A numerical method for a nonlinear structured population model with an indefinite growth rate coupled with the environment

A numerical method is developed for a general structured population model coupled with the environment dynamics over a bounded domain where the individual growth rate changes sign. Sign changes notably exhibit nonlocal dependence on the population density and environmental factors (e.g., resource availability and other habitat variables). This leads to a highly nonlinear PDE describing the time‐evolution of the population density coupled with a nonlinear‐nonlocal system of ODEs describing the environmental time‐dynamics. Stability of the finite‐difference numerical scheme and its convergence to the unique weak solution are proved. Numerical experiments are provided to demonstrate the performance of the finite difference scheme and to illustrate a range of biologically relevant potential applications.     

POPULATION‐LEVEL ANALYSIS AND VALIDATION OF AN INDIVIDUAL‐BASED CUTTHROAT TROUT MODEL

ABSTRACT. An individual‐based model of stream trout is analyzed by testing its ability to reproduce patterns of population‐level behavior observed in real trout: (1) “self‐thinning,” a negative power relation between weight and abundance; (2) a “critical period” of density‐dependent mortality in young‐of‐the‐year; (3) high and age‐specific inter‐annual variability in abundance; (4) density dependence in growth; and (5) fewer large trout when pool habitat is eliminated. The trout model successfully reproduced these patterns and was useful for evaluating their theoretical basis. The model analyses produced new explanations for some field observations and indicated that some patterns are less general than field studies indicate. The model did not reproduce field‐observed patterns of population variability by age class, discrepancies potentially explained by site differences, predation mortality being more stochastic than the model assumes, or uncertainty in the field study's age estimates.     

A METAPOPULATION MODEL FOR FERAL HOGS IN GREAT SMOKY MOUNTAINS NATIONAL PARK

Feral Hogs (Sus scrofa) are an invasive species that have occupied the Great Smoky Mountains National Park since the early 1900s. Recent studies on vegetation, mast, and harvest history were important for our work. Using these data, a model with discrete time and space was formulated to represent the feral hog dynamics in the Park. Management strategies and key characteristics of the population were investigated. The model uses observed mast variation to help govern population dynamics and results indicate that Park control efforts have limited the growth of the population.     

AN INDIVIDUAL‐BASED MODEL FOR FERAL HOGS IN GREAT SMOKY MOUNTAINS NATIONAL PARK

The expansion of feral hog (Sus scrofa) populations in the United States has resulted in increased efforts to develop and implement control strategies designed to minimize the impacts done by this invasive species. We describe an individual‐based model for feral hogs in Great Smoky Mountains National Park (GSMNP). The objectives of the model are to provide an understanding of the population dynamics of this feral hog population and to determine the efficacy of the annual harvest as a population control method. Results suggest that the dynamics of the population are driven by fall hard mast production and the GSMNP harvests currently limit growth of the population, but these control efforts have not reduced the population.     

MODELING THE DEPLETION OF A RENEWABLE RESOURCE BY POPULATION AND INDUSTRIALIZATION: EFFECT OF TECHNOLOGY ON ITS CONSERVATION

Abstract In this paper, a nonlinear mathematical model is proposed and analyzed to study the depletion of a renewable resource by population and industrialization with resource‐dependent migration. The effect of technology on resource conservation is also considered. In the modeling process, four variables are considered, namely, density of a renewable resource, population density, density of industrialization, and technological effort. Both the growth rate and carrying capacity of resource biomass, which follows logistic model, are assumed to be simultaneously depleted by densities of population and industrialization but it is conserved by technological effort. It is further assumed that densities of population and industrialization increase due to increase in the density of renewable resource. The growth rate of technological effort is assumed to be proportional to the difference of carrying capacity of resource biomass and its current density. The model is analyzed by using the stability theory of differential equations and computer simulation. The model analysis shows that the biomass density decreases due to increase in densities of population and industrialization. It decreases further as the resource‐dependent industrial migration increases. But the resource may never become extinct due to population and industrialization, if technological effort is applied appropriately for its conservation.     

