Single-patch Model Research Articles

Dynamics of the generalized Rosenzweig–MacArthur model in a changing and patchy environment

In this paper, we propose a generalized Rosenzweig–MacArthur predator–prey model with two patches and Allee effects in predators. In a constant environment, for the single-patch model we show the existence of degenerate Bogdanov–Takens bifurcation with codimension up to 4 and degenerate Hopf bifurcation with codimension up to 3, which indicate that the nilpotent cusp of codimension 4 is the organizing center of the bifurcation set. Our results show that Allee effects in predators can not only induce much more complex bifurcations and dynamics (such as three limit cycles, multitype tristability), but also serve as a destabilizing factor to make predators extinction (deterministically) when the environment is sufficiently enriched, which provides strong support for the so-called “paradox of enrichment”. For the two-patch model, we numerically reveal the existence of eight positive steady states including two new coexisting attractors, which means that population dispersal can help species coexist or increase the population abundance. In a changing environment, for the single-patch model we find that the population can track unstable states, fast positive or slow negative environmental change can be beneficial to predators by delaying or even avoiding extinction; while for the two-patch model we observe some interesting phenomena: fast negative (or slow positive) environmental change can make predators in different patches trend to extinction synchronously, and rate-induced different regime shifts.

Inference on a Multi-Patch Epidemic Model with Partial Mobility, Residency, and Demography: Case of the 2020 COVID-19 Outbreak in Hermosillo, Mexico.

Most studies modeling population mobility and the spread of infectious diseases, particularly those using meta-population multi-patch models, tend to focus on the theoretical properties and numerical simulation of such models. As such, there is relatively scant literature focused on numerical fit, inference, and uncertainty quantification of epidemic models with population mobility. In this research, we use three estimation techniques to solve an inverse problem and quantify its uncertainty for a human-mobility-based multi-patch epidemic model using mobile phone sensing data and confirmed COVID-19-positive cases in Hermosillo, Mexico. First, we utilize a Brownian bridge model using mobile phone GPS data to estimate the residence and mobility parameters of the epidemic model. In the second step, we estimate the optimal model epidemiological parameters by deterministically inverting the model using a Darwinian-inspired evolutionary algorithm (EA)-that is, a genetic algorithm (GA). The third part of the analysis involves performing inference and uncertainty quantification in the epidemic model using two Bayesian Monte Carlo sampling methods: t-walk and Hamiltonian Monte Carlo (HMC). The results demonstrate that the estimated model parameters and incidence adequately fit the observed daily COVID-19 incidence in Hermosillo. Moreover, the estimated parameters from the HMC method yield large credible intervals, improving their coverage for the observed and predicted daily incidences. Furthermore, we observe that the use of a multi-patch model with mobility yields improved predictions when compared to a single-patch model.

Effects of travel frequency on the persistence of mosquito-borne diseases

<p style='text-indent:20px;'>Travel frequency of people varies widely with occupation, age, gender, ethnicity, income, climate and other factors. Meanwhile, the distribution of the numbers of times people in different regions or with different travel behaviors bitten by mosquitoes may be nonuniform. To reflect these two heterogeneities, we develop a multipatch model to study the impact of travel frequency and human biting rate on the spatial spread of mosquito-borne diseases. The human population in each patch is divided into four classes: susceptible unfrequent, infectious unfrequent, susceptible frequent, and infectious frequent. The basic reproduction number <inline-formula><tex-math id="M1">\begin{document}$ \mathcal{R}_0 $\end{document}</tex-math></inline-formula> is defined. It is shown that the disease-free equilibrium is globally asymptotically stable if <inline-formula><tex-math id="M2">\begin{document}$ \mathcal{R}_0\leq 1 $\end{document}</tex-math></inline-formula>, and there is a unique endemic equilibrium that is globally asymptotically stable if <inline-formula><tex-math id="M3">\begin{document}$ \mathcal{R}_0&gt;1 $\end{document}</tex-math></inline-formula>. A more detailed study is conducted on the single patch model. We use analytical and numerical methods to demonstrate that the model without considering the difference of humans in travel frequency mostly underestimates the risk of infection. Numerical simulations suggest that the greater the difference in travel frequency, the larger the underestimate of the transmission potential. In addition, the basic reproduction number <inline-formula><tex-math id="M4">\begin{document}$ \mathcal{R}_0 $\end{document}</tex-math></inline-formula> may decreasingly, or increasingly, or nonmonotonically vary when more people travel frequently.

