A Catastrophe Model for Water Bloom Prediction: A Case Study of China's Lake Chaohu

Using a case study of Lake Chaohu, the fifth largest lake in China, we constructed a cusp model for water bloom prediction that used TP (total phosphorus), T (temperature), Chla (chlorophyll-a), and DO (dissolved oxygen). These four parameters were assumed to be the most important factors in eutrophication and water bloom of the lake. The model was found to be accurate, because its relative error was around 10%. What is more convincing, according to the catastrophe discriminant of the cusp model, it could be judged that a discontinuous jump of the aquatic ecosystem occurred in July 2004, in Lake Chaohu. This conclusion is consistent with the fact that water blooms arose in August 2004. The cusp model also showed satisfactory precision when applied to forecast the eutrophication trend and prediction of water bloom in Lake Chaohu in 2005. The case study found that water bloom brought on by eutrophication can be fit and predicted by a catastrophe model. We suggest that catastrophe models would be a constructive approach to forecast and judge the outbreak of water bloom in lakes. In addition, by constructing and studying such catastrophe models, lake managers would be able to simulate the effects of different protection and mitigation projects and enrich the scientific basis for the optimization of these projects as well.     

水华爆发的突变模型——以巢湖为例

面对湖泊水库富营养化的严峻形势,水华防治是当务之急,而根据富营养化现状对水华进行准确的判定和预报更是重中之重.从水生态系统的角度出发,综合考虑TP、T、Chla和DO等4个对巢湖富营养化乃至水华影响最为突出的因子,构建了巢湖水华突变的尖点模型,通过模型来模拟巢湖富营养化引发水华的情势.在对巢湖2000～2003年时间序列的监测数据进行数据拟合的基础上,发现南淝河入湖区断面处逐月TP、T、Chla和DO数据的演变规律符合突变理论的尖点模型特征.经检验,模型的相对误差控制在10%左右,具有较好的拟合精度.根据模型的突变判别,巢湖的水生态系统2003年7月在南淝河入湖区断面处发生了突变,这一模拟结论与该断面2003年8月爆发水华的实际情况相一致.研究表明,水华突变模型的建立,准确地反映了巢湖富营养化引发水华的实际情况,系统地勾勒出富营养化状态下水生态系统各要素间的动态响应模式,揭示了水华爆发的突变机理.通过水华突变模型的构建,不仅能够对巢湖水华的发生进行判断和预报,还可以对各项防治措施的实施效果进行预测和模拟,进而为综合整治方案的优化和统筹提供科学依据.水华的爆发是多诱因的综合作用结果,是营养物质长期累积、由量变到质变的演化过程,作为目前唯一的一种研究由渐变引起突变的系统理论,突变理论满足水华研究的数理要求,为水华现象的数值模拟提供了可行的解决方案.     

Ecological outbreak dynamics and the cusp catastrophe

Many ecological processes exhibit trajectories which can be suitably represented by stable equilibria or smooth limit cycles. However, a third kind of ecological process involves intermittent, abrupt, and drastic changes in densities, here termed outbreak dynamics, which require different modelling frameworks. One such framework, the cusp catastrophe, is used here in a modelling study of a particular outbreak insect, the forest tent caterpillar. This model is then generalized to cover a set of related ecological systems. The particular form of the model for each system depends on whether the major controlling ecological variables are externally imposed, or are incorporated in the model equations. It is concluded that the simple cusp catastrophe is an appropriate metaphor for understanding outbreak dynamics.     

On a global representation of dynamical characteristics in ESS-models

ESS-models are considered in the frame of adequate dynamical systems. This leads to a method of global representation of the dynamical characteristics, which allows for a comprehensive view on the relations among ESS-matrices. As an extension of ESS-models, the concept of ESS-models with changing matrix is introduced, which, in applications, makes use of such global representations. In particular, structurally stable and “catastrophic” cases can be identified and their significance in the context of ESS-models can be clarified. As a complete result for the case of two strategies a surface representation is obtained, which is an analogue to the elementary “cusp catastrophe”. These concepts are applied in a model that describes directed evolution of a behavioural component to a certain limit.     

