Nonlinear Population Dynamics Research Articles

The Status of Sanitation in Malawi: Is SDG6.2 Achievable?

Ensuring access to adequate and equitable sanitation and ending open defecation by 2030 is the focus of Sustainable Development Goal 6.2 (SDG6.2). We evaluated Malawi's progress towards SDG 6.2 (specifically the goal to end open defecation), presenting the results of a national survey of over 200,000 sanitary facilities and evaluating their management. Based on non-linear population dynamics, we used a linear model to evaluate the reduction in open defecation between 1992-2018, and to project whether Malawi can meet the SDG target to end open defecation by 2030 under multiple scenarios of population growth. Whilst Malawi has made considerable progress in providing sanitary provision for the population, we estimate that, at the current rate of the provision of sanitary facilities, Malawi will not reach SDG 6.2 by 2030 under any of the modelled socioeconomic scenarios. Furthermore, we compare the estimates of the extent of sanitary provision classed as improved from multiple surveys, including the USAID Demographic and Health (DHS) Surveys and Government of Malawi Census data. We conclude that some of the surveys (particularly the 2015/16 DHS) may be overestimating the level of improved sanitary provision, and we hypothesize that this is due to how pit-latrines with earth/sand slabs are classed. Furthermore, we examine the long-term sustainability of pit-latrine use, investigating the challenge of pit-latrine abandonment and identifying pit-latrine filling as a cause of the abandonment in 30.2% of cases. We estimate that between 2020-2070, 31.8 (range 2.8 to 3320) million pit-latrines will be filled and abandoned, representing a major challenge for the safe management of abandoned latrines, a potential for long-term impacts on the groundwater quality, and a significant loss of investment in sanitary infrastructure. For Malawi to reach SDG 6.2, improvements are needed in both the quantity and quality of its sanitary facilities.

Existence results and behavior analysis for nonlinear age-dependent population dynamics problems with conformable fractional operator

In this paper, we investigate the existence of solutions and analyze the large-time behavior for Gurtin–Maccamy population model involving conformable fractional derivatives. As a preliminary step, we construct a generic structure of the solution associated with our proposed model by utilizing some basic properties and tools of conformable fractional calculus. We establish the existence of a unique solution of the given model with the given initial conditions. At last, by using the upper and lower solutions for the characteristic equation, we define the upper and lower boundaries for the obtained solution and describe the large-time behavior of the total population.

시스템다이내믹스 관점에서 본 SIR-기반 실업 모델링 연구

Unemployment is a macroeconomic indicator that reflects productivity changes and employment status in the labor market. In fact, the unemployment rate is defined in the relationship of non-linear population dynamics in which the number of unemployed and employed, and the number of job seekers and jobs are affected by macroeconomic influences. This paper investigates the unemployment dynamics by establishing an unemployment rate model based on the SIR (Susceptible-Infected- Recovered) model as a system dynamics method. The unemployment model of this study is built on the basis of actual data values from 1990 to 2019, and as a result of the simulation, this model can verify the validity of the working-age population, the economically active population, the unemployed and the employed, all showing similar trends with the actual data values. In the unemployment model of system dynamics, the trend of economic growth reflects the characteristics of the unemployment rate.

A bifurcation threshold for contact-induced language change

One proposed mechanism of language change concerns the role played by second-language (L2) learners in situations of language contact. If sufficiently many L2 speakers are present in a speech community in relation to the number of first-language (L1) speakers, then those features which present a difficulty in L2 acquisition may be prone to disappearing from the language. This paper presents a mathematical account of such contact situations based on a stochastic model of learning and nonlinear population dynamics. The equilibria of a deterministic reduction of the model, describing a mixed population of L1 and L2 speakers, are fully characterized. Whether or not the language changes in response to the introduction of L2 learners turns out to depend on three factors: the overall proportion of L2 learners in the population, the strength of the difficulty speakers face in acquiring the language as an L2, and the language-internal utilities of the competing linguistic variants. These factors are related by a mathematical formula describing a phase transition from retention of the L2-difficult feature to its loss from both speaker populations. This supplies predictions that can be tested against empirical data. Here, the model is evaluated with the help of two case studies, morphological levelling in Afrikaans and the erosion of null subjects in Afro-Peruvian Spanish; the model is found to be broadly in agreement with the historical development in both cases.

