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We use a mixed time model to study the dynamics of a system consisting of two consumers that reproduce only in annual birth pulses, possibly at different times, with interaction limited to competition for a resource that reproduces continuously. Ecological theory predicts competitive exclusion; this expectation is met under most circumstances, the winner being the species with the greater 'power', defined as the time average consumer level at the fixed point. Instability of that fixed point for the stronger competitor slightly weakens its domination, so that a resident species with an unstable fixed point can sometimes be invaded by a slightly weaker species, leading ultimately to coexistence. Differences in birth pulse times can lead to qualitatively different long-term coexistence behaviour, including cycles of different lengths or chaos. We also determine conditions under which the timing of an annual pulse of a toxin can change the balance of power.

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.

Dynamic observers are considered in the context of structured-population modeling and management. Roughly, observers combine a known measured variable of some process with a model of that process to asymptotically reconstruct the unknown state variable of the model. We investigate the potential use of observers for reconstructing population distributions described by density-independent (linear) models and a class of density-dependent (nonlinear) models. In both the density-dependent and -independent cases, we show, in several ecologically reasonable circumstances, that there is a natural, optimal construction of these observers. Further, we describe the robustness these observers exhibit with respect to disturbances and uncertainty in measurement.

Assessing and managing risks of anthropogenic activities to ecological systems is necessary to ensure sustained delivery of ecosystem services for future generations. Ecological models provide a means of quantitatively linking measured risk assessment endpoints with protection goals, by integrating potential chemical effects with species life history, ecological interactions, environmental drivers and other potential stressors. Here we demonstrate how an ecosystem modeling approach can be used to quantify insecticide-induced impacts on ecosystem services provided by a lake from toxicity data for organism-level endpoints. We used a publicly available aquatic ecosystem model AQUATOX that integrates environmental fate of chemicals and their impacts on food webs in aquatic environments. By simulating a range of exposure patterns, we illustrated how exposure to a hypothetical insecticide could affect aquatic species populations (e.g., recreational fish abundance) and environmental properties (e.g., water clarity) that would in turn affect delivery of ecosystem services. Different results were observed for different species of fish, thus the decision to manage the use of the insecticide for ecosystem services derived by anglers depends upon the favored species of fish. In our hypothetical shallow reservoir, water clarity was mostly driven by changes in food web dynamics, specifically the presence of zooplankton. In contrast to the complex response by fishing value, water clarity increased with reduced insecticide use, which produced a monotonic increase in value by waders and swimmers. Our study clearly showed the importance of considering nonlinear ecosystem feedbacks where the presence of insecticide changed the modeled food-web dynamics in unexpected ways. Our study highlights one of the main advantages of using ecological models for risk assessment, namely the ability to generalize to meaningful levels of organization and to facilitate quantitative comparisons among alternative scenarios and associated trade-offs among them while explicitly accounting for different groups of beneficiaries.

We present a length-structured matrix model for fish populations in which the probability that a fish grows into the next length class is a decreasing nonlinear function of the total biomass of the population. We present mathematical results classifying the dynamics that this density-dependent model predicts. We illustrate these results with numerical simulations for an invasive white perch population and show how the mathematical results can be used to predict the persistence and/or boundedness of the population as well as an equilibrium structure that is dominated by small fish. We illustrate the results with management recommendations for an invasive white perch population.

