Hopf bifurcation in a stomatal oscillator

Stomata are microscopic openings in leaves of green plants which permit gas exchange. This paper presents a parameter study of a model of a stomatal oscillator first derived by Delwiche and Cooke in 1977. We prove the existence of an unstable limit cycle by using the theory of the Hopf bifurcation. Other bifurcations exhibited by the model are also discussed.     

Maintenance of polygenic variation through mutation-selection balance: bifurcation analysis of a biallelic model

Biallelic models which ignore linkage disequilibrium have been used to study variability maintained by mutation in the presence of Gaussian stabilizing selection. Recent work of Barton (1986) showed that these models have stable equilibria at which the mean phenotype differed from the optimum, and that the variability maintained at such equilibria would be higher than at the symmetric equilibria calculated by Bulmer (1980) and others. Here I determine the bifurcation structure of this model, and confirm and extend Barton's results. The form of the bifurcations gives information about the domains of attraction of various equilibria, and shows why the nonsymmetric equilibria may not be observed. The techniques may prove useful in the analysis of other population genetic models.     

动态IP3－Ca2+振荡模型的数值分析

通过改进J.W.Shuai和P.Jung钙振荡模型,得到与IP3浓度相关的动态IP3-Ca^2＋振荡模型.利用改进模型,数值分析依赖性参数λ和钙通道数目N对Ca^2＋振荡的影响,得到Ca^2＋振荡关于参数λ的分叉图、Ca^2＋振荡与IP3振荡的一致性、钙通道数目N对Ca^2＋振荡的影响等.这些模型结果显示了Ca^2＋振荡的特性.     

Equilibrium analysis and phase synchronization of two coupled HR neurons with gap junction

The properties of equilibria and phase synchronization involving burst synchronization and spike synchronization of two electrically coupled HR neurons are studied in this paper. The findings reveal that in the non-delayed system the existence of equilibria can be turned into intersection of two odd functions, and two types of equilibria with symmetry and non-symmetry can be found. With the stability and bifurcation analysis, the bifurcations of equilibria are investigated. For the delayed system, the equilibria remain unchanged. However, the Hopf bifurcation point is drastically affected by time delay. For the phase synchronization, we focus on the synchronization transition from burst synchronization to spike synchronization in the non-delayed system and the effect of coupling strength and time delay on spike synchronization in delayed system. In addition, corresponding firing rhythms and spike synchronized regions are obtained in the two parameters plane. The results allow us to better understand the properties of equilibria, multi-time-scale properties of synchronization and temporal encoding scheme in neuronal systems.     

Thermodynamic view on decision-making process: emotions as a potential power vector of realization of the choice

This research is devoted to possible mechanisms of decision-making in frames of thermodynamic principles. It is also shown that the decision-making system in reply to emotion includes vector component which seems to be often a necessary condition to transfer system from one state to another. The phases of decision-making system can be described as supposed to be nonequilibrium and irreversible to which thermodynamics laws are applied. The mathematical model of a decision choice, proceeding from principles of the nonlinear dynamics considering instability of movement and bifurcation is offered. The thermodynamic component of decision-making process on the basis of vector transfer of energy induced by emotion at the given time is surveyed. It is proposed a three-modular model of decision making based on principles of thermodynamics. Here it is suggested that at entropy impact due to effect of emotion, on the closed system—the human brain,—initially arises chaos, then after fluctuations of possible alternatives which were going on—reactions of brain zones in reply to external influence, an order is forming and there is choice of alternatives, according to primary entrance conditions and a state of the closed system. Entropy calculation of a choice expectation of negative and positive emotion shows judgment possibility of existence of “the law of emotion conservation” in accordance with several experimental data.     

Mathematical modeling of the population dynamics of a distinct interactions type system with local dispersal

Distinct biotic interactions in multi-species communities are a ubiquitous force in the natural ecosystem, and this force is an essential determinant of community stability and species coexistence outcomes. We conduct numerical simulations and bifurcation analysis of partial differential equations to gain better understanding and ecological insights into how predation (a), predator handling time (h), and local dispersal affect multi-species community dynamics. This system consists of resource-mutualist-exploiter-competitor interactions and local dispersal. From the inspection of our numerical simulations and co-dimension one bifurcation analysis findings, we discover several critical values that correspond to transcritical bifurcation, subcritical and supercritical Hopf bifurcations. This occurs as we vary the bifurcation parameters a and h in this complex ecological system under symmetric and asymmetric dispersal scenarios. Furthermore, the interplay between these local bifurcation points results in an exciting co-dimension two bifurcations, i.e., Bogdanov-Takens and cusp bifurcation points, respectively, which act as the synchronization points in this complex ecological system. From an ecological viewpoint, we find that (i) the effect of the no-dispersal scenario supports the maintenance of species biodiversity when the predation strength is moderate; (ii) symmetric dispersal induces both subcritical and supercritical Hopf bifurcation and support species diversity for moderate predation strength; and (iii) asymmetric dispersal promotes species diversity as it simplifies the bifurcation changes in dynamics by eliminating the subcritical bifurcations that trigger uncertainty, and this dispersal mechanism mediates species coexistence outcomes. Fundamentally, stable limit cycles have been reported as predator handling time varies in some ecological models; however, we observed in our bifurcation analysis the emergence of the unstable limit cycle as predator handling time changes. We discover that intense predator handling time destabilizes this complex ecological community. In general, our results demonstrate the influential roles of predation, predator handling time, and local dispersal in determining this system’s coexistence dynamics. This knowledge provides a better understanding of species conservation and biological control management.     

一类自催化反应扩散模型正解的唯一性与稳定性

主要考虑了一类三分子自催化反应扩散系统．在齐次Dirichlet和Robin边界条件下,当反应率c适当小,系统没有共存态;当c适当大,系统至少有一个共存态;当c充分大,系统有唯一渐近稳定的共存态．特别地,在一维空间上共存态是唯一的．在齐次Neumann边界条件下系统是一个简单系统．     

海洋生态系统受海洋病毒影响的研究

该文对一个海洋病毒感染模型进行研究,分析其动力学行为,发现系统的动力性受病毒复制因素的影响较大,找到病毒复制因素的阈值,研究系统平衡点的存在性,稳定性,系统持久性及分支的产生.这对海洋生态研究具有一定意义.     

未搅拌恒化器中单食物链模型的反应扩散方程组

本文研究一类描写未搅拌恒化器中单食物链模型的反应扩散方程组,用分定理证明正稳态解的存在性,并给出种群绝灭及持续生存的条件。     

损伤神经自发放电节律分岔与频率变化的非线性特征

为了研究神经放电节律回周分岔与放电频率变化之间的关系,采用大鼠坐骨神经慢性结扎模型,记录损伤区的自发放电,观察放电节律转化的动力学规律,分析相应的放电频率的变化,并用理论模型进行数值模拟。结果表明,与放电节律加周分岔相对应,放电频率的变化呈现非线性的特征,数值模拟支持实验的发现。研究提示：神经放电的频率变化与刺激强度的改变并非呈简单的线性相关,可能具有更复杂的关系。