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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.

Natural resource managers and ecologists often desire an understanding of spatial dynamics such as migration, dispersion, and meta-population dynamics. Network-node models can capture these salient features. Additionally, the state-variable used with many species may be appropriately modeled as a continuous variable (e.g., length) and management activities sometimes can only target individuals of certain sizes. Integral projection models (IPMs) can capture this life history characteristic and allow for the examination of size-specific management. We combined an IPM with a network-node model to capture both of these salient features. We then demonstrated how this model could be used to understand and manage populations of invasive species focusing on grass carp as an example. Grass carp disrupt ecosystems outside of their native range and have spread around much of the world, including North America. The impacts of grass carp include adversely changing aquatic plant communities, which in turn affect a wide range of endpoints ranging from water quality to waterfowl recruitment. We specifically examined two theoretical systems using parameters from the literature. First, we modeled a lake with two tributaries and examined how modified sterile males could be used as a control tool. We found that modified sterile males may be a feasible control tool to limit population growth. Second, we modeled a series of river pools and examined how harvest and deterrents could be used to decrease the risk of expanding grass carp's range within a river system. Within this system, we also compared the impacts of size specific harvest and uniform harvest across all sizes. We found that targeting the largest, spawning populations may be more important than targeting the populations close to the invasion front for reducing the risk of spreading grass carp. We also demonstrate that size of harvested fish was important for controlling populations.

Invasive fish species disrupt ecosystems and cause economic damage. Several methods have been discussed to control populations of invasive fish including the release of YY-males. YY-males are fish that have 2 male chromosomes compared to a XY-male. When YY-males mate, they only produce male (XY) offspring. This decreases the female proportion of the population and can, in theory, eradicate local populations by biasing the sex-ratio. YY-males have been used as a population control tool for brook trout in montane streams and lakes in Idaho, USA. The YY-male control method has been discussed for grass carp in Lake Erie, North America. We developed and presented an integral projection model for grass carp to model the use of YY-males as a control method for populations in this lake. Using only the YY-male control method, we found that high levels of YY-males would need to be release annually to control the species. Specifically, these levels were the same order of magnitude as the baseline adult population (e.g., 1000 YY-males needed to be released annual for 20 years to control a baseline adult population of 2500 grass carp). These levels may not be reasonable or obtainable for fisheries managers given the impacts of YY-males on aquatic vegetation and other constraints of natural resource management.

Trend estimates are often used as part of environmental monitoring programs. These trends inform managers (e.g., are desired species increasing or undesired species decreasing?). Data collected from environmental monitoring programs is often aggregated (i.e., averaged), which confounds sampling and process variation. State-space models allow sampling variation and process variations to be separated. We used simulated time-series to compare linear trend estimations from three state-space models, a simple linear regression model, and an auto-regressive model. We also compared the performance of these five models to estimate trends from a long term monitoring program. We specifically estimated trends for two species of fish and four species of aquatic vegetation from the Upper Mississippi River system. We found that the simple linear regression had the best performance of all the given models because it was best able to recover parameters and had consistent numerical convergence. Conversely, the simple linear regression did the worst job estimating populations in a given year. The state-space models did not estimate trends well, but estimated population sizes best when the models converged. We found that a simple linear regression performed better than more complex autoregression and state-space models when used to analyze aggregated environmental monitoring data.

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.

Biological populations are strongly influenced by random variations in their environment, which are often autocorrelated in time. For disturbance specialist plant populations, the frequency and intensity of environmental stochasticity (via disturbances) can drive the qualitative nature of their population dynamics. In this article, we extended our earlier model to explore the effect of temporally autocorrelated disturbances on population persistence. In our earlier work, we only assumed disturbances were independent and identically distributed in time. We proved that the plant seed bank population converges in distribution, and we showed that the mean and variance in seed bank population size were both increasing functions of the autocorrelation coefficient for all parameter values considered, but the interplay between increasing population size and increasing variability caused interesting relationships between quasi-extinction probability and autocorrelation. For example, for populations with low seed survival, fecundity, and disturbance frequency, increasingly positive autocorrelated disturbances decreased quasi-extinction probability. Higher disturbance frequency coupled with low seed survival and fecundity caused a nonmontone relationship between autocorrelation and quasi-extinction, where increasingly positive autocorrelations eventually caused an increase in quasi-extinction probability. For higher seed survival, fecundity, and/or disturbance frequency, quasi-extinction probability was generally a monotonically increasing function of the autocorrelation coefficient.

