Spectral properties of a non-compact operator in ecology

Ecologists have recently used integral projection models (IPMs) to study fish and other animals which continue to grow throughout their lives. Such animals cannot shrink, since they have bony skeletons; a mathematical consequence of this is that the kernel of the integral projection operator T is unbounded, and the operator is not compact. To our knowledge, all theoretical work done on IPMs has assumed the operator is compact, and in particular has a bounded kernel. A priori, it is unclear whether these IPMs have an asymptotic growth rate λ, or a stable-stage distribution ψ. In the case of a compact operator, these quantities are its spectral radius and the associated eigenvector, respectively. Under biologically reasonable assumptions, we prove that the non-compact operators in these IPMs share some important traits with their compact counterparts: the operator T has a unique positive eigenvector ψ corresponding to its spectral radius λ, this λ is strictly greater than the supremum of the modulus of all other spectral values, and for any nonnegative initial population ϕ0, there is a c \u3e 0 such that T nϕ0/λn → c · ψ

Lack of Time-Delay Robustness for Stabilization of a Structural Acoustics Model

In this paper we consider a natural robustness question for a model for structural acoustics. This model, which has been of great interest in recent years, is represented by a wave equation in R^2 coupled to a Kelvin--Voigt beam; the coupling is natural physically, and is represented mathematically by highly unbounded operators. We assume that the observation consists of point evaluation of the beam position, the beam velocity, and the wave velocity. We are interested in the effect of arbitrarily small delays in the feedback loop on a controller that uses these observations. We show that it is not possible to construct a dynamic stabilizer of a very general form--including static feedback--such that the stabilization is robust with respect to delays in the feedback loop. In order to do this we need to carefully analyze the input-to-output map. Finally, we relate these results to already existing numerical results obtained for a Galerkin approximation of the system. AMS Subject Classifications. 93C20 , 93D09 , 93D15 , 93D25 , 35M1

Towards understanding factors influencing the benefit of diversity in predator communities for prey suppression

It is generally assumed that high biodiversity is key to sustaining critical ecosystem services, including prey suppression by natural predator guilds. Prey suppression is driven by complex interactions between members of predator and prey communities, as well as their shared environment. Because of this, empirical studies have found both positive and negative effects of high predator diversity on prey suppression. However, we lack an understanding of when these different prey suppression outcomes will occur. In this work, we use a mechanistic, trait-based model to unravel how intraguild interactions, species body mass, predator foraging area, and ambient temperature can combine to produce different levels of prey suppression. Surprisingly, we find that prey suppression is only improved by high biodiversity under a limited set of conditions. The most important factor in determining whether diversity improves prey sup- pression is the amount of overlap between predators’ foraging areas. The degree of overlap in foraging areas shapes species interactions, and as the overlap between species increases, we see decreasing benefits from species-rich communities. In contrast, diversity in body mass only improves prey suppression when there is significant variation in temperature

Promoting Undergraduate Research in Mathematics at the University of Nebraska – Lincoln

The Department of Mathematics at the University of Nebraska – Lincoln (UNL) has several programs which promote undergraduate research in a variety of ways. Two of these are summer programs which draw from a national applicant pool: The Nebraska REU in Applied Mathematics (Section 1) is a traditional NSF-funded REU site, and Nebraska IMMERSE (Section 2) offers a summer “bridge” program (with a research bent) for students about to start graduate school in mathematics. IMMERSE is a relatively new program, started in 2004 as part of the department’s Mentoring through Critical Transition Points (MCTP) grant from NSF. The MCTP grant also is now the primary source of funding for two conferences involving undergraduate research which the department launched in 1999: The Nebraska Conference for Undergraduate Women in Mathematics (NCUWM) (Section 3) and the Regional Workshop in the Mathematical Sciences (Section 4). The bulk of the program at NCUWM consists of talks by undergraduates on their own research, and, while the original goal of the Regional Workshop was to forge and maintain ties between faculty at smaller college and universities, it has recently been expanded to provide a forum for undergraduates to present their research. Finally, we offer several opportunities for our own undergraduates to do research: the MCTP Undergraduate Scholars program (Section 5), the Research for Undergraduates in Theoretical Ecology (RUTE) program (Section 6), the Undergraduate Creative Activities and Research Experiences (UCARE) program (Section 7), and two upper-level undergraduate courses which aim to give students a taste of mathematics research (Section 8). This article provides a brief overview of each of these programs; more details can be found online at http://www.math.unl.edu

