Departamento de Matemática, Universidade Federal de Santa Maria, RS, Brasil.
Bull Math Biol. 2011 Aug;73(8):1812-40. doi: 10.1007/s11538-010-9593-5. Epub 2010 Oct 23.
Understanding of population dynamics in a fragmented habitat is an issue of considerable importance. A natural modelling framework for these systems is spatially discrete. In this paper, we consider a predator-prey system that is discrete both in space and time, and is described by a Coupled Map Lattice (CML). The prey growth is assumed to be affected by a weak Allee effect and the predator dynamics includes intra-specific competition. We first reveal the bifurcation structure of the corresponding non-spatial system. We then obtain the conditions of diffusive instability on the lattice. In order to reveal the properties of the emerging patterns, we perform extensive numerical simulations. We pay a special attention to the system properties in a vicinity of the Turing-Hopf bifurcation, which is widely regarded as a mechanism of pattern formation and spatiotemporal chaos in space-continuous systems. Counter-intuitively, we obtain that the spatial patterns arising in the CML are more typically stationary, even when the local dynamics is oscillatory. We also obtain that, for some parameter values, the system's dynamics is dominated by long-term transients, so that the asymptotical stationary pattern arises as a sudden transition between two different patterns. Finally, we argue that our findings may have important ecological implications.
理解破碎生境中的种群动态是一个非常重要的问题。这些系统的自然建模框架是离散的。在本文中，我们考虑了一个时空离散的捕食者-被捕食者系统，该系统由耦合映射晶格（CML）描述。假设被捕食者的生长受到微弱的阿利效应的影响，而捕食者的动态包括种内竞争。我们首先揭示了相应非空间系统的分岔结构。然后我们得到了格子上扩散不稳定性的条件。为了揭示新兴模式的性质，我们进行了广泛的数值模拟。我们特别关注在图灵-霍普夫分岔附近的系统性质，该分岔被广泛认为是空间连续系统中模式形成和时空混沌的机制。与直觉相反，我们发现 CML 中出现的空间模式更典型地是静止的，即使局部动力学是振荡的。我们还发现，对于某些参数值，系统的动力学主要由长期瞬态主导，因此渐近稳定模式是在两种不同模式之间的突然转变。最后，我们认为我们的发现可能具有重要的生态意义。
