David J. Wollkind

Metastability in a temperature-dependent model system for predator-prey mite outbreak interactions on fruit trees

Abstract The nonlinear behavior of particular Kolmogorov-type exploitation differential equation system assembled by May (1973 , Stability and Complexity in Model Ecosystems , Princeton University Press) from predator and prey components developed by Leslie (1948 , Biometrica 35 , 213–245) and Holling (1973 , Mem. Entomol. Soc. Can. 45 , 1–60), respectively, is re-examined by means of the numerical bifurcation code AUTO 86 with model parameters chosen appropriately for a temperature dependent mite interaction on fruit trees. The most significant result of this analysis is that, in addition to the temperature ranges over which the single community equilibrium point of the system is either globally stable or gives rise to a globally stable limit cycle, there can also exist a range wherein multiple stable states occur. These stable states consist of a focus (spiral point) and a limit cycle, separated from each other in the phase plane by an unstable limit cycle. The ecological implications of such metastability, hysteresis and threshold behavior for the occurrence of outbreaks, the persistence of oscillations, the resiliency of the system and the biological control of mite populations are discussed. It is further suggested that a model of this sort which possesses a single community equilibrium point may be more useful for representing outbreak phenomena, especially in the presence of oscillations, than the non-Kolmogorov predator-prey systems possessing three community equilibrium points, two of which are stable and the other a saddle point, traditionally employed for this purpose.

Temperature-dependent predator-prey mite ecosystem on apple tree foliage

SummaryThe biological control of the McDaniel spider mite, which is a pest on apple tree foliage, by a predacious mite species, which feeds upon it, is represented by a continuous-discrete time hybrid model incorporating temperature effects explicityland mite metamorphosis implicitly. The results of this model can be shown to provide good agreement when compared with relevant field data.

Modeling of bone formation and resorption mediated by parathyroid hormone: response to estrogen/PTH therapy

Bone, a major reservoir of body calcium, is under the hormonal control of the parathyroid hormone (PTH). Several aspects of its growth, turnover, and mechanism, occur in the absence of gonadal hormones. Sex steroids such as estrogen, nonetheless, play an important role in bone physiology, and are extremely essential to maintain bone balance in adults. In order to provide a basis for understanding the underlying mechanisms of bone remodeling as it is mediated by PTH, we propose here a mathematical model of the process. The nonlinear system model is then utilized to study the temporal effect of PTH as well as the action of estrogen replacement therapy on bone turnover. Analysis of the model is done on the assumption, supported by reported clinical evidence, that the process is characterized by highly diversified dynamics, which warrants the use of singular perturbation arguments. The model is shown to exhibit limit cycle behavior, which can develop into chaotic dynamics for certain ranges of the systems parametric values. Effects of estrogen and PTH administrations are then investigated by extending on the core model. Analysis of the model seems to indicate that the paradoxical observation that intermittent PTH administration causes net bone deposition while continuous administration causes net bone loss, and certain other reported phenomena may be attributed to the highly diversified dynamics which characterizes this nonlinear remodeling process.

Metastability of forest ecosystems infested by bark beetles

A simple two-species differential equation model is used to investigate the intrinsic metastability of forest ecosystems subjected to bark beetle infestations. We demonstrate that only one globally stable node or limit cycle is likely under biologically plausible conditions, but that, in the former case, this equilibrium is very sensitive to external perturbation.

A theoretical investigation of the development of interfacial cells during the solidification of a dilute binary alloy: Comparison with the experiments of Morris and Winegard

Abstract An investigation is made of the development of hexagonal cells on an initially planar interface of solid growing uniformly into the liquid phase during the unidirectional solidification of a dilute binary alloy. The model employed postulates three-dimensional diffusion of solute and heat under the simplifying assumptions that there is no convection in the liquid phase, no solute diffusion in the solid phase, both these phases are infinite in extent, latent heat is negligible, and the thermal fields satisfy Laplaces equation in each phase. A nonlinear stability analysis is performed analogous to that originally developed for the study of Benard convection cells. The main results of this analysis are that, for a fixed value of the liquid temperature gradient, the structure of the planar interface evolves from nodes (circular depressions) to bands (elongated cells) to dome-shaped hexagonal cells as the solidification speed is increased. These results, which can only be obtained provided the variation of interfacial surface free energy with solute concentration is taken into account, are in excellent agreement with the experimental observations of Morris and Winegard [J. Inst. Metals 97 (1969) 220; J. Crystal Growth 5 (1969) 361] in regard to interface morphologies and solute segregation under their solidification conditions.