Global dynamics of a delayed two-patch discrete SIR disease model

In this paper, we propose a delayed discrete SIR disease model with a saturate incidence rate and extend it to a patchy environment by taking the dispersal of susceptible individuals from one patch to the other into consideration. For the single-patch model, we establish the global threshold dynamics by the method of Lyapunov functionals. For the two-patch model, we show that the global dynamics of the disease-free equilibrium, two boundary endemic equilibria and the interior endemic equilibrium are determined by several threshold quantities. We also explore the impacts of the dispersal on the disease dynamics. Our interesting findings may provide some useful insights on how to properly manage the dispersal between different regions to control the spread of diseases.

Habitat fragmentation promotes malaria persistence.

Based on a Ross-Macdonald type model with a number of identical patches, we study the role of the movement of humans and/or mosquitoes on the persistence of malaria and many other vector-borne diseases. By using a theorem on line-sum symmetric matrices, we establish an eigenvalue inequality on the product of a class of nonnegative matrices and then apply it to prove that the basic reproduction number of the multipatch model is always greater than or equal to that of the single patch model. Biologically, this means that habitat fragmentation or patchiness promotes disease outbreaks and intensifies disease persistence. The risk of infection is minimized when the distribution of mosquitoes is proportional to that of humans. Numerical examples for the two-patch submodel are given to investigate how the multipatch reproduction number varies with human and/or mosquito movement. The reproduction number can surpass any given value whenever an appropriate travel pattern is chosen. Fast human and/or mosquito movement decreases the infection risk, but may increase the total number of infected humans.

Spatial panorama of malaria prevalence in Africa under climate change and interventions scenarios

BackgroundMalaria is highly sensitive to climatic variables and is strongly influenced by the presence of vectors in a region that further contribute to parasite development and sustained disease transmission. Mathematical analysis of malaria transmission through the use and application of the value of the basic reproduction number (R0) threshold is an important and useful tool for the understanding of disease patterns.MethodsTemperature dependence aspect of R0 obtained from dynamical mathematical network model was used to derive the spatial distribution maps for malaria transmission under different climatic and intervention scenarios. Model validation was conducted using MARA map and the Annual Plasmodium falciparum Entomological Inoculation Rates for Africa.ResultsThe inclusion of the coupling between patches in dynamical model seems to have no effects on the estimate of the optimal temperature (about 25 °C) for malaria transmission. In patches environment, we were able to establish a threshold value (about α = 5) representing the ratio between the migration rates from one patch to another that has no effect on the magnitude of R0. Such findings allow us to limit the production of the spatial distribution map of R0 to a single patch model. Future projections using temperature changes indicated a shift in malaria transmission areas towards the southern and northern areas of Africa and the application of the interventions scenario yielded a considerable reduction in transmission within malaria endemic areas of the continent.ConclusionsThe approach employed here is a sole study that defined the limits of contemporary malaria transmission, using R0 derived from a dynamical mathematical model. It has offered a unique prospect for measuring the impacts of interventions through simple manipulation of model parameters. Projections at scale provide options to visualize and query the results, when linked to the human population could potentially deliver adequate highlight on the number of individuals at risk of malaria infection across Africa. The findings provide a reasonable basis for understanding the fundamental effects of malaria control and could contribute towards disease elimination, which is considered as a challenge especially in the context of climate change.

An algebraic study of the matrix meta-population model

In studying the population living in some patches, a single patch model usually generalizes into a meta-population model. Roughly speaking a meta-population is population consisting of some sub-populations, such as sub-population reflecting different life stages of the population or different habitat of patches. In a meta-population framework the dynamics of population within and between patches can be easily studied. This paper will discuss the matrix meta-population model that describes the growth and reproduction within the patches denoted by F. The matrix F is obtained by multiplying three matrices; the transpose of a permutation matrix, a block diagonal matrix in which its diagonal blocks is a matrix Aj, and the permutation matrix. The matrix Aj is the real non-negative matrix that represents the dynamics of population growth in the patch j. The permutation matrix has a special form that has properties that are more specific than the general permutation matrix. This paper will also gives a brief application for the matrix F. This will provide a space for exploring the characterization of the matrix F from algebraic point of view.