A mathematical model of an anthrax epizoötic in the Kruger National Park

An outbreak of anthrax in a single homogeneous geographical region is modelled. The spread of the disease to neigbouring regions is also simulated and the model is compared with data from the 1970 epizoötic in the Kruger National Park.Two possible driving mechanisms for the epizoötic are proposed and examined, viz. environmental contamination or direct contact between vulnerable animals and fresh carcases. In the latter case, it is shown that the model exhibits a threshold effect, in that an epizoötic will only arise if the vulnerable population exceeds a threshold value which is determined in terms of one of the parameters of the model.     

A FULL ANNUAL CYCLE MODELING FRAMEWORK FOR AMERICAN BLACK DUCKS

American black ducks (Anas rubripes) are a harvested, international migratory waterfowl species in eastern North America. Despite an extended period of restrictive harvest regulations, the black duck population is still below the population goal identified in the North American Waterfowl Management Plan (NAWMP). It has been hypothesized that density‐dependent factors restrict population growth in the black duck population and that habitat management (increases, improvements, etc.) may be a key component of growing black duck populations and reaching the prescribed NAWMP population goal. Using banding data from 1951 to 2011 and breeding population survey data from 1990 to 2014, we developed a full annual cycle population model for the American black duck. This model uses the seven management units as set by the Black Duck Joint Venture, allows movement into and out of each unit during each season, and models survival and fecundity for each region separately. We compare model population trajectories with observed population data and abundance estimates from the breeding season counts to show the accuracy of this full annual cycle model. With this model, we then show how to simulate the effects of habitat management on the continental black duck population.     

A general model of size-dependent population dynamics with nonlinear growth rate

We study a general model of size-dependent population dynamics with nonlinear growth rate. The existence of a local solution is shown by using Schauder's fixed point theorem. Uniqueness and continuous dependence on initial data are also established under additional conditions.     

Fast game dynamics coupled to slow population dynamics: A single population with hawk-dove strategies

We study a model of a population subdivided into two subpopulations corresponding to hawk and dove tactics. It is assumed that the hawk and dove individuals compete for a resource every Day, I.e., at a fast time scale. This fast part of the model is coupled to a slow part which describes the growth of the subpopulations and the long term effects of the encounters between the individuals which must fight to have an access to the resource. We aggregate the model into a single equation for the total population. It is shown that in the case of a constant game matrix, the total population grows according to a logistic curve whose τ and K parameters are related to the coefficients of the hawk-dove game matrix. Our result shows that high equilibrium density populations are mainly doves, whereas low equilibrium density populations are mainly hawks. We also study the case of a density dependent game matrix for which the gain is linearly decreasing with the total density.     

GRAY WOLF POPULATION PROJECTION WITH INTRASPECIFIC COMPETITION

Competition effects are incorporated into a model of wolf‐population dynamics. A classic single‐state model is augmented into a dual‐state mapping of the evolution of the size of wolf packs and the number of wolf packs. This dual‐state model, unlike the single‐state density dependent model, is amenable to analyzing intraspecific competition. The single‐state, dual‐state and dual‐state with competition models are estimated using Yellowstone National Park (YNP) data on wolf populations and pack structures from 1996 to 2011. The dynamic properties of each model are examined under an array of harvesting policies. Results suggest that intraspecific competition matters when projecting wolf populations. Wolf pack removal has competition‐reducing effects from added territory availability, making populations more sensitive to pack size reduction than reduction in the number of packs. This research suggests that wildlife managers may consider monitoring the composition of wolf kills throughout a harvesting season, adaptively adjusting harvesting quotas and delineating harvesting zones over a few pack territories rather than spreading these effects evenly across all packs.     

A stage-structured food chain model with stage dependent predation: Existence of codimension one and codimension two bifurcations

Stage-structured predator–prey models exhibit rich and interesting dynamics compared to homogeneous population models. The objective of this paper is to study the bifurcation behavior of stage-structured prey–predator models that admit stage-restricted predation. It is shown that the model with juvenile-only predation exhibits Hopf bifurcation with the growth rate of the adult prey as the bifurcation parameter; also, depending on parameter values, a stable limit cycle will emerge, that is, the bifurcation will be of supercritical