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首页> 外文期刊>Journal of dynamics and differential equations >Existence and Stability of a Spike in the Central Component for a Consumer Chain Model
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Existence and Stability of a Spike in the Central Component for a Consumer Chain Model

机译：消费者链模型中心组件中尖峰的存在和稳定性

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摘要

We study a three-component consumer chain model which is based on Schnakenberg type kinetics. In this model there is one consumer feeding on the producer and a second consumer feeding on the first consumer. This means that the first consumer (central component) plays a hybrid role: it acts both as consumer and producer. The model is an extension of the Schnakenberg model suggested in Gierer and Meinhardt (Kybernetik 12:30-39, 1972) and Schnakenberg (J Theoret Biol 81:389-400, 1979) for which there is only one producer and one consumer. It is assumed that both the producer and second consumer diffuse much faster than the central component. We construct single spike solutions on an interval for which the profile of the first consumer is that of a spike. The profiles of the producer and the second consumer only vary on a much larger spatial scale due to faster diffusion of these components. It is shown that there exist two different single spike solutions if the feed rates are small enough: a large-amplitude and a small-amplitude spike. We study the stability properties of these solutions in terms of the system parameters. We use a rigorous analysis for the linearized operator around single spike solutions based on nonlocal eigenvalue problems. The following result is established: If the time-relaxation constants for both producer and second consumer vanish, the large-amplitude spike solution is stable and the small-amplitude spike solution is unstable. We also derive results on the stability of solutions when these two time-relaxation constants are small. We show a new effect: if the time-relaxation constant of the second consumer is very small, the large-amplitude spike solution becomes unstable. To the best of our knowledge this phenomenon has not been observed before for the stability of spike patterns. It seems that this behavior is not possible for two-component reaction-diffusion systems but that at least three components are required. Our main motivation to study this system is mathematical since the novel interaction of a spike in the central component with two other components results in new types of conditions for the existence and stability of a spike. This model is realistic if several assumptions are made: (i) cooperation of consumers is prevalent in the system, (ii) the producer and the second consumer diffuse much faster than the first consumer, and (iii) there is practically an unlimited pool of producer. The first assumption has been proven to be correct in many types of consumer groups or populations, the second assumption occurs if the central component has a much smaller mobility than the other two, the third assumption is realistic if the consumers do not feel the impact of the limited amount of producer due to its large quantity. This chain model plays a role in population biology, where consumer and producer are often called predator and prey. This system can also be used as a model for a sequence of irreversible autocatalytic reactions in a container which is in contact with a well-stirred reservoir.
机译：我们研究了基于Schnakenberg型动力学的三要素消费者链模型。在此模型中，有一个消费者以生产者为食，第二个消费者以第一个消费者为食。这意味着第一个消费者（中心部分）扮演着混合角色：既充当消费者又充当生产者。该模型是Gierer和Meinhardt（Kybernetik 12：30-39，1972）和Schnakenberg（J Theoret Biol 81：389-400，1979）建议的Schnakenberg模型的扩展，该模型只有一个生产者和一个消费者。假设生产者和第二消费者的扩散速度都快于中央环节。我们在第一个使用者的配置文件是峰值的时间间隔上构造单个峰值解决方案。由于这些成分的扩散较快，因此生产者和第二消费者的分布只在更大的空间尺度上变化。结果表明，如果进给速度足够小，则存在两种不同的单尖峰解决方案：大幅度尖峰和小幅度尖峰。我们根据系统参数研究这些解决方案的稳定性。我们基于非局部特征值问题对单峰解决方案的线性化算子进行了严格的分析。建立了以下结果：如果生产者和第二个消费者的时间松弛常数都消失，则大幅度峰值解决方案稳定，而小幅度峰值解决方案不稳定。当这两个时间松弛常数都小时，我们还可以得出溶液稳定性的结果。我们显示出一种新的效果：如果第二个使用者的时间松弛常数非常小，则大幅度尖峰解决方案将变得不稳定。据我们所知，这种现象以前从未观察到以确保尖峰模式的稳定性。对于两组分反应扩散系统，似乎无法实现这种行为，但是至少需要三个组分。我们研究此系统的主要动机是数学的，因为中心组件中尖峰与其他两个组件之间的新颖相互作用会导致出现尖峰的存在和稳定性的新型条件。如果做出几个假设，此模型将是现实的：（i）系统中普遍存在消费者合作，（ii）生产者和第二个消费者的扩散速度比第一个消费者快得多，并且（iii）实际上存在无限数量的消费者。生产者。第一个假设已被证明在许多类型的消费群体或人群中都是正​​确的，第二个假设是如果中心部分的流动性比其他两个要小得多，则第三个假设是现实的，如果消费者没有感受到生产者数量庞大，数量有限。这种链模型在人口生物学中扮演着重要角色，在那里消费者和生产者通常被称为掠食者和猎物。该系统还可用作与搅拌良好的容器接触的容器中一系列不可逆的自催化反应的模型。

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