Cavity molecular dynamics of vibrational modes enhanced non-linear absorption and population dynamics

We present a quantum description of SERRS, in analogy to the energy spectrum dynamics for a system that comprises two molecules interacting with a single plasmonic cavity, taking into account the effect of molecular vibrations and Van der Waals interactions between the two molecules. It is shown that the splitting of the transparency peak can easily distinguish two adjacent molecules with similar chemical structures and nearly equal vibrational frequencies. We also found that vibrational pumping can selectively activate a particular vibrational mode, thus providing a mechanism to control its population dynamics and drives plasmon-assisted chemistry. Our results will provide a new approach for understanding the role of cavity photons in a two-level system comprising interacting polaritons and phonons.

A nonlinear population dynamics model of patient diagnosis and treatment involving in two level medical institutions and its qualitative analysis of positive singularity.

In this article, we firstly establish a nonlinear population dynamical model to describe the changes and interaction of the density of patient population of China's primary medical institutions (PHCIs) and hospitals in China's medical system. Next we get some sufficient conditions of existence of positive singularity by utilising homotopy invariance theorem of topological degree. Meanwhile, we study the qualitative properties of positive singularity based on Perron's first theorem. Furthermore, we briefly analyze the significance and function of the mathematical results obtained in this paper in practical application. As verifications, some numerical examples are ultimately exploited the correctness of our main results. Combined with the numerical simulation results and practical application, we give some corresponding suggestions. Our research can provide a certain theoretical basis for government departments to formulate relevant policies.

Timing outweighs magnitude of rainfall in shaping population dynamics of a small mammal species in steppe grassland.

Climate change-induced shifts in species phenology differ widely across trophic levels, which may lead to consumer-resource mismatches with cascading population and ecosystem consequences. Here, we examined the effects of different rainfall patterns (i.e., timing and amount) on the phenological asynchrony of population of a generalist herbivore and their food sources in semiarid steppe grassland in Inner Mongolia. We conducted a 10-y (2010 to 2019) rainfall manipulation experiment in 12 0.48-ha field enclosures and found that moderate rainfall increases during the early rather than late growing season advanced the timing of peak reproduction and drove marked increases in population size through increasing the biomass of preferred plant species. By contrast, greatly increased rainfall produced no further increases in vole population growth due to the potential negative effect of the flooding of burrows. The increases in vole population size were more coupled with increased reproduction of overwintered voles and increased body mass of young-of-year than with better survival. Our results provide experimental evidence for the fitness consequences of phenological mismatches at the population level and highlight the importance of rainfall timing on the population dynamics of small herbivores in the steppe grassland environment.

On uniqueness and reconstruction of a nonlinear diffusion term in a parabolic equation

The problem of recovering coefficients in a diffusion equation is one of the basic inverse problems. Perhaps the most important term is the one that couples the length and time scales and is often referred to as the diffusion coefficient a in ut−∇(a∇u)=f. In this paper we seek the unknown a assuming that a=a(u) depends only on the value of the solution at a given point. Such diffusion models are the basic of a wide range of physical phenomena such as nonlinear heat conduction, chemical mixing and population dynamics. We shall look at two types of overposed data in order to effect recovery of a(u): the value of a time trace u(x0,t) for some fixed point x0 on the boundary of the region Ω; or the value of u on an interior curve Σ lying within Ω. As examples, these might represent a temperature measurement on the boundary or a census of the population in some subset of Ω taken at a fixed time T>0. In the latter case we shall show a uniqueness result that leads to a constructive method for recovery of a. Indeed, for both types of measured data we shall show reconstructions based on the iterative algorithms developed in the paper.

