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.     

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.     

The control of ciliary beat frequency

Ciliary movement is powered by axonemal dynein. This article considers how a signal transduction cascade initiated at the cell membrane may activate outer dynein arms to change the velocity of microtubule sliding and the swimming speed of ciliated cells. For Paramecium, a critical event in the cascade is the cAMP-dependent phosphorylation of a 29 kDa polypeptide that is associated with the outer dynein arm.     

The control of oscillatory movements of the forearm

The patterns of EMG activity in the biceps and triceps muscles were recorded during horizontal oscillatory movements of the forearm. Subjects showed increased frequency of oscillation as they voluntarily reduced movement amplitude. EMG burst duration was significantly correlated with wavelength of oscillation in every case. In almost half the cases burst intensity was also positively correlated with wavelength. Subjects seemed to be using one or both these methods to control amplitude. A model was developed in three stages which satisfactorily accounted for the data.     

Directed movements of ciliary and flagellar membrane components: a review.

The ability to rapidly translocate polystyrene microspheres attached to the surface of a plasma membrane domain reflects a unique form of cellular force transduction occurring in association with the plasma membrane of microtubule based cell extensions. This unusual form of cell motility can be utilized by protistan organisms for whole cell locomotion, the early events in mating, and transport of food organisms along the cell surface, and possibly intracellular transport of certain organelles. Since surface motility is observed in association with cilia and flagella of algae, sea urchin embryos and cultured mammalian cells, it is likely that it serves an additional role beyond those already cited; this is likely to be the transport of precursors for the assembly and turnover of ciliary and flagellar membranes and axonemes. In the case of the Chlamydomonas flagellum, where surface motility has been most extensively studied, it appears that cross-linking of flagellar surface exposed proteins induces a transmembrane signaling pathway that activates machinery for moving flagellar membrane proteins in the plane of the flagellar membrane. This signaling pathway in vegetative Chlamydomonas reinhardtii appears to involve an influx of calcium, a rise in intraflagellar free calcium concentration and a change in the level of phosphorylation of specific membrane-matrix proteins. It is hypothesized that flagellar surface contact with a solid substrate (during gliding), a polystyrene microsphere or another flagellum (during mating) will all activate a signaling pathway similar to the one artificially activated by the use of monoclonal antibodies to flagellar membrane glycoproteins. A somewhat different signaling pathway, involving a transient rise in intracellular cAMP level, may be associated with the mating of Chlamydomonas gametes, which is initiated by flagellum-flagellum contact. The hypothesis that the widespread observation of microsphere movements on various ciliary and flagellar surfaces may reflect a mechanism normally utilized to transport axonemal and membrane subunits along the internal surface of the organelle membrane presents a paradox in that one would expect this to be a constitutive mechanism, not one necessarily activated by a signaling pathway.     

The control of mandible movements in the ant Odontomachus

Ants use their mandibles to manipulate many different objects including food, brood and nestmates. Different tasks require the modification of mandibular force and speed. Besides normal mandible movements the trap-jaw ant Odontomachus features a particularly fast mandible reflex during which both mandibles close synchronously within 3 ms. The mandibular muscles that govern mandible performance are controlled by four opener and eight closer motor neurons. During slow mandible movements different motor units can be activated successively, and fine tuning is assisted by co-activation of the antagonistic muscles. Fast and powerful movements are generated by the additional activation of two particular motor units which also contribute to the mandible strike. The trap-jaw reflex is triggered by a fast trigger muscle which is derived from the mandible closer. Intracellular recording reveals that trigger motor neurons can generate regular as well as particularly large postsynaptic potentials, which might be passively propagated over the short distance to the trigger muscle. The trigger motor neurons are dye-coupled and receive input from both sides of the body without delay, which ensures the synchronous release of both mandibles.     

The control of multi-muscle systems: human jaw and hyoid movements

A model is presented of sagittal plane jaw and hyoid motion based on the model of motor control. The model, which is implemented as a computer simulation, includes central neural control signals, position- and velocity-dependent reflexes, reflex delays, and muscle properties such as the dependence of force on muscle length and velocity. The model has seven muscles (or muscle groups) attached to the jaw and hyoid as well as separate jaw and hyoid bone dynamics. According to the model, movements result from changes in neurophysiological control variables which shift the equilibrium state of the motor system. One such control variable is an independent change in the membrane potential of -motoneurons (MNs); this variable establishes a threshold muscle length () at which MN recruitment begins. Motor functions may be specified by various combinations of s. One combination of s is associated with the level of coactivation of muscles. Others are associated with motions in specific degrees of freedom. Using the model, we study the mapping between control variables specified at the level of degrees of freedom and control variables corresponding to individual muscles. We demonstrate that commands can be defined involving linear combinations of change which produce essentially independent movements in each of the four kinematic degrees of freedom represented in the model (jaw orientation, jaw position, vertical and horizontal hyoid position). These linear combinations are represented by vectors in space which may be scaled in magnitude. The vector directions are constant over the jaw/hyoid workspace and result in essentially the same motion from any workspace position. The demonstration that it is not necessary to adjust control signals to produce the same movements in different parts of the workspace supports the idea that the nervous system need not take explicit account of musculo-skeletal geometry in planning movements.This article was processed by the author using the LAT_EX style file pljour2 from Springer-Verlag.     

