Mathematical Biosciences & Engineering
2007 , Volume 4 , Issue 1
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This Special Issue of Mathematical Biosciences and Engineering (MBE) contains a few of the papers presented at a special session in Mathematical Biology at the Regional Meeting of the American Mathematical Society, held in Lincoln, Nebraska, on October 21-23, 2005. The AMS meeting was preceded by a 2-day workshop on educational issues in mathematical biology, organized by Glenn Ledder. Many of the visitors attended both meetings, and therefore the five-day period was an enriching, double-barreled, math biology experience for nearly everyone. Near the end of the session the Editor-in-Chief of MBE, Yang Kuang, asked if we would be willing to collect together some of the papers from the special session for an issue of MBE.
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Much ecological research involves identifying connections between abiotic forcing and population densities or distributions. We present theory that describes this relationship for populations in media with strong unidirectional flow (e.g., aquatic organisms in streams and rivers). Typically, equilibrium populations change in very different ways in response to changes in demographic versus dispersal rates and to changes over local versus larger spatial scales. For populations in a mildly heterogeneous environment, there is a population ''response length'' that characterizes the distance downstream over which the impact of a point source perturbation is felt. The response length is also an important parameter for characterizing the response to non-point source disturbances at different spatial scales. In the absence of density dependence, the response length is close to the mean distance traveled by an organism in its lifetime. Density-dependent demographic rates are likely to increase the response length from this default value, and density-dependent dispersal will reduce it. Indirect density dependence, mediated by predation, may also change the response length, the direction of change depending on the strength of the prey's tendency to flee the predator.
We consider the mathematical model originally created by Ludwig, Jones, and Holling to model the infestation of spruce forests in New Brunswick by the spruce budworm. With biologically plausible parameter values, the dimensionless version of the model contains small parameters derived from the time scales of the state variables and smaller parameters derived from the relative importance of different population change mechanisms. The small time-scale parameters introduce a singular perturbation structure to solutions, with one variable changing on a slow time scale and two changing on a fast time scale. The smaller process-scale parameters allow for the existence of equilibria at vastly different orders of magnitude. These changes in scale of the state variables result in fast dynamics not associated with the time scales. For any given set of parameters, the observed dynamics is a mixture of time-scale effects with process-scale effects. We identify and analyze the different scenarios that can occur and indicate the relevant regions in the parameter space corresponding to each.
Stoichiometry-based models brought into sharp focus the importance of the nutritional quality of plant for herbivore-plant dynamics. Plant quality can dramatically affect the growth rate of the herbivores and may even lead to its extinction. These results stem from models continuous in time, which raises the question of how robust they are to time discretization. Discrete time can be more appropriate for herbivores with non-overlapping generations, annual plants, and experimental data collected periodically. We analyze a continuous stoichiometric plant-herbivore model that is mechanistically formulated in . We then introduce its discrete analog and compare the dynamics of the continuous and discrete models. This discrete model includes the discrete LKE model (Loladze, Kuang and Elser (2000)) as a limiting case.
We model the development of an individual insect, a grasshopper, through its nymphal period as a function of a trade-off between prey vigilance and nutrient intake in a changing environment. Both temperature and food quality may be variable. We scale up to the population level using natural mortality and a predation risk that is mass, vigilance, and temperature dependent. Simulations reveal the sensitivity of both survivorship and development time to risk and nutrient intake, including food quality and temperature variations. The model quantifies the crucial role of temperature in trophic interactions and development, which is an important issue in assessing the effects of global climate change on complex environmental interactions.
We model the effects of both stochastic and deterministic temperature variations on arthropod predator-prey systems. Specifically, we study the stochastic dynamics of arthropod predator-prey interactions under a varying temperature regime, and we develop an individual model of a prey under pressure from a predator, with vigilance (or foraging effort), search rates, attack rates, and other predation parameters dependent on daily temperature variations. Simulations suggest that an increase in the daily average temperature may benefit both predator and prey. Furthermore, simulations show that anti-predator behavior may indeed decrease predation but at the expense of reduced prey survivorship because of a greater increase in other types of mortality.
Many natural population growths and interactions are affected by seasonal changes, suggesting that these natural population dynamics should be modeled by nonautonomous differential equations instead of autonomous differential equations. Through a series of carefully derived models of the well documented high-amplitude, large-period fluctuations of lemming populations, we argue that when appropriately formulated, autonomous differential equations may capture much of the desirable rich dynamics, such as the existence of a periodic solution with period and amplitude close to that of approximately periodic solutions produced by the more natural but mathematically daunting nonautonomous models. We start this series of models from the Barrow model, a well formulated model for the dynamics of food-lemming interaction at Point Barrow (Alaska, USA) with sufficient experimental data. Our work suggests that an autonomous system can indeed be a good approximation to the moss-lemming dynamics at Point Barrow. This, together with our bifurcation analysis, indicates that neither seasonal factors (expressed by time-dependent moss growth rate and lemming death rate in the Barrow model) nor the moss growth rate and lemming death rate are the main culprits of the observed multi-year lemming cycles. We suspect that the main culprits may include high lemming predation rate, high lemming birth rate, and low lemming self-limitation rate.
Invasive plant populations typically consist of a large (main) focus and several smaller outlier populations. Management of the spread of invasives requires repeated control measures, constrained by limited funding and effort. Posing this as a control problem, we investigate whether it is best to apply control to the main focus, the outlier populations, or some combination of these. We first formulate and solve a discrete-time optimal control problem to determine where control is best applied over a finite time horizon. However, if limited funds are available for control, this optimal solution may not be feasible. In this case, we add an additional constraint to account for the fixed budget and solve the new optimality system. Our results have a variety of practical implications for invasive species management.
We consider the stability of single-front stationary solutions to a spatially discrete reaction-diffusion equation which models front propagation in a nerve axon. The solution's stability depends on the coupling parameter, changing from stable to unstable and from unstable to stable at a countably infinite number of values of this diffusion coefficient.
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