Controllability of a Family of Nonlinear Population Dynamics Models

Considering a nonlinear dynamical system, we study the nonlinear infinite-dimensional system obtained by grafting an operatorAand an age structure. This system is such that the nonlinearity is at the level of births. We show that there is a timeTdependent on the constraints on the age and the observability minimal timeT0of the pairA,B(Bis the control operator), from which the system is null controllable. We first establish an observability inequality useful for the proof of the null controllability of an auxiliary system. We also apply Schauder’s fixed point in the proof of the null controllability of the nonlinear system..

A Multi-Region Nonlinear Size-Structured Population Model with Coagulation and Vertical Effects

A general nonlinear model describing the evolution of size-structured populations influenced by coagulation and vertical effects is presented. The nonlinear population dynamics occur in a multi-region setting in which the transfer, coagulation, and vital rates of an individual depend on a vector describing conditions in the environment. The model for these dynamics consists of a system of two-dimensional nonlinear-nonlocal hyperbolic partial differential equations coupled with a system of one-dimensional nonlocal differential equations parametrized by a vertical location coordinate $$${z}$$$ describing the environmental time-dynamics. A finite difference approximation approach is employed to study the wellposedness of the model and convergence of the scheme to the unique weak solution is established. Several examples are presented to illustrate the generality of the model and to motivate applications.

Nonlinear self-organized population dynamics induced by external selective nonlocal processes

Self-organization evolution of a population is studied considering generalized reaction-diffusion equations. We proposed a model based on non-local operators that has several of the equations traditionally used in research on population dynamics as particular cases. Then, employing a relatively simple functional form of the non-local kernel, we determined the conditions under which the analyzed population develops spatial patterns, as well as their main characteristics. Finally, we established a relationship between the developed model and real systems by making simulations of bacterial populations subjected to non-homogeneous lighting conditions. Our proposal reproduces some of the experimental results that other approaches considered previously had not been able to obtain.

A competing-risks model explains hierarchical spatial coupling of measles epidemics en route to national elimination.

Apart from its global health importance, measles is a paradigm for the low-dimensional mechanistic understanding of local nonlinear population interactions. A central question for spatio-temporal dynamics is the relative roles of hierarchical spread from large cities to small towns and metapopulation transmission among local small population clusters in measles persistence. Quantifying this balance is critical to planning the regional elimination and global eradication of measles. Yet, current gravity models do not allow a formal comparison of hierarchical versus metapopulation spread. We address this gap with a competing-risks framework, capturing the relative importance of competing sources of reintroductions of infection. We apply the method to the uniquely spatio-temporally detailed urban incidence dataset for measles in England and Wales, from 1944 to the infection's vaccine-induced nadir in the 1990s. We find that despite the regional influence of a few large cities (for example, London and Liverpool), metapopulation aggregation in neighbouring towns and cities played an important role in driving national dynamics in the prevaccination era. As vaccination levels increased in the 1970s and 1980s, the signature of spatially predictable spread diminished: increasingly, infection was introduced from unidentifiable random sources possibly outside regional metapopulations. The resulting erratic dynamics highlight the challenges of identifying shifting sources of infection and characterizing patterns of incidence in times of high vaccination coverage. More broadly, the underlying incidence and demographic data, accompanying this paper, will also provide an important resource for exploring nonlinear spatiotemporal population dynamics.

Nonlinear population dynamics are ubiquitous in animals.