We demonstrate how mechanistic modeling can be used to predict whether and how biological responses to chemicals at (sub)organismal levels in model species (i.e., what we typically measure) translate into impacts on ecosystem service delivery (i.e., what we care about). We consider a hypothetical case study of two species of trout, brown trout (Salmo trutta; BT) and greenback cutthroat trout (Oncorhynchus clarkii stomias; GCT). These hypothetical populations live in a high-altitude river system and are exposed to human-derived estrogen (17α‑ethinyl estradiol, EE2), which is the bioactive estrogen in many contraceptives. We use the individual-based model inSTREAM to explore how seasonally varying concentrations of EE2 could influence male spawning and sperm quality. Resulting impacts on trout recruitment and the consequences of such for anglers and for the continued viability of populations of GCT (the state fish of Colorado) are explored. inSTREAM incorporates seasonally varying river flow and temperature, fishing pressure, the influence of EE2 on species-specific demography, and inter-specific competition. The model facilitates quantitative exploration of the relative importance of endocrine disruption and inter-species competition on trout population dynamics. Simulations predicted constant EE2 loading to have more impacts on GCT than BT. However, increasing removal of BT by anglers can enhance the persistence of GCT and offset some of the negative effects of EE2. We demonstrate how models that quantitatively link impacts of chemicals and other stressors on individual survival, growth, and reproduction to consequences for populations and ecosystem service delivery, can be coupled with ecosystem service valuation. The approach facilitates interpretation of toxicity data in an ecological context and gives beneficiaries of ecosystem services a more explicit role in management decisions. Although challenges remain, this type of approach may be particularly helpful for site-specific risk assessments and those in which tradeoffs and synergies among ecosystem services need to be considered.

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 $$\lambda $$ , or a stable-stage distribution $$\psi $$ . 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 $$\psi $$ corresponding to its spectral radius $$\lambda $$ , this $$\lambda $$ is strictly greater than the supremum of the modulus of all other spectral values, and for any nonnegative initial population $$\varphi _0$$ , there is a $$c>0$$ such that $$T^n\varphi _0/\lambda ^n \rightarrow c \cdot \psi $$ .

We discuss the use of optimal control theory to determine the most cost-effective management strategies for insect pests. We use a stage-structured linear population projection model where the modeled control action increases the mortality in one of the stage-classes. We illustrate the method by using a published model for the root weevil Diaprepes abbreviatus, an invasive insect species having a substantial negative impact on citrus trees in regions such as Florida and California in the United States. Here control corresponds to the application of inundative biological control agents (entomopathogenic nematodes as biopesticides) which increases the mortality of the larval stage. Our approach determines levels and timing of control to minimize the economic loss caused by D. abbreviatus. We use two numerical methods to approximate the optimal control, and compare their effectiveness.

a b s t r a c t When vital rates depend on population structure (e.g., relative frequencies of males or females), an important question is how the long-term population growth rate λ responds to changes in rates. For instance, availability of mates may depend on the sex ratio of the population and hence reproductive rates could be frequency-dependent. In such cases change in any vital rate alters the structure, which in turn, affect frequency-dependent rates. We show that the elasticity of λ to a rate is the sum of (i) the effect of the linear change in the rate and (ii) the effect of nonlinear changes in frequency-dependent rates. The first component is always positive and is the classical elasticity in density-independent models obtained directly from the population projection matrix. The second component can be positive or negative and is absent in density-independent models. We explicitly express each component of the elasticity as a function of vital rates, eigenvalues and eigenvectors of the population projection matrix. We apply this result to a two-sex model, where male and female fertilities depend on adult sex ratio α (ratio of females to males) and the mating system (e.g., polygyny) through a harmonic mating function. We show that the nonlinear component of elasticity to a survival rate is negligible only when the average number of mates (per male) is close to α. In a strictly monogamous species, elasticity to female survival is larger than elasticity to male survival when α < 1 (less females). In a polygynous species, elasticity to female survival can be larger than that of male survival even when sex ratio is female biased. Our results show how demography and mating system together determine the response to selection on sex-specific vital rates.

We present a novel management methodology for restocking a declining population. The strategy uses integral control, a concept ubiquitous in control the- ory which has not been applied to population dynamics. Integral control is based on dynamic feedback—using measurements of the population to inform management strategies and is robust to model uncertainty, an important consideration for ecologi- cal models. We demonstrate from first principles why such an approach to population management is suitable via theory and examples.