CHEMICAL TURING PATTERN FORMATION ANALYSES: COMPARISON OF THEORY WITH EXPERIMENT ∗

The development of one- and two-dimensional Turing patterns characteristic of the chlorite-iodide-malonic acid/indicator reaction occurring in an open gel continuously fed unstirred reactor is investigated by means of various weakly nonlinear stability analyses applied to the appropriately scaled governing chlorine dioxide-iodine-malonic acid/indicator reaction-diffusion model system. Then the theoretical predictions deduced from these pattern formation studies are compared with experimental evidence relevant to the diffusive instabilities under examination. The latter consist of stripes, rhombic arrays of rectangles, and hexagonal arrays of spots or nets. Here, starch (for the case of a polyacrylamide gel) or the gel itself (for a polyvinyl alcohol gel) serves as the Turing pattern indicator. The main purpose of these analyses is to explain more fully thetransition to such stationary symmetry-breaking structures when the malonic acid reservoir concentration is decreased.

Interfacial patterns during plane front alloy solidification

Abstract An investigation is made of the development of hexagonal cells on an initially planar interface of solid growing uniformly into the liquid phase during the unidirectional solidification of a dilute binary alloy. The model employed postulates three-dimensional diffusion of solute and heat under the simplifying assumptions that there is no convection in the liquid phase, no solute diffusion in the solid phase, both these phases are infinite in extent, latent heat is negligible, and the thermal fields satisfy Laplaces equation in each phase. A nonlinear stability analysis is performed analogous to that originally developed for the study of Benard convection cells. The main results of this analysis are that, for a fixed value of the liquid temperature gradient, the structure of the planar interface evolves from nodes (circular depressions) to bands (elongated cells) to dome-shaped hexagonal cells as the solidification speed is increased. These results, which can only be obtained provided the variation of interfacial surface free energy with solute concentration is taken into account, are in excellent agreement with the experimental observations of Morris and Winegard [8, 9] in regard to interface morphologies and solute segregation under their solidification conditions.

Age structure in predator-prey systems. I. A general model and a specific example

Abstract To study the effects of age structure in predator-prey systems, a general, analytically tractable model is formulated and solved. We demonstrate the usefulness of the model in a study of a specific system of two mites. We show that to maintain stable equilibrium between the herbaceous (pest) mite and the predacious mite, the nonintuitive strategy of reducing the growth rate of the predator may be necessary. The modelling technique allows a determination of the magnitude of the effect of age structure on stability.

The effect of latent heat on weakly non-linear morphological stability

Abstract During directional solidification of a dilute binary alloy, the release of latent heat plays an important role in the determination of the weakly non-linear morphological stability of the solid-liquid interface. The non-linear stability of an initial planar interface is examined by using a Stuart-Watson type of approach, according to which the planar interface is subject to a two-dimensional perturbation that is predicted to be marginally stable by linear theory but no longer has an infinitesimal amplitude. In previous work of this nature, the latent heat of fusion, which appears explicitly in the equation describing the local balance of energy across the solid-liquid interface, was neglected. Allowing for a finite latent heat results in positive values of the Landau coefficient, and hence two-dimensional bands of small stable amplitude, for a much enhanced range of growth conditions, especially at low solute concentrations. We also find that the magnitude of n , the ratio of the thermal conductivities of the solid and fluid, can either augment or diminish the latent heat effect. For values of n less than unity there are values of the distribution coefficient for which no regions of weakly non-linear instability exist, while n greater than unity always admits the possibility of weakly non-linear instability. In general, it appears that two-dimensional bands with small stable amplitudes are favoured by a steep thermal gradient in the solid relative to that in the liquid.

Age structure in predator-prey systems. II. Functional response and stability and the paradox of enrichment

Abstract We employ the general model of predator-prey systems incorporating age structure in the predator, developed in the previous paper, to study the role of functional response in stability and the paradox of enrichment. The destabilizing effect of age structure leads to both qualitatively and quantitatively new results, including a lower bound to prey density for a stable equilibrium, a feature not present in models without age structure.