Nonlinear dynamics, where a change in the input is not proportional to a change in the output, are often found throughout nature, for example in biochemical kinetics. Because of the complex suite of interacting abiotic and biotic variables present in ecosystems, animal population dynamics are often thought to be driven in a nonlinear, state-dependent fashion. However, so far these have only been identified in model organisms and some natural systems. Here we show that nonlinear population dynamics are ubiquitous in nature. We use nonlinear forecasting to analyse 747 datasets of 228 species to find that insect population trends were highly nonlinear (74%), followed by mammals (58%), bony fish (49%) and birds (35%). This indicates that linear, equilibrium-based model assumptions may fail at predicting population dynamics across a wide range of animal taxa. We show that faster-reproducing animals are more likely to have nonlinear and high-dimensional dynamics, supporting past ecological theory. Lastly, only a third of time series were predictable beyond two years; therefore, the ability to predict animal population trends using these methods may be limited. Our results suggest that the complex dynamics necessary to cause regime shifts and other transitions may be inherent in a wide variety of animals.

Boundary Null Controllability With Constrained Control for a Nonlinear Two Stroke System: Application to Boundary Sentinel and Identification of Parameters in a Nonlinear Population Dynamics Model

We first prove a new controllability result for a nonlinear two stroke system. The key to solve this controllability problem is an adapted Carleman inequality. Next, the obtained result is used to build a boundary sentinel to identify unknown parameters in a nonlinear population dynamics model with incomplete data.

Differentiation of True Nonlinear and Incoherent Mixing of Linear Signals in Action-Detected 2D Spectroscopy.

Phase modulation and phase cycling schemes have been commonly used in electronic two-dimensional (2D) spectroscopy where the observables are incoherent signals such as fluorescence or photocurrent. Although the methods have distinct advantages compared to the coherent signal-detected 2D spectroscopy in sensitivity, possibility to measure spectra from isolated quantum systems and direct visualization of the contributions from the different states to the action signals, and ambiguities in interpreting the spectra have emerged. Recent reports have shown that apart from the nonlinear signals from the four pulse interactions, mixing of the linear signals due to nonlinear population dynamics during the long measurement time of the action signals can also contribute to the measured 2D spectra. Exciton-exciton annihilation has been considered to play a major role in the mixing of the linear signals. Thus, it has become important to further characterize the origin of the measured signals. Here, using a nonperturbative simulation of the 2D spectra based on the time evolution of the density matrix in the Lindblad form, we show that the exciton-exciton annihilation contributes to the measured signal only if the quantum yields of the different excited states are not the same. In these cases, the mixed signals can be distinguished from the true nonlinear signals if the phases are measured with respect to the linear signals. In action-detected 2D spectra, the mixed signals have a π phase shift relative to the true nonlinear signals. A detailed discussion on the experimental implementations of the schemes used in the simulations is also provided.

Null controllability via comparison results for nonlinear age-structured population dynamics

We consider an infinite dimensional nonlinear controlled system describing age-structured population dynamics, where the birth and the mortality rates are nonlinear functions of the population size. The control being active on some age range, we give sharp conditions subject to the age range and the control time horizon to get the null controllability of the nonlinear controlled population dynamics. The main novelty is that we use here as a main ingredient the comparison principle for age-structured population dynamics, and in case of null controllability we provide a feedback control with a very simple structure, while preserving the nonnegativity of the state trajectory. Finally, we establish the lack of the null controllability for the linear Lotka-McKendrick equation with spatial diffusion when the control acts in a subset of the habitat and we want to preserve the positivity of the state trajectory.