马尾松林节肢动物群落的稳定性

依据广西大青山马尾松林区马尾松毛虫一个暴发周期中食叶、捕食、寄生、刺吸四大类群多样性指数(H′)、丰富度指数(MD)、均匀度指数(J)和害虫与天敌的数量比例的数据,建立突变模型对马尾松林节肢动物群落的稳定性进行研究,并利用湖南桂阳数据对模型进行了检验.结果表明:在马尾松毛虫的整个暴发过程中,暴发区在暴发年的昆虫生物群落处于明显的不稳定状态,而在非暴发年时,群落则处于亚稳定的状态;非暴发区在暴发年处于亚稳定的状态,非暴发年时处于稳定状态.在松毛虫各个发生阶段,当上升期处于不稳定状态时,有可能导致松毛虫的大发生;松毛虫暴发后,节肢动物群落依旧处于不稳定状态,有可能会造成松毛虫的连续暴发,这主要依赖于各种因子的影响.马尾松毛虫暴发呈现非线性、突变性等特点,食叶类群的影响为马尾松毛虫暴发的关键因子,各类群作用均衡的系统比较稳定,虫害暴发是群落结构趋稳调节过程中的强烈外部表现.研究表明,突变模型在检测群落结构稳定性和预测松毛虫发生上有很好的应用前景.     

Catastrophe Theory Enables Moves to be Detected Towards and Away from Self-Organization: The Example of Epileptic Seizure Onset

Macroscopic systems with many interacting subunits, when driven far from equilibrium, exhibit self-organization, for example when a pathological rhythm appears suddenly in an epileptic patient. Sudden changes occurring while conditions vary smoothly have, in cases of interest, underlying mathematics that are the subject of Thom’s catastrophe theory. The assumption made herein that the system’s state variables, akin to order parameters, reduce in practice to only one single real variable, ensures that the system derives from a potential function, and warrants recourse to the catastrophe theory. The order parameter is, furthermore, interpreted as a measure of the electropathophysiological activity in the brain, increasing monotonously with the degree of neuronal synchronism. With two neuronal influences, excitatory and inhibitory, as control parameters, the catastrophe is the archetypal cusp. Implementation of catastrophe theory leads to equations showing that fluctuations in a system’s dynamics may be utilised for signalling steps precursory to oncoming catastrophes. Pre-seizure dynamics in epileptic patients exhibit steps towards and away from catastrophe; the steps away are interpreted in terms of homeostatic feedback, consequent on changing patterns of neuronal activity. A number of characteristics of epileptic seizures of differing types merely follow from the geometry of the cusp equilibrium surface. In particular, types of seizures are distinguished by their angle of final approach to onset in parameter space. The measurable parameters by which approach to catastrophe is characterized, may be of use in investigations of the organism’s plasticity in epileptic patients, and in tests of therapeutic means for preventing seizures. There is no need to resort to a model, in the usual sense of the word, and therefore no differential equation needs to be set up.     

Spatial aspects of nest defence by pumpkinseed sunfish (Lepomis gibbosus): Stimulus features and an application of catastrophe theory

Using fixed stimulus dummies as territory intruders, we studied the spatial distributions of the aggressive responses made by nesting male pumpkinseed sunfish. Results replicated over three field seasons indicate that males defend two territory boundaries concurrently, depending on whether an intruder is approaching (“the defence perimeter’) or withdrawing (‘the attack perimeter’). This finding supports Zeeman's (1976) ‘cusp catastrophe model’ of nest defence. In addition, the defence perimeter remains constant over the breeding cycle while the attack perimeter varies. We discuss the ecological costs and benefits of this variation and present a motivational interpretation of the cusp catastrophe model. By varying the speed at which dummies intruded into nests, we determined that males react with a fixed latency to intruders at a fixed distance from the nest. Finally, the spatial distributions of male defence responses were partially determined by the location of the nest rim, but were unaffected by dummy size (contrary to Zeeman's model) or posture.     

Application of catastrophe theory to corneal swelling

Stromal swelling in human, cat, and rabbit cornea is biphasic, interpretable as an elementary cusp catastrophe proposed by Thom, with t* = log t and Q* = log Q (stromal charge Q, time t) as control parameters, and H0.5 (hydration H) as the state variable. A thermodynamic potential with two attractor regions, each with a local minimum, governs corneal stromal swelling. Transitions follow a 'saturation convention' whereby the second minimum is preferred upon availability. Corneal swelling is an example of a space-equivalent unfolding, where the transition plane moves in time. It is proposed that the transition plane coincides with the uncoupling of interfibrillary linkages or 'springs' in the corneal stroma, and is associated with a critical hydration of ca. 10 kg H2O per kilogram dry mass, and stromal charge ca. 1 x 10(-7) mol electrons.     

