Dynamical study of nutrient-phytoplankton model with toxicity: Effect of diffusion and time delay

A harmful algal bloom is one of the significant concerns of marine biodiversity. The control of such algal bloom is required for the conservation of marine ecology. The proposed work discusses a model for interacting nutrient phytoplankton systems with the effect of toxic chemicals released by phytoplankton and time delay in toxin liberation. Our main aim is to provide a great insight into the impact of toxins and time delay on the dynamics of nutrient-phytoplankton. We investigate the stability of system dynamics, and the condition for the existence of Turing instability is obtained. The role of time delay is also investigated for the proposed system. The numerical simulation shows that toxin release rate, the diffusion coefficient of nutrients, and time delay significantly impact system dynamics. We observe that increasing values of toxin release results in the system dynamics show stable and oscillatory behavior without diffusion. The spatial and spatio-temporal patterns show that the higher value of the rate of toxin release lead to periodic and standing waves. Finally, we observe that the time delay in toxin distribution term stabilizes and destabilizes the system dynamics.     

Turing incomparability in Scott sets

For every Scott set and every nonrecursive set in , there is a such that and are Turing incomparable.

     

金属胶束催化体系中经Turing分支形成的空间有序结构与催化活性

摘要基于金属胶束催化反应的三元复合物动力学模型, 分析了金属胶束作为一类特殊的超分子体系, 其参与催化反应的特殊动力学行为与体系中可能出现的自组织结构的关系. 研究结果表明, 在一定的边界条件下, 即使基元反应是线性化学反应步骤, 由于扩散步骤与化学反应的耦合, 以及非理想性的组分热力学行为, 也将不可避免地出现经Turing分支形成的浓度场空间自组织. 计算结果表明, 这种宏观有序结构的形成将使金属胶束催化体系中呈现出以存在反应活性极高的位点为特征的反应活性的空间有序分布; 从自组织理论的角度对这类胶束体系催化活性的时空有序特征给予一定程度的阐释.     

Stability and Hopf bifurcation of a diffusive predator–prey model with predator saturation and competition

This article discusses a predator–prey system with predator saturation and competition functional response. The local stability, existence of a Hopf bifurcation at the coexistence equilibrium and stability of bifurcating periodic solutions are obtained in the absence of diffusion. Further, we discuss the diffusion-driven instability, Hopf bifurcation for corresponding diffusion system with zero flux boundary condition and Turing instability region regarding the parameters are established. Finally, numerical simulations supporting the theoretical analysis are also included.     

Modeling ternary mixtures by mean-field theory of polyelectrolytes: Coupled Ginzburg-Landau and Swift-Hohenberg equations

The purpose of this work is to model ternary mixtures using the theory of pattern formation and of polyelectrolytes, with mean-field approximations. The model has two local, non-conserved order parameters. In the free energy short-range and long-range nonlocal interactions between elements of the mixture are considered. The spatiotemporal dynamics of the system is described by coupling the time-dependent Ginzburg-Landau equation and the Swift-Hohenberg equation. These non-linear partial differential equations are solved with numerical methods to study the emergent spatially stable configurations. The model shows a large diversity of patterns, which permit an interpretation of the behavior of some biological systems and presents different growth lengths within its spatial structures.     

Turing instability for a semi-discrete Gierer-Meinhardt system

In this paper, we investigate the spatial patterns of a Gierer-Meinhardt system where the space is discrete in two dimensions with the periodic boundary condition and time is continuous, in contrast to the continuum models. The conditions of Turing instability are obtained by linear analysis and a series of numerical simulations are performed. In the instability region, we have shown that this system can produce a number of different patterns such as stripes and snowflake pattern, other than ubiquitous fix-spotted patterns. As mentioned, the results are substantiated only by means of snapshots of the apatial grid. However, we also give some analysis by using the time series at three random grids and of the average value of states, that is, the stable state patterns can be observed. On the other hand, the effects of varying parameters on pattern formation are also discussed. Moreover, simulations for fixed parameters and special initial conditions indicate that the initial conditions can alter the structure of patterns. The patterns can form as a consequence of cellular interaction. So patterns arising from a semi-discrete model can present simulations on a geometrically accurate representation in biology. As a result, our work is interesting and important in ecology.     

图灵机计算实数函数的稳定性

定义在全体实数上的可计算函数是一个很重要的概念．在这以前定义可计算的实数函数有两个途径．第一个途径是首先要定义可计算实数的指标．想要确定实数函数y=f（x）是不是可以计算就要看是否存在一个自然数的（部分）递归函数将可计算实数x的指标对应到可计算实数y的指标．这样一来对实数函数的研究依赖于对自然数函数的研究．第二个定义可计算的实数函数的途径是以逼近为基础的．一个实数函数是可以计算的如果它既是序列可计算的同时也是一致连续的．用这个途径来定义可计算实数函数使用的条件过强以至于很多有用的实数函数成为不可计算的实数函数．例如“〈”和“=”的命题函数就是不可以计算的因为它们是不连续的命题函数．本文讨论了图灵机的稳定性并且给出了一个基于稳定图灵机的可计算实数函数的定义．我们的定义不需要用到自然数的（部分）递归函数．根据我们的定义很多常用实数函数特别是一些不连续的常用实数函数都是可以计算的．用我们的定义来讨论可计算实数函数的性质比原来的定义要方便得多．     

Relations between algorithmic reducibilities of algebraic systems

In this paper we adduce examples of various algorithmic reducibilities of algebraic systems.