Nonlinear population dynamics in a bounded habitat

A key issue in ecology is whether a population will survive long term or go extinct. This is the question we address in this paper for a population in a bounded habitat. We will restrict our study to the case of a single species in a one-dimensional habitat of length L. The evolution of the population density distribution ρ(x, t), where x is the position and t the time, is governed by elementary processes such as growth and dispersal, which, in standard models, are typically described by a constant per capita growth rate and normal diffusion, respectively. However, feedbacks in the regulatory mechanisms and external factors can produce density-dependent rates. Therefore, we consider a generalization of the standard evolution equation, which, after dimensional scaling and assuming large carrying capacity, becomes ∂tρ=∂x(ρν−1∂xρ)+ρμ, where μ,ν∈R. This equation is complemented by absorbing boundaries, mimicking adverse conditions outside the habitat. For this nonlinear problem, we obtain, analytically, exact expressions of the critical habitat size Lc for population survival, as a function of the exponents and initial conditions. We find that depending on the values of the exponents (ν, μ), population survival can occur for either L ≥ Lc, L ≤ Lc or for any L. This generalizes the usual statement that Lc represents the minimum habitat size. In addition, nonlinearities introduce dependence on the initial conditions, affecting Lc.

Structural stability of nonlinear population dynamics.

In population dynamics, the concept of structural stability has been used to quantify the tolerance of a system to environmental perturbations. Yet, measuring the structural stability of nonlinear dynamical systems remains a challenging task. Focusing on the classic Lotka-Volterra dynamics, because of the linearity of the functional response, it has been possible to measure the conditions compatible with a structurally stable system. However, the functional response of biological communities is not always well approximated by deterministic linear functions. Thus, it is unclear the extent to which this linear approach can be generalized to other population dynamics models. Here, we show that the same approach used to investigate the classic Lotka-Volterra dynamics, which is called the structural approach, can be applied to a much larger class of nonlinear models. This class covers a large number of nonlinear functional responses that have been intensively investigated both theoretically and experimentally. We also investigate the applicability of the structural approach to stochastic dynamical systems and we provide a measure of structural stability for finite populations. Overall, we show that the structural approach can provide reliable and tractable information about the qualitative behavior of many nonlinear dynamical systems.

Incoherent population mixing contributions to phase-modulation two-dimensional coherent excitation spectra.

We present theoretical and experimental results showing the effects of incoherent population mixing on two-dimensional (2D) coherent excitation spectra that are measured via a time-integrated population and phase-sensitive detection. The technique uses four collinear ultrashort pulses and phase modulation to acquire two-dimensional spectra by isolating specific nonlinear contributions to the photoluminescence or photocurrent excitation signal. We demonstrate that an incoherent contribution to the measured line shape, arising from nonlinear population dynamics over the entire photoexcitation lifetime, generates a similar line shape to the expected 2D coherent spectra in condensed-phase systems. In those systems, photoexcitations are mobile such that inter-particle interactions are important on any time scale, including those long compared with the 2D coherent experiment. Measurements on a semicrystalline polymeric semiconductor film at low temperatures show that, in some conditions in which multi-exciton interactions are suppressed, the technique predominantly detects coherent signals and can be used, in our example, to extract homogeneous line widths. The same method used on a lead-halide perovskite photovoltaic cell shows that incoherent population mixing of mobile photocarriers can dominate the measured signal since carrier-carrier bimolecular scattering is active even at low excitation densities, which hides the coherent contribution to the spectral line shape. In this example, the intensity dependence of the signal matches the theoretical predictions over more than two orders of magnitude, confirming the incoherent nature of the signal. While these effects are typically not significant in dilute solution environments, we demonstrate the necessity to characterize, in condensed-phase materials systems, the extent of nonlinear population dynamics of photoexcitations (excitons, charge carriers, etc.) in the execution of this powerful population-detected coherent spectroscopy technique.

Sparse model selection via integral terms.

Model selection and parameter estimation are important for the effective integration of experimental data, scientific theory, and precise simulations. In this work, we develop a learning approach for the selection and identification of a dynamical system directly from noisy data. The learning is performed by extracting a small subset of important features from an overdetermined set of possible features using a nonconvex sparse regression model. The sparse regression model is constructed to fit the noisy data to the trajectory of the dynamical system while using the smallest number of active terms. Computational experiments detail the model's stability, robustness to noise, and recovery accuracy. Examples include nonlinear equations, population dynamics, chaotic systems, and fast-slow systems.