Teleosemantic modeling of cognitive representations

Naturalistic theories of representation seek to specify the conditions that must be met for an entity to represent another entity. Although these approaches have been relatively successful in certain areas, such as communication theory or genetics, many doubt that they can be employed to naturalize complex cognitive representations. In this essay I identify some of the difficulties for developing a teleosemantic theory of cognitive representations and provide a strategy for accommodating them: to look into models of signaling in evolutionary game theory. I show how these models can be used to formulate teleosemantics and expand it in new directions.     

An approach to the nonlinear dynamics of Russian wheat aphid population growth with the cusp catastrophe model

Many insect field populations, especially aphids, often exhibit irregular and even catastrophic fluctuations. The objective of the present study is to explore whether or not the population intrinsic rates of growth ( r _m) obtained under laboratory conditions can shed some light on the irregular changes of insect field populations. We propose to use the catastrophe theory, one of the earliest nonlinear dynamics theories, to answer the question. To collect the necessary data, we conducted a laboratory experiment to investigate population growth of the Russian wheat aphid (RWA), Diuraphis noxia (Mordvilko), in growth chambers. The experiment was designed as the factorial combinations of five temperatures and five host plant-growth stages (25 treatments in total): 1800 newly born RWA nymphs arranged in the 25 treatments (each treatment with 72 repetitions) were observed for their development, reproduction and survival through their entire lifetimes. After obtaining the population intrinsic rates of growth ( r _m) from the experimental data under various environmental conditions, we built a cusp catastrophe model for RWA population growth by utilizing r _m as the system state variable, and temperature and host plant-growth stage as control variables. The cusp catastrophe model suggests that RWA population growth is intrinsically catastrophic , and dramatic jumps from one state to another might occur even if the temperature and plant-growth stage change smoothly . Other basic behaviors of the cusp catastrophe model, such as catastrophic jumps , hystersis and divergence , are also expected in RWA populations. These results suggest that the answer to the previously proposed question should be "yes".     

Discrete Model of Opinion Changes Using Knowledge and Emotions as Control Variables

We present a new model of opinion changes dependent on the agents emotional state and their information about the issue in question. Our goal is to construct a simple, yet nontrivial and flexible representation of individual attitude dynamics for agent based simulations, that could be used in a variety of social environments. The model is a discrete version of the cusp catastrophe model of opinion dynamics in which information is treated as the normal factor while emotional arousal (agitation level determining agent receptiveness and rationality) is treated as the splitting factor. Both variables determine the resulting agent opinion, which itself can be in favor of the studied position, against it, or neutral. Thanks to the flexibility of implementing communication between the agents, the model is potentially applicable in a wide range of situations. As an example of the model application, we study the dynamics of a set of agents communicating among themselves via messages. In the example, we chose the simplest, fully connected communication topology, to focus on the effects of the individual opinion dynamics, and to look for stable final distributions of agents with different emotions, information and opinions. Even for such simplified system, the model shows complex behavior, including phase transitions due to symmetry breaking by external propaganda.     

Switching effect of predation on competitive prey species

The fact that the predation pressure has a stabilizing effect on the community of competitive species is demonstrated by a mathematical model of two-preys and one-predator system which has the switching property of predation. By analyzing a dynamical system for these three species populations, it is shown that, in a wide range of parameter space, the system has stable coexisting equilibrium states and the manifold of stable stationary points exhibits a cusp catastrophe and there exist two stable stationary points in the cusp region in the parameter space. Thus, it has been shown that Cause's competitive exclusion is actually relaxed by the switching mechanism of predation.     