Seed banks are critically important for disturbance specialist plants because seeds of these species germinate only in disturbed soil. Disturbance and seed depth affect the survival and germination probability of seeds in the seed bank, which in turn affect population dynamics. We develop a density-dependent stochastic integral projection model to evaluate the effect of stochastic soil disturbances on plant population dynamics with an emphasis on mimicking how disturbances vertically redistribute seeds within the seed bank. We perform a simulation analysis of the effect of the frequency and mean depth of disturbances on the population's quasi-extinction probability, as well as the long-term mean and variance of the total density of seeds in the seed bank. We show that increasing the frequency of disturbances increases the long-term viability of the population, but the relationship between the mean depth of disturbance and the long-term viability of the population are not necessarily monotonic for all parameter combinations. Specifically, an increase in the probability of disturbance increases the long-term viability of the total seed bank population. However, if the probability of disturbance is too low, a shallower mean depth of disturbance can increase long-term viability, a relationship that switches as the probability of disturbance increases. However, a shallow disturbance depth is beneficial only in scenarios with low survival in the seed bank.

Pest management is vitally important for modern arable farming, but models for pest species are often highly uncertain. In the context of pest management, control actions are naturally described by a nonlinear feedback that is generally unknown, which thus motivates a robust control approach. We argue that adaptive approaches are well suited for the management of pests and propose a simple high-gain adaptive tuning mechanism so that the nonlinear feedback achieves exponential stabilization. Furthermore, a switched adaptive controller is proposed, cycling through a set of given control actions, that also achieves global asymptotic stability. Such a model in practice allows for the possibility of rotating between different courses of management action. In developing our control strategies we appeal to comparison and monotonicity arguments. Interestingly, componentwise nonnegativity of the model, combined with an irreducibility assumption, implies that several issues typically associated with high-gain adaptive controllers do not arise and usual high-gain structural assumptions are not required.

We use feedback control methods to prove a trichotomy of stability for nonlinear (density dependent) discrete-time population dynamics defined on a natural state space of non-negative vectors. Specifically, using comparison results and small gain techniques we obtain a computable formula for parameter ranges when one of the following must hold: there is a positive, globally asymptotically stable equilibrium; zero is globally asymptotically stable or all solutions with non-zero initial conditions diverge. We apply our results to a model for Chinook Salmon.

Tree height and crown allometries reflect adaptations for resource acquisition and structural stability, as well as plas tic responses to a heterogeneous environment. While both light and soil resources limit growth and influence compet itive responses in tropical forests, the effects of belowground resources on allometries are less understood, especially within the understory. To characterize outcomes of tree competition along an edaphic resource gradient, we quantified variation in height and crown allometries of six Bornean tree species from contrasting regeneration niches (lightdemanding vs. shade-tolerant) on two soil habitats (clay-fine loam and sandy loam) within a 52-ha forest dynamics plot. Using empiricallyfit allometric parameters and diameter growth rates from plot census data, we modeled tree height and crown area growth over the projected life span of each species. Based on resource competition theory, we hypothesized that tree species specializing on and populations of generalist species growing on the relatively moister, more fertile clay-fine loam soil habitat would have faster height and crown growth rates, compared to those on the sandy loam habitat, regardless of regeneration niche. Among soil specialists and within generalists of both genera, trees growing on clay-fine loam had taller stems and larger crowns at a given age and faster height and crown area growth rates at most sizes than trees on sandy loam. Differences in height and crown growth were driven by the faster diameter growth rates of trees on clay-fine loam, not by differences in height- and crown-diameter allometries, as trees on sandy loam were significantly taller at a given diameter, and differences in crown allometry were not consistent across soil habitats. Characterizing the height and crown growth responses of trees along resource gradients provides insight into the mechanisms that maintain diversity in tropical forests. Our results point to the importance of adaptive and plastic responses to both above- and belowground resource availability in determining the allometric growth of trees and suggest that this diversity of responses may contribute tree species coexistence through competition-based trade-off mechanisms and variation in growth among individuals. Additionally, as the importance estimating natural carbon sequestration increases with the escalating effects of anthropogenic climate change, differences in tree growth and architecture across soil habitats also have implications for the approximation of forest carbon storage on heterogeneous tropical soils.