Biological conservation through marine protected areas in the presence of alternative stable states

This article addresses how depleted stock can be restored by creation of marine reserve and species mobility when alternative stable states persist in a marine ecosystem. To understand the role of a marine protected area, we develop a two-patch version of an originally single-patch model. In the two-patch model, we prove that some of the locally stable equilibria are not stable equilibria from an ecological viewpoint. Similarly, some unstable equilibria determined classically from the mathematical model are no longer equilibria. It is shown that increasing reserve size may produce three alternative stable states in the presence of harvesting. Dynamic solutions have a tendency to reach an upper stable state from a lower stable state when reserve size is increased, but the opposite phenomenon (i.e., shifting to a lower stable state from an upper one) never occurs. This suggests that MPAs always have a positive effect in stock conservation even when alternative stable states inherently persist in marine ecosystems.

Modeling 2-D Lens-Based Forward-Scan Sonar Imagery for Targets With Diffuse Reflectance

Sonar images are formed by transmitting acoustical pulses and measuring the reflected sound power from the scene surfaces. The recorded signal by a sonar device encodes information about the shape and material properties of these surfaces. In this paper, we present a detailed derivation of an image model for a new class of high-resolution lens-based 2-D forward-looking sonar systems, when the diffuse reflectance of scene/target can be characterized by the Lambertian model. A simplified single patch model is generalized to account for the finite pulse width of the transmitted beam and the simultaneous arrival of scattered signals from multiple patches at the same range in a given azimuthal direction. Validating the model using intensity measurements of isolated cylindrical targets, we then demonstrate application for multipath reflections from bottom surfaces with cylindrical and spherical targets.

Nitsche method for isogeometric analysis of Reissner–Mindlin plate with non-conforming multi-patches

Nitsche method application in non-conforming plate is presented in the context of isogeometric analysis. Reissner–Mindlin plate theory is employed to build governing equation and stiffness matrix. We use this theory to solve the elasticity problems of various classical plate models, and compare the obtained results to those from single-patch models and the exact solutions in Kirchhoff theory. The solutions of problem involving the use of complex model are as well obtained using the same Reissner–Mindlin theory and compared to the results from finite element method. All models are built with NURBS (non-uniform rational B-spline) patches with non-conforming mesh along the common boundaries. The algorithms of knot insertion and order elevation are applied to enrich the basis functions of NURBS patches. The results of numerical examples show the accuracy, robustness and high convergence rate of this method.

Quantitative analogy between polymer-grafted nanoparticles and patchy particles.

We establish a quantitative analogy between polymer grafted nanoparticles (PGNPs) and patchy nanoparticles (NPs). Over much of the experimentally relevant parameter space, we show that PGNPs behave quantitatively like Janus NPs, with the patch size having a universal dependence on the number of grafts and the ratio of the size of the NPs to the grafted chain size. The widely observed anisotropic self-assembly of PGNP into superstructures can thus be understood through simple geometric considerations of single patch models, in the same spirit as the geometry-based surfactant models of Israelachvili.

CAN CULLING TO PREVENT MONKEYPOX INFECTION BE COUNTER-PRODUCTIVE? SCENARIOS FROM A THEORETICAL MODEL

In the last two decades, monkeypox outbreaks in human populations in Africa and North America have reminded us that smallpox is not the only poxvirus with potential to cause harm in human populations. Monkeypox transmission is sustained in animal reservoirs, and animal–human contacts are responsible for sporadic outbreaks in humans. Here, we develop and analyze a deterministic epizootic (animal-based) transmission model capturing disease dynamics in an animal population, disease dynamics in an age-structured human population, and their coupling through animal–human contacts. We develop a single-patch model as well as a two-patch meta-population extension. We derive mathematical expressions for the basic reproduction number, which governs the likelihood of a large outbreak. We also investigate the effectiveness of culling strategies and the impact of changes in the animal–human contact rate. Numerical analysis of the model suggests that, for some parameter values, culling can actually have the counter-productive outcome of increasing monkeypox infection in children, if animal reproduction is a density-dependent process. The likelihood of this happening, as well as the prevalence of monkeypox in humans, depends sensitively on the animal–human contact rate. We also find that ignoring age structure in human populations can lead to overestimating the transmissibility of monkeypox in humans. The effectiveness of monkeypox control strategies such as culling can strongly depend on the details of demography and epidemiology in the animal reservoirs that sustain it. Therefore, to better understand how to prevent and control monkeypox outbreaks in humans, better empirical data from wild animal populations where monkeypox is endemic must be collected, and these data must be incorporated into highly structured theoretical models.