STABILIZATION BY ADAPTIVE FEEDBACK CONTROL FOR POSITIVE DIFFERENCE EQUATIONS WITH APPLICATIONS IN PEST MANAGEMENT

An adaptive feedback control scheme is proposed for stabilizing a class of forced nonlinear positive difference equations. The adaptive scheme is based on so-called high-gain adaptive controllers and contains substantial robustness with respect to model uncertainty as well as with respect to persistent forcing signals, including measurement errors. Our results take advantage of the underlying positive systems structure and ideas from input-to-state stability from nonlinear control theory. Our motivating application is to pest or weed control, and in this context the present work substantially strengthens previous work by the authors. The theory is illustrated with examples

Low‐gain integral control for a class of discrete‐time Lur'e systems with applications to sampled‐data control

We study low-gain (P)roportional (I)ntegral control of multivariate discrete-time, forced Lur’e systems to solve the output-tracking problem for constant reference signals. We formulate an incremental sector condition which is sufficient for a usual linear low-gain PI controller to achieve exponential disturbance-to-state and disturbance-to-tracking-error stability in closed-loop, for all sufficiently small integrator gains. Output tracking is achieved in the absence of exogenous disturbance (noise) terms. Our line of argument invokes a recent circle criterion for exponential incremental input-to-state stability. The discrete-time theory facilitates a similar result for a continuous-time forced Lur’e system in feedback with sampled-data low-gain integral control. The theory is illustrated by two examples

Modeling the evolution of herbicide resistance in weed species with a complex life cycle

A growing number of weed species have evolved resistance to herbicides in recent years, which causes an immense financial burden to farmers. An increasingly popular method of weed control is the adoption of crops that are resistant to specific herbicides, which allows farmers to apply the herbicide during the growing season without harming the crop. If such crops are planted in the presence of closely related weed species, it is possible that resistance genes could transfer from the crop species to feral populations of the wild species via gene flow and become stably introgressed under ongoing selective pressure by the herbicide. We use a density-dependent matrix model to evaluate the effect of planting such crops on the evolution of herbicide resistance under a range of management scenarios. Our model expands on previous simulation studies by considering weed species with a more complex life cycle (perennial, rhizomatous weed species), studying the effect of environmental variation in herbicide effectiveness, and evaluating the role of common simplifying genetic assumptions on resistance evolution. Our model predictions are qualitatively similar to previous modeling studies using species with a simpler life cycle, which is, crop rotation in combination with rotation of herbicide site of action effectively controls weed populations and slows the evolution of herbicide resistance. We find that ignoring the effect of environmental variation can lead to an over- or under-prediction of the speed of resistance evolution. The effect of environmental variation in herbicide effectiveness depends on the resistance allele frequency in the weed population at the beginning of the simulation. Finally, we find that degree of dominance and ploidy level have a much larger effect on the predicted speed of resistance evolution compared to the rate of gene flow. Includes 5 supplemental file

Global Attracting Equilibria for Coupled Systems with Ceiling Density Dependence

In this paper, we present a system of two difference equations modeling the dynamics of a coupled population with two patches. Each patch can house only a limited number of individuals (called a carrying capacity) because resources like food and breeding sites are limited in each patch. We assume that the population in each patch is governed by a linear model until reaching a carrying capacity in each patch, resulting in map which is nonlinear and not sublinear. We analyze the global attractors of this model

Stabilisation by adaptive feedback control for positive difference equations with applications in pest management

An adaptive feedback control scheme is proposed for stabilising a class of forced nonlinear positive difference equations. The adaptive scheme is based on so-called high-gain adaptive controllers, and contains substantial robustness with respect to model uncertainty as well as with respect to persistent forcing signals, including measurement errors. Our results take advantage of the underlying positive systems structure and ideas from input-to-state stability from nonlinear control theory. Our motivating application is to pest or weed control, and in this context the present work substantially strengthens previous work by the authors. The theory is illustrated with examples