Jesse A. Logan

A model for diurnal variation in soil and air temperature

Abstract A model for predicting diurnal changes in soil and air temperatures given the maximum and minimum temperatures has been developed. The model uses a truncated sine wave to predict daytime temperature changes and an exponential function to predict nighttime temperatures. The model is based upon hourly soil and air temperatures for 1977 at a shortgrass prairie site and is parameterized for 150-cm and 10-cm air temperatures and for soil-surface and 10-cm soil temperatures. The absolute mean error for the model ranged from a maximum of 2.64°C for the 10-cm air temperature to a minimum of 1.20°C for the 10-cm soil temperature. The model was also parameterized for hourly air temperature data for Denver, Colorado. Comparison of the model with other models showed that it did a superior job of fitting the data with a smaller number of parameters.

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.

A unifying hypothesis of temperature effects on egg development and diapause of the migratory grasshopper, Melanoplus sanguinipes (Orthoptera: Acrididae)*

A conceptual model of diapause and egg development for Melanoplus sanguinipes (F) is presented. The processes regulating development are represented as non-linear functions of temperature. A diapause regulating process is hypothesized which is permitted to occur at any time during morphogenesis but must be completed before morphological development can progress beyond 80%. Under this hypothesis, the mean rates of the diapause regulating process and morphological development relative to each other determine whether or not morphogenesis is interrupted and the length of the resulting diapause. The model provides an excellent fit to an extensive data set collected for M. sanguinipes and can account for the observed occurrence of bivoltine southern populations, as well as of the more typical univoltine situation in the northern populations.

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.

Functional response, numerical response, and stability in arthropod predator-prey ecosystems involving age structure.

A general model of arthropod predator-prey systems incorporating age structure in the predator is employed to study the role of functional and numerical responses on stability and the paradox of enrichment. The destabilizing effect of age structure leads to both qualitatively and quantitatively new results for an environment which has an infinite prey carrying capacity, including a lower bound to prey density for a stable equilibrium, a feature not present in models without age structure. When applied to an environment with finite prey carrying capacity, the effect of age structure is to reinforce the arguments implicit to the paradox of enrichment originally developed for traditional models lacking age structure.

Simple order of prey preference technique for modelling the predator functional response

The predator functional response to several prey types and densities may be conceptualized as a multi-dimensional version of the one-dimensional Holling functional-response curves; however, this empirical approach requires inordinate amounts of data to develop and test. A simulation method of modelling this functional response is to consider the behavior of a predator faced with the choice of several prey types. In this model, when all prey are available the predator’s selection will depend on the absolute abundance of the most-preferred prey type, irrespective of the abundances of the less-preferred prey types. Consequently, the predator will consume only the most-preferred prey types while that type is available in sufficient numbers. When abundance of the most-preferred type declines below a certain level, the predator will begin to include in its diet the second-most-preferred prey type along with the most-preferred prey type. This order-of-preference technique holds up well when the model is compared to population data fromOligonychus pratensis (Acarina: Tetranychidae)/Neoseiulus fallacis (Acarina: Phytoseiidae), and is consistent with optimal foraging theory. Implementation is simple, and the data requirements are reduced to determining the predator’s order of preference and normalizing the nutritional values of the prey types to a single type.

Temperature-Mediated Stability of the Interaction Between Spider Mites and Predatory Mites in Orchards

The nonlinear behavior of the Holling-Tanner predatory-prey differential equation system, employed by R.M. May to illustrate the apparent robustness of Kolmogorov’s Theorem when applied to such exploitation systems, 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 there exists a temperature range wherein multiple stable states can occur, in direct violation of May’s interpretation of this system’s satisfaction of Kolmogorov’s Theorem: namely, that linear stability predictions have global consequences. In particular 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, all of which are associated with the single community equilibrium point of the system. 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.

Mathematical stability analysis for an age-structured population with finite life span

The system ρ 1 + ρ a = − λ L − a ρ P ( t ) = ∫ 0 L ρ ( a , t ) d a ρ ( 0 , t ) = ∫ 0 L β ( a , P ) ρ ( a , t ) d a is used to model population density, π, of age a individuals at time t. The maximum life span is allowed to depend on net population density L = L(P). Equilibrium, time independent solutions are shown to exist, provided L, P, λ and β are suitably related. Linearized stability analysis for constant L, shows βp