Dispersal effects on a discrete two-patch model for plant–insect interactions

A two-patch discrete time plant–insect model coupled through insect dispersal is studied. The model is based on three different phases: Plant growth is followed by the dispersal of insects followed by insect attacks. Our objective is to understand how different intensities of dispersal impact both local and global population dynamics of the two-patch model. Special attention is paid to two situations: When the single-patch model (i.e., in the absence of dispersal) is permanent and when the single-patch model exhibits Allee-like effects. The existence and stability of synchronous and asynchronous dynamics between two patches is explored. If the single-patch system is permanent, the permanence of the system in two patches is destroyed by extremely large dispersals and large attacking rates of insects, thus creating multiple attractors. If the single-patch model exhibits Allee-like effects, analytical and numerical results indicate that small intensity of dispersals can generate source–sink dynamics between two patches, while intermediate intensity of dispersals promote the extinction of insects in both patches for certain parameter ranges. Our study suggests a possible biology control strategy to stop the invasion of a pest by controlling its migration between patches.

Asynchronous and Synchronous Dispersals in Spatially Discrete Population Models

This study is on the role of synchronous and asynchronous dispersals in a discrete-time single-species population model with dispersal between two patches, where predispersal dynamics are compen- satory or overcompensatory and dispersal is synchronous or asynchronous or mixed synchronous and asynchronous. It is known that single-species dispersal-linked population models behave as single- species single-patch models whenever all predispersal local dynamics are compensatory and dispersal is synchronous. However, the dynamics of the corresponding model connected by asynchronous and mixed synchronous-asynchronous dispersals depend on the dispersal rates, intrinsic growth rates, and the parameter that models the possible modes of dispersal. The species becomes extinct on at least one patch when the asynchronous dispersal rates are high, while it persists when the rates are low. In mixed synchronous-asynchronous systems, depending on the model parameters, the pioneer species either becomes extinct on all patches or persists on all patches. Overcompensatory predispersal dynamics with synchronous dispersal can lead to multiple attractors with fractal basin boundaries. However, the associated models with either asynchronous or mixed synchronous and asynchronous dispersals exhibit multiple attractors with fewer numbers of distinct attractors. That is, the long-term dynamics of synchronous dispersal-linked systems can be more sensitive to initial population sizes than that of the corresponding asynchronous and mixed synchronous-asynchronous systems. Also, synchronous, asynchronous, and mixed synchronous-asynchronous dispersals can stabilize the local patch dynamics from overcompensatory to compensatory dynamics. In our mixed synchronous-asynchronous model, the dominant mode of dispersal usually drives the dynam- ics of the full system.

Is mean squared error a consistent indicator of accuracy for spatially structured demographic models?

Animal demographic models are used in many disciplines and it is well-understood that their results and interpretation depend greatly on their structure and complexity. However, when models are constructed using data compiled from multiple sources it is difficult to objectively assess optimal structure and complexity. We explore the use of mean squared error (MSE) to identify optimal model structure. To illustrate, we compare the performance of a single-patch model versus a two-patch model for a migratory songbird in a spatially structured environment. In the single-patch model, mean fecundity across patches is estimated, and movement between patches is ignored. In the two-patch model, patch-specific reproductive rates are estimated, which requires the estimation and use of rates of movement between patches. MSE performed well as an indicator of optimal model structure, consistently conforming to our intuition about which model should be preferred under different combinations of model parameters. We show that (1) when dispersal offsets patch-specific demography, the single-patch model will always be favored, (2) with good estimates of vital rates, only modest sample sizes for dispersal are required to make the two-patch model more accurate, and (3) the same factors that result in large bias in the simple model also result in large sampling error for the two-patch model. Finally, our analyses also suggest that erroneous conclusions about optimal model complexity can be reached when additional structure is added to an already poorly parameterized model and we recommend a stepwise approach to the study of model complexity.