The logic of catastrophe

It has been assumed that public or collective goods, such as resource conservation or pollution reduction, will not be created or maintained since rational actors in the international system will choose to consume these goods while avoiding the cost of preserving them. I argue that on the contrary private and public goods are so interrelated that cooperative coalition behavior over their generation is possible. The problem of the maintenance of such cooperative coalitions, via an internal bargaining process, is considered. It is suggested that local optimizing behavior by a coalition may bring about qualitative change in the system, to the extent that the stability of the coalition is destroyed. Such a catastrophic change could lead to entirely new patterns of cooperation, and hence to qualitatively different development paths for the economic and ecological system.An earlier version of this paper, entitled The Logic of Collective Action and the Law of the Commons, was presented at the XIX International Meeting of the Institute of Management Science, on Management Science and the Quality of Life, Houston, Texas, April, 1972.     

The contribution of muscle properties in the control of explosive movements

Explosive movements such as throwing, kicking, and jumping are characterized by high velocity and short movement time. Due to the fact that latencies of neural feedback loops are long in comparison to movement times, correction of deviations cannot be achieved on the basis of neural feedback. In other words, the control signals must be largely preprogrammed. Furthermore, in many explosive movements the skeletal system is mechanically analogous to an inverted pendulum; in such a system, disturbances tend to be amplified as time proceeds. It is difficult to understand how an inverted-pendulum-like system can be controlled on the basis of some form of open loop control (albeit during a finite period of time only). To investigate if actuator properties, specifically the force-length-velocity relationship of muscle, reduce the control problem associated with explosive movement tasks such as human vertical jumping, a direct dynamics modeling and simulation approach was adopted. In order to identify the role of muscle properties, two types of open loop control signals were applied: STIM(t), representing the stimulation of muscles, and MOM(t), representing net joint moments. In case of STIM control, muscle properties influence the joint moments exerted on the skeleton; in case of MOM control, these moments are directly prescribed. By applying perturbations and comparing the deviations from a reference movement for both types of control, the reduction of the effect of disturbances due to muscle properties was calculated. It was found that the system is very sensitive to perturbations in case of MOM control; the sensitivity to perturbations is markedly less in case of STIM control. It was concluded that muscle properties constitute a peripheral feedback system that has the advantage of zero time delay. This feedback system reduces the effect of perturbations during human vertical jumping to such a degree that when perturbations are not too large, the task may be performed successfully without any adaptation of the muscle stimulation pattern.     

The control of walking movements in the leg of the rock lobster

1. Experiments with rock lobsters walking on a treadmill were undertaken to obtain information upon the system controlling the movement of the legs. Results show that the position of the leg is an important parameter affecting the cyclic movement of the walking leg. Stepping can be interrupted when the geometrical conditions for terminating either a return stroke or a power stroke are not fullfilled. 2. The mean value of anterior and posterior extreme positions (AEP and PEP respectively) of the walking legs do not depend on the walking speed (Fig. 1). 3. When one leg is isolated from the other walking legs by placing it on a platform the AEPs and PEPs of the other legs show a broader distribution compared to controls (Figs. 2 and 3). 4. Force measurements (Fig. 4) are in agreement with the hypothesis that the movement of the leg is controlled by a position servomechanism. 5. When one leg stands on a stationary force transducer this leg develops forces which oscillate with the step rhythm of the other legs (Fig. 5). 6. A posteriorly directed influence is found, by which the return stroke of a leg can be started when the anterior leg performs a backward directed movement. 7. Results are compared with those obtained from stick insects. The systems controlling the movement of the individual leg are similar in both, lobster and stick insect but the influences between the legs seem to be considerably different.     

Cortical control of reaching movements

Recent studies provide further support for the hypothesis that spatial representations of limb position, target locations, and potential motor actions are expressed in the neuronal activity in parietal cortex. In contrast, precentral cortical activity more strongly expresses processes involved in the selection and execution of motor actions. As a general conceptual framework, these processes may be interpreted in terms of such formalisms as sensorimotor transformation and ‘internal models’.     

Various characteristics of the control of cyclic movements

It was shown by studying control forces and phasic trajectories during oscillation of human forearm, locomotion and rocking of the body on the support that there was an image of accomplished movement in the central nervous system. This image seems to be realized by linear connected displacements of the muscle tension level and threshold of tonic stretch reflex. During the control process, velocity of the threshold and tension level is similarly transformed to that of the body part. The piece constant similarity coefficient is regulated centrally. The main result of such control is moving of the body along energy optimal trajectories.     

A theory on the control of arbitrary movements

A theory dealing with the control of human, arbitrary movements is proposed. A schema is set up to suggest how the relevant information flows and what kind of operations affect it. A number of successive steps are distinguished in the production of a movement. It is assumed that the intended movement is carried out in the imagination, and that this imaginary movement is composed of a spatial trajectory and an intensity course, which are considered to be independent features of the intended movement. The spatial trajectory will be encoded in a special coding, which is related to the lengths of the muscles that effect the movement. From this special coding of the intended movement static and dynamic control signals can be derived. Because afferent and efferent signals are encoded in the same way in this schema, the evaluation and correction of the performed movement is quite simple. The higher levels in the control schema may function in an abstract way, i.e. the signals at these levels are barely concerned with details of the peripheral motor system. This abstract functioning of the higher levels is based on the numerous feedback mechanisms involved at all levels of control and in the peripheral motor system. Nevertheless, it is possible to incorporate specific peripheral properties in the generation of the control signals. The assumptions in this theory will be discussed and aspects of the proposed control schema will be compared with general control principles.