈的传统微分方程模型也不能应用于细菌素系统，因为它涉及正相互作用项。本文提出了一个合理的RSP博弈竞争型洛特卡 - 沃尔泰拉系统模型，并分析了该模型产生的动力学。首次建立了一个连续的、空间均匀的模型，该模型描述了产细菌素、耐药和敏感细菌之间的竞争相互作用。相互作用项具有负系数。例如，在一些实验中，如小鼠培养实验，迁移似乎对RSP循环中的再次感染至关重要。通常，诸如迁移和突变等统计和空间效应被认为对周期性至关重要。我们的模型在没有这些效应的情况下在RSP博弈中产生了振荡动力学。这里，推导了极限环的范式描述及其稳定性条件。细菌素的毒性被用作分岔参数。获得了精确的参数范围，在该范围内，三种群博弈中存在稳定（鲁棒）极限环和稳定异宿环。这些参数与观察到的大肠杆菌菌株关系非常吻合。讨论了三种菌株的生长速率和生长产量的作用。数值计算表明，可能被视为最弱的敏感菌株可以具有最长的停留时间。