Two-patch dispersal-linked compensatory-overcompensatory spatially discrete population models

We study the role of asynchronous and synchronous dispersals on discrete-time two-patch dispersal-linked population models, where the pre-dispersal local patch dynamics are of mixed compensatory and overcompensatory types. Single-species dispersal-linked models behave as single-species single-patch models whenever all pre-dispersal local patch dynamics are compensatory and dispersal is synchronous. However, the dynamics of the corresponding two-patch population model connected by asynchronous dispersal depends on the dispersal rates. The species goes extinct on at least one patch when the asynchronous dispersal rates are high, while it persists when the rates are low. We use numerical simulations to show that in both synchronous and asynchronous mixed compensatory and overcompensatory systems, symmetric and asymmetric dispersals can control and impede the onset of cyclic population oscillations via period-doubling reversal bifurcations. Also, we show that in mixed systems both asynchronous and synchronous dispersals are capable of altering the pre-dispersal local patch dynamics from overcompensatory to compensatory dynamics. Dispersal-linked population models with ‘unstructured’ overcompensatory pre-dispersal local dynamics connected by synchronous dispersal can generate multiple attractors with fractal basin boundaries. However, mixed compensatory and overcompensatory systems appear to exhibit single attractors and not coexisting (multiple) attractors.

SIS Epidemic Models with Multiple Pathogen Strains

The dynamics of discrete-time SIS epidemic models with multiple pathogen strains are studied. The population infected with these strains may be confined to one geographic region or patch or may disperse between two patches. The models are systems of difference equations. It is the purpose of this investigation to study the persistence and extinction dynamics of multiple pathogen strains in a single patch and in two patches. It is shown for the single patch model that the basic reproduction number determines which strain dominates and persists. The strain with the largest basic reproduction number is the one that persists and outcompetes all other strains, provided its magnitude is greater than one. However, in the two-patch epidemic model, both the dispersal probabilities and the basic reproduction numbers for each strain determine whether a strain persists. With two patches, there is a greater chance that more than one strain will coexist. Analytical results are complemented with numerical simulations to help illustrate both competitive exclusion and coexistence of pathogens strains within the host population.

Multiple Attractors in Juvenile-adult Single Species Models

The role of age-structure and the Allee effect in generating multiple attractors in juvenile-adult single species single patch discrete-time models without dispersal are studied. In the presence of the Allee effect juvenile-adult single patch models support multiple attractors. However, in the absence of the Allee effect single attractors are supported when the dynamics are compensatory while multiple attractors are supported under overcompensatory dynamics. When the governing dynamics are compensatory, the boundaries of the basins of attraction have simple structure while complicated fractal basin boundaries are supported under overcompensatory dynamics.

Discrete-time Metapopulation Dynamics and Unidirectional Dispersal

The effects of unidirectional dispersal on single pioneer species discrete-time metapopulations where the pre-dispersal local patch dynamics are of the same (compensatory or overcompensatory) or mixed (compensatory and overcompensatory) types are studied. Single-species unidirectional metapopulation models behave as single-species single-patch models whenever all pre-dispersal local patch dynamics are compensatory and the dispersal rate is low. The pioneer species goes extinct in at least one patch when the dispersal rate is high, while it persists when the rate is low. Unidirectional dispersal can generate multiple attractors with fractal basin boundaries whenever the pre-dispersal local patch dynamics are overcompensatory, and is capable of altering the local patch dynamics in mixed systems from compensatory to overcompensatory dynamics and vice versa.

Interplay between Local Dynamics and Dispersal in Discrete-time Metapopulation Models

The effects of synchronous dispersal on discrete-time metapopulation dynamics with local (patch) dynamics of the same (compensatory or overcompensatory) or mixed (compensatory and overcompensatory) types are explored. Single-species metapopulation models behave as single-species single-patch models, whenever all local patches are governed by compensatory dynamics. Dispersal gives rise to multiple attractors with complex basin structures, whenever some local patches are under overcompensatory dynamics. In mixed systems, dispersal is capable of altering the local dynamics from compensatory to overcompensatory dynamics and vice versa. Examples are provided of metapopulation models supporting multiple attractors with intermingled basins of attraction.