The Problem of Colliding Networks and its Relation to Cell Fusion and?Cancer

Cell fusion, a process that merges two or more cells into one, is required for normal development and has been explored as a tool for stem cell therapy. It has also been proposed that cell fusion causes cancer and contributes to its progression. These functions rely on a poorly understood ability of cell fusion to create new cell types. We suggest that this ability can be understood by considering cells as attractor networks whose basic property is to adopt a set of distinct, stable, self-maintaining states called attractors. According to this view, fusion of two cell types is a collision of two networks that have adopted distinct attractors. To learn how these networks reach a consensus, we model cell fusion computationally. To do so, we simulate patterns of gene activities using a formalism developed to simulate patterns of memory in neural networks. We find that the hybrid networks can assume attractors that are unrelated to parental attractors, implying that cell fusion can create new cell types by nearly instantaneously moving cells between attractors. We also show that hybrid networks are prone to assume spurious attractors, which are emergent and sporadic network states. This finding means that cell fusion can produce abnormal cell types, including cancerous types, by placing cells into normally inaccessible spurious states. Finally, we suggest that the problem of colliding networks has general significance in many processes represented by attractor networks, including biological, social, and political phenomena.     

A Cayley tree immune network model with antibody dynamics

A Cayley tree model of idiotypic networks that includes both B cell and antibody dynamics is formulated and analysed. As in models with B cells only, localized states exist in the network with limited numbers of activated clones surrounded by virgin or near-virgin clones. The existence and stability of these localized network states are explored as a function of model parameters. As in previous models that have included antibody, the stability of immune and tolerant localized states are shown to depend on the ratio of antibody to B cell lifetimes as well as the rate of antibody complex removal. As model parameters are varied, localized steady-states can break down via two routes: dynamically, into chaotic attractors, or structurally into percolation attractors. For a given set of parameters percolation and chaotic attractors can coexist with localized attractors, and thus there do not exist clear cut boundaries in parameter space that separate regions of localized attractors from regions of percolation and chaotic attractors. Stable limit cycles, which are frequent in the two-clone antibody B cell (AB) model, are only observed in highly connected networks. Also found in highly connected networks are localized chaotic attractors. As in experiments by Lundkvistet al. (1989.Proc. natn. Acad. Sci. U.S.A. 86, 5074–5078), injection ofAb ₁ antibodies into a system operating in the chaotic regime can cause a cessation of fluctuations ofAb ₁ andAb ₂ antibodies, a phenomenon already observed in the two-clone AB model. Interestingly, chaotic fluctuations continue at higher levels of the tree, a phenomenon observed by Lundkvistet al. but not accounted for previously.     

Encoding certainty in bump attractors

Persistent activity in neuronal populations has been shown to represent the spatial position of remembered stimuli. Networks that support bump attractors are often used to model such persistent activity. Such models usually exhibit translational symmetry. Thus activity bumps are neutrally stable, and perturbations in position do not decay away. We extend previous work on bump attractors by constructing model networks capable of encoding the certainty or salience of a stimulus stored in memory. Such networks support bumps that are not only neutrally stable to perturbations in position, but also perturbations in amplitude. Possible bump solutions then lie on a two-dimensional attractor, determined by a continuum of positions and amplitudes. Such an attractor requires precisely balancing the strength of recurrent synaptic connections. The amplitude of activity bumps represents certainty, and is determined by the initial input to the system. Moreover, bumps with larger amplitudes are more robust to noise, and over time provide a more faithful representation of the stored stimulus. In networks with separate excitatory and inhibitory populations, generating bumps with a continuum of possible amplitudes, requires tuning the strength of inhibition to precisely cancel background excitation.     

Biophysical interpretation of membrane fluidity by catastrophe theory

We have regrouped the data of two examples where membrane fluidity was progressively modified by pharmacological and physiological agents. In our approach, each initial condition is determined by control parameters (depth of the membrane bilayer explored, concentration of agents). The fluidity is expressed as a state parameters followed on the control space. Then, according to Catastrophe Theory, the results are depicted as tridimensional patterns which can be recognized as bifurcation sets. Consequently, the fluidity is considered as resulting in a compromise phenomenon (normal factor) between two structurally attracting configurations (of hydrophilic and hydrophobic nature). The concepts of membrane activity and membrane function are then discussed on the basis of physiological functionality of biomembranes. The main application of this research interests the pharmacological domain. Indeed, a new classification of drugs could be proposed. According to the loss of membrane functionality, some drugs could imply a too high differentiation of attractors (splitting factor) and others could imply the destruction of the compromise. The first type is characterized by the physical destruction of the membrane. In the second type, the entity of the bilayer is preserved but the membrane is destructurated.