Potential biological removal (PBR) is an approach used to calculate sustainable harvest and “take” limits for populations. PBR was originally derived assuming logistic growth while ignoring the effects of small population size (i.e., an Allee effect). We derived a version of PBR that includes an Allee effect (i.e., small population size or densities limiting population growth rates). We found that PBR becomes less conservative when it fails to consider an Allee effect. Specifically, sustainable harvest and take levels based upon PBR with an Allee effect were between approximately 51% and 66% of levels based upon PBR without an Allee effect. Managers and biologists using PBR may need to consider the limitations if an Allee effect may be present in the species being modeled. Considerations for Management Based upon our finding, management considerations may include: acknowledging that populations under stress may also be subject to Allee effects; recognizing limitations of approaches such as PBR when applying them to small populations; and broadly considering the Allee effects when using population models for natural resource management.

Understanding the complexities of environmental issues requires individuals to bring together ideas and data from different disciplines, including ecology and mathematics. With funding from the national science foundation (NSF), scientists from the University of Wisconsin-La Crosse and the US geological survey held a research experience for undergraduates (REU) program in the summer of 2016. The goals of the program were to expose students to open problems in the area of mathematical ecology, motivate students to pursue STEM-related positions, and to prepare students for research within interdisciplinary, collaborative settings. Based on backgrounds and interests, eight students were selected to participate in one of two research projects: wind energy and wildlife conservation or the establishment and spread of waterfowl diseases. Each research program was overseen by a mathematician and a biologist. Regardless of the research focus, the program first began with formal lectures to provide students with foundational knowledge followed by student-driven research projects. Throughout this period, student teams worked in close association with their mentors to create, parameterize and evaluate ecological models to better understand their systems of interest. Students then disseminated their results at local, regional, and international meetings and through publications (one in press and one in progress). Direct and indirect measures of student development revealed that our REU program fostered a deep appreciation for and understanding of mathematical ecology. Finally, the program allowed students to gain experiences working with individuals with different backgrounds and perspectives. Taken together, this REU program allowed us to successfully excite, motivate and prepare students for future positions in the area of mathematical biology, and because of this it can be used as a model for interdisciplinary programs at other institutions.

Underserved minority groups are disproportionately absent from the pursuit of careers in science, technology, engineering, and mathematics (STEM) fields. One such underserved population, Native Americans, are particularly underrepresented in STEM fields. Although recent advocacy and outreach designed toward increasing minority involvement in health care-related occupations have been mostly successful, little is known about the efficacy of outreach programs in increasing minority enthusiasm toward careers in traditional scientific professions. Furthermore, very little is known about outreach among Native American schools toward increasing involvement in STEM. We collaborated with tribal middle and high schools in South Dakota and Nebraska through a National Institutes of Health Science Education Partnership Award to hold a day-long physiology, activity-based event to increase both understanding of physiology and enthusiasm to scientific careers. We recruited volunteer biomedical scientists and trainees from the University of Nebraska Medical Center, Nebraska Wesleyan University, and University of South Dakota. To evaluate the effectiveness of the day of activities, 224 of the ~275–300 participating students completed both a pre- and postevent evaluation assessment. We observed increases in both students self-perceived knowledge of physiology and enthusiasm toward scientific career opportunities after the day of outreach activities. We conclude that activity-based learning opportunities in underserved populations are effective in increasing both knowledge of science and interest in scientific careers.

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.

We develop a mathematical and statistical framework to model the actions of underlying metabolites for carbon dioxide assimilation in photosynthesis. This study was motivated by a challenge posed by Syngenta Biotechnology to use modeling to better characterize photosynthesis in plants and subsequently better understand growth and crop yield. We use a dynamical system model proposed by Zhu., et al. [16], which describes the Calvin Cycle in spinach plants through changes in the concentrations of 38 metabolites (state variables) using non-linear enzyme kinetic ordinary differential equations and mass-balance laws that contain a total of 165 parameters. In our study of the CO2 assimilation rate, we pose our research questions with this dynamical system mathematical model and a statistical model to describe the observation process. In particular, we address the research question "Once a subset of parameters is fixed and the times at which data is collected are determined, can we identify which metabolites we should measure in order to optimize the confidence in our parameter estimates?" This problem of choosing the best subset of metabolites to measure is not well-explored in the existing literature. Here, we propose two methods. The first, the ad hoc Simple Linear Regression (SLR) method, models carbon assimilation as a linear function of the metabolites. Using simulated data, we implement a step-wise selection algorithm to determine the best subset of metabolites at each of 10 fixed time points. The second method, the Optimal Design Criterion method, fixes the number of metabolites to be observed and searches for the best combination by calculating a measure of the size of the Fisher Information Matrix (FIM) associated with measuring only the selected metabolites. Both methods suggest promise in determining appropriate sets of metabolites to observe for successful implementation of inverse problems. Our conclusions represent a paradigm for estimating parameters and designing experiments more efficiently in plant biological modeling. Ultimately, our results may be used to help engineer seeds that maximize carbon dioxide assimilation in photosynthesis and hence promote plant growth.

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.

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.

On Monday January 19, 2015 a story broke that the National Football League (NFL) had started an investigation into whether the New England Patriots deliberately deflated the footballs they used during their championship win over the Indianapolis Colts. Like an infectious disease, discussion regarding Deflategate grew rapidly on social media sites in the hours and days after the release of the story. However, after the Super Bowl was over, the scandal slowly began to dissipate and lost much of the attention it had originally had, as interest in the NFL wained at the completion of its season. We construct a simple epidemic model for the infectiousness of the Deflategate news story. We then use data from the social media site Twitter to estimate the parameters of this model using standard techniques from the study of inverse problems. We find that the infectiousness (as measured by the basic reproduction number) of Deflategate rivals that of any infectious disease that we are aware of, and is actually more infectious than recent news stories of greater importance - both in terms of the basic reproduction number and in terms of the average amount of time the average tweeter continued to tweet about the news story., 12 pages, 4 figures

In many plant species dormant seeds can persist in the soil for one to several years. The formation of these seed banks is especially important for disturbance specialist plants, as seeds of these species germinate only in disturbed soil. Seed movement caused by disturbances affects the survival and germination probability of seeds in the seed bank, which subsequently affect population dynamics. In this paper, we develop a stochastic integral projection model for a general disturbance specialist plant-seed bank population that takes into account both the frequency and intensity of random disturbances, as well as vertical seed movement and density-dependent seedling establishment. We show that the probability measures associated with the plant-seed bank population converge weakly to a unique measure, independent of initial population. We also show that the population either persists with probability one or goes extinct with probability one, and provides a sharp criteria for this dichotomy. We apply our results to an example motivated by wild sunflower (Helianthus annuus) populations, and explore how the presence or absence of a “storage effect” impacts how a population responds to different disturbance scenarios.

Many plant populations have persistent seed banks, which consist of viable seeds that remain dormant in the soil for many years. Seed banks are important for plant population dynamics because they buffer against environmental perturbations and reduce the probability of extinction. Viability of the seeds in the seed bank can depend on the seed’s age, hence it is important to keep track of the age distribution of seeds in the seed bank. In this paper we construct a general density-dependent plant-seed bank model where the seed bank is age-structured. We consider density dependence in both seedling establishment and seed production, since previous work has highlighted that overcrowding can suppress both of these processes. Under certain assumptions on the density dependence, we prove that there is a globally stable equilibrium population vector which is independent of the initial state. We derive an analytical formula for the equilibrium population using methods from feedback control theory. We apply these results to a model for the plant species Cirsium palustre and its seed bank.

Quantifying the impact of anthropogenic development on local populations is important for conservation biology and wildlife management. However, these local populations are often subject to demographic stochasticity because of their small population size. Traditional modeling efforts such as population projection matrices do not consider this source of variation whereas individual-based models, which include demographic stochasticity, are computationally intense and lack analytical tractability. One compromise between approaches is branching process models because they accommodate demographic stochasticity and are easily calculated. These models are known within some sub-fields of probability and mathematical ecology but are not often applied in conservation biology and applied ecology. We applied branching process models to quantitatively compare and prioritize species locally vulnerable to the development of wind energy facilities. Specifically, we examined species vulnerability using branching process models for four representative species: A cave bat (a long-lived, low fecundity species), a tree bat (short-lived, moderate fecundity species), a grassland songbird (a short-lived, high fecundity species), and an eagle (a long-lived, slow maturation species). Wind turbine-induced mortality has been observed for all of these species types, raising conservation concerns. We simulated different mortality rates from wind farms while calculating local extinction probabilities. The longer-lived species types (e.g., cave bats and eagles) had much more pronounced transitions from low extinction risk to high extinction risk than short-lived species types (e.g., tree bats and grassland songbirds). High-offspring-producing species types had a much greater variability in baseline risk of extinction than the lower-offspring-producing species types. Long-lived species types may appear stable until a critical level of incidental mortality occurs. After this threshold, the risk of extirpation for a local population may rapidly increase with only minimal increases in wind mortality. Conservation biologists and wildlife managers may need to consider this mortality pattern when issuing take permits and developing monitoring protocols for wind facilities. We also describe how our branching process models may be generalized across a wider range of species for a larger assessment project and then describe how our methods may be applied to other stressors in addition to wind.