Modelers have to make choices about which functional forms to use for representing model components, such as the relationship between the state of individuals and their vital rates. Even though these choices significantly influence model predictions, this type of structural uncertainty has been largely ignored in theoretical ecology. In this paper, we use integral projection models (IPMs) for Platte thistle as a case study to illustrate that the choice of functional form characterizing density dependence in seedling recruitment has important implications for predicting transient dynamics (short-term population dynamics following disturbances). In one case, the seedling recruitment function is modeled as a power function, and in the other case, we derive density dependence in seedling recruitment from biological first principles. We chose parameter values for the recruitment functions such that both IPMs predicted identical equilibrium population densities and both recruitment functions fit the empirical recruitment data sufficiently well. We find that the recovery from a transient attenuation, and the magnitude of transient amplification, can vary tremendously depending on which function is used to model density-dependent seedling recruitment. When we loosen the restriction of having identical equilibrium densities, model predictions not only differ in the short term but also in the long term. We derive some mathematical properties of the IPMs to explain why the short-term differences occur.

A single species is often modeled as a structured population. In a matrix projection model, individuals in the population are partitioned into a finite number of stage classes. For example, an inse...

In this paper we solve tracking and disturbance rejection problems for stable infinite-dimensional systems using a simple low-gain controller suggested by the internal model principle. For stable discrete-time systems, it is shown that the application of a low-gain controller (depending on only one gain parameter) leads to a stable closed-loop system which asymptotically tracks reference signals $r$ of the form $r(k)=\sum_{j=1}^N \lambda_j^k\,\text{{\mathfrakr r}}_j$, where $\text{{\mathfrakr r}}_j\in\mathbb{C}^p$ and $\lambda_j\in\mathbb{C}$ with $|\lambda_j|=1$ for $j=1,\ldots,N$. The closed-loop system also rejects disturbance signals which are asymptotically of this form. The discrete-time result is used to derive results on approximate tracking and disturbance rejection for a large class of infinite-dimensional sampled-data feedback systems, with reference signals which are finite sums of sinusoids, and disturbance signals which are asymptotic to finite sums of sinusoids. The results are given for both input-output systems and state-space systems.

Matrix population models are a common tool for evaluating different management strategies. In general, under deterministic analyses, management strategies are recommended that improve those matrix transitions that are most sensitive or elastic with respect to the asymptotic population growth rate, λ. These recommendations usually ignore the biological limit for these transitions. In this paper we use the endangered Serengeti cheetah (Acinonyx jubatus) as a case study to illustrate that ignoring biological limits leads to a recommendation that will not always achieve the desired goal of an asymptotic population growth rate, λ ≥ 1. We estimate the survivorships of adult cheetahs in captivity using cheetah studbook data, which is a conservative estimate of the biological limit for the adult survivorship of wild cheetahs. Our analysis suggests that adult survival sharply decreases after 8.5 years. In addition, captive cheetahs older than 18 years do not reproduce. We modify a previously published population projection matrix to include the effect of senescence on survival and fecundity. Our model suggests that increasing adult survivorship alone is not sufficient to reverse population decline. However, an additional small increase in newborn survival is likely to result in a viable cheetah population. We show these conclusions hold even in the presence of relatively large parameter perturbations.

We consider a linear single-input single-output system on a Hilbert space X, with infinitesimal generator A, bounded control element b, and bounded observation element c. We address the problem of finding the largest feedback invariant subspace of X that is in the space c ⊥ perpendicular to c. If b is not in c ⊥, we show this subspace is c ⊥. If b is in c ⊥, a number of situations may occur, depending on the relationship between b and c.

Sensitivity and elasticity analyzes have become central to the analysis of models in population biology and ecology. While much work has been done applying sensitivity and elasticity analysis to study density-independent (linear) matrix and integral projection models, little work has been done to study the sensitivity and elasticity of density-dependent models, especially integral projection models. In this paper we derive sensitivity and elasticity formulas for the equilibrium population n* of a structured population modeled by a Lur’e system, which consists of a linear system plus a nonlinearity modeling density-dependent fecundity. Sensitivity and elasticity formulas are easy to interpret ecologically, and we apply these formulas to published models for Chinook Salmon and Platte thistle (Cirsium canescens). In the C. canescens example we show that models with identical equilibrium populations can have sensitivities that are an order-of-magnitude apart, depending on the functional form for the nonlinearity.

We suppose that a continuous-time feedback is input-output stabilizing for an infinite-dimensional system. We address the question of whether the sampled-data controller obtained by applying idealized sample-and-hold to this continuous-time feedback is also input-output stabilizing if the sampling time is small enough. This question has been previously addressed for fairly general systems under various conditions. In this note, we restrict our attention to Riesz spectral systems, for which we generalize the existing results. Specifically, we give two relatively simple conditions which, combined, are sufficient for the sampled-data controller to be stabilizing. The first condition is a spectrum decomposition for the open-loop system generator, which by itself is necessary, but not sufficient, for the system to be stabilizable by sampled-data control. The second is a summability condition relating the real part of the spectrum of the generator and the expansion coefficients for the input and feedback operators

We consider well-posed linear infinite-dimensional systems, the outputs of which are sampled in a generalized sense using a suitable weighting function. Under certain natural assump- tions on the system, the weighting function, and the sampling period, we show that there exists a generalized hold function such that unity sampled-data feedback renders the closed-loop system exponentially stable (in the state-space sense) as well as L 2 -stable (in the input-output sense). To illustrate our main result, we describe an application to a structurally damped Euler-Bernoulli beam.

We consider two problems in boundary controllability of coupled wave/Kirchoff systems. Let Ω be a bounded region in R n , n ⩾2, with Lipschitz continuous boundary Γ . In the motivating structural acoustics application, Ω represents an acoustic cavity. Let Γ 0 be a flat subset of Γ which represents a flexible wall of the cavity. Let z denote the acoustic velocity potential, which satisfies a wave equation in Ω , and let v denote the displacement on Γ 0 , which satisfies a Kirchoff plate equation on Γ 0 . These equations are coupled via ∂z / ∂ν = v t on Γ 0 (where ν is the exterior unit normal to Γ 0 ), and the backpressure— z t appears in the Kirchoff equation. In the first problem, we consider a control u 0 in the Kirchoff equation on Γ 0 , and an additional control u 1 in the Neumann conditions on a subset Γ 1 of Γ , where Γ ⧹ Γ 1 satisfies geometric conditions. Using both controls, we obtain exact controllability of the wave and plate components in the natural state space. In the second problem we consider only the control u 0 . Without geometric conditions, exact controllability is not possible, but we show that for any initial data, we can steer the plate component exactly, and the wave component approximately.

In this paper we solve the tracking and disturbance rejection problem for infinite-dimensional linear systems, with reference and disturbance signals that are finite superpositions of sinusoids. We explore two approaches, both based on the internal model principle. In the first approach, we use a low gain controller, and here our results are a partial extension of results by Hamalainen and Pohjolainen. In their papers, the plant is required to have an exponentially stable transfer function in the Callier-Desoer algebra, while in this paper we only require the plant to be well-posed and exponentially stable. These conditions are sufficiently unrestrictive to be verifiable for many partial differential equations in more than one space variable. Our second approach concerns the case when the second component of the plant transfer function (from control input to tracking error) is positive. In this case, we identify a very simple stabilizing controller which is again an internal model, but which does not require low gain. We apply our results to two problems involving systems modeled by partial differential equations: the problem of rejecting external noise in a model for structure/acoustics interactions, and a similar problem for two coupled beams.