Assessing critical population thresholds under periodic disturbances

Population responses to repeated environmental or anthropogenic disturbances depend on complicated interactions between the disturbance regime, population structure, and differential stage susceptibility. Using a matrix modeling approach, we develop a methodological framework to explore how the interplay of these factors impacts critical population thresholds. To illustrate the wide applicability of this approach, we present two case studies pertaining to agroecosystems and conservation science. We apply sensitivity analysis to the two case studies to examine how population and disturbance properties affect these thresholds. Contrasting outcomes between these two applications, including differences in how factors such as disturbance intensity and pre-disturbance population distributions impact population responses, highlight the importance of accounting for demographic features when performing ecological risk assessments

Sensitivity equations for measure-valued solutions to transport equations

We consider the following transport equation in the space of bounded, nonnegative Radon measures M+(Rd): θtμt + θx(v(x)μt) = 0: We study the sensitivity of the solution μt with respect to a perturbation in the vector field, v(x). In particular, we replace the vector field v with a perturbation of the form vh = v0(x) + hv1(x) and let μh t be the solution of θtμh t + θx(vh(x)μh t) = 0: We derive a partial differential equation that is satisfied by the derivative of μh t with respect to h, θh(μh t). We show that this equation has a unique very weak solution on the space Z, being the closure of M(Rd) endowed with the dual norm (C1,α(Rd))*. We also extend the result to the nonlinear case where the vector field depends on μt, i.e., v = v[μt](x).Fil: Ackleh, Azmy S.. State University of Louisiana; Estados UnidosFil: Saintier, Nicolas Bernard Claude. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Skrzeczkowski, Jakub. University of Warsaw; Poloni

Ecosystem Modeling of College Drinking: Parameter Estimation and Comparing Models to Data

Recently we developed a model composed of five impulsive differential equations that describes the changes in drinking patterns (that persist at epidemic level) amongst college students. Many of the model parameters cannot be measured directly from data; thus, an inverse problem approach, which chooses the set of parameters that results in the “best” model to data fit, is crucial for using this model as a predictive tool. The purpose of this paper is to present the procedure and results of an unconventional approach to parameter estimation that we developed after more common approaches were unsuccessful for our specific problem. The results show that our model provides a good fit to survey data for 32 campuses. Using these parameter estimates, we examined the effect of two hypothetical intervention policies: 1) reducing environmental wetness, and 2) penalizing students who are caught drinking. The results suggest that reducing campus wetness may be a very effective way of reducing heavy episodic (binge) drinking on a college campus, while a policy that penalizes students who drink is not nearly as effective

On the net reproduction rate of continuous structured populations with distributed states at birth

We consider a nonlinear structured population model with a distributed recruitment term. The question of the existence of non-trivial steady states can be treated (at least!) in three different ways. One approach is to study spectral properties of a parametrized family of unbounded operators. The alternative approach, on which we focus here, is based on the reformulation of the problem as an integral equation. In this context we introduce a density dependent net reproduction rate and discuss its relationship to a biologically meaningful quantity. Finally, we briefly discuss a third approach, which is based on the finite rank approximation of the recruitment operator.Comment: To appear in Computers and Mathematics with Application

Heavy episodic drinking on college campuses: Does changing the legal drinking age make a difference

ABSTRACT. Objective: This article extends the compartmental model previously developed by Scribner et al. in the context of college drinking to a mathematical model of the consequences of lowering the legal drinking age. Method: Using data available from 32 U.S. campuses, the analyses separate underage and legal age drinking groups into an eight-compartment model with different alcohol availability (wetness) for the underage and legal age groups. The model evaluates the likelihood that underage students will incorrectly perceive normative drinking levels to be higher than they actually are (i.e., misperception) and adjust their drinking accordingly by varying the interaction between underage students in social and heavy episodic drinking compartments. Results: The results evaluate the total heavy episodic drinker population and its dependence on the difference in misperception, as well as its dependence on underage wetness, legal age wetness, and drinking age. Conclusions: Results suggest that an unrealistically extreme combination of high wetness and low enforcement would be needed for the policies related to lowering the drinking age to be effective. (J. Stud. Alcohol Drugs, 72

Analysis of lethal and sublethal impacts of environmental disasters on sperm whales using stochastic modeling

Mathematical models are essential for combining data from multiple sources to quantify population endpoints. This is especially true for species, such as marine mammals, for which data on vital rates are difficult to obtain. Since the effects of an environmental disaster are not fixed, we develop time-varying (nonautonomous) matrix population models that account for the eventual recovery of the environment to the pre-disaster state. We use these models to investigate how lethal and sublethal impacts (in the form of reductions in the survival and fecundity, respectively) affect the population’s recovery process. We explore two scenarios of the environmental recovery process and include the effect of demographic stochasticity. Our results provide insights into the relationship between the magnitude of the disaster, the duration of the disaster, and the probability that the population recovers to pre-disaster levels or a biologically relevant threshold level. To illustrate this modeling methodology, we provide an application to a sperm whale population. This application was motivated by the 2010 Deepwater Horizon oil rig explosion in the Gulf of Mexico that has impacted a wide variety of species populations including oysters, fish, corals, and whales

Parameter Estimation In A Structured Algal Coagulation-Fragmentation Model

Coagulation of phytoplankton is a fundamental mechanism for vertical flux of carbon in the ocean. This process is dependent on parameters that are not available from experimental data, such as the encounter rate of particles, the contact efficiency of unlike particles and the probability of sticking upon contact. Fragmentation, the breakup of large particles into two smaller ones has been observed in the ocean, but very little modeling effort for incorporating this process in the dynamics of phytoplankton has been attempted. In this paper we incorporate fragmentation process into a nonlinear hyperbolic equation that describes the evolution of a size structured algal population with the aggregation model. We examine through numerical simulation the effect of fragmentation on the dynamics of phytoplankton. We present convergence theory for estimating parameters in this model using nonlinear least squares fit. The least square method is then tested numerically in ideal cases where the dat..

Monotone method for nonlinear nonlocal hyperbolic problems

We present recent results concerning the application of the monotone method for studying existence and uniqueness of solutions to general first-order nonlinear nonlocal hyperbolic problems. The limitations of comparison principles for such nonlocal problems are discussed. To overcome these limitations, we introduce new definitions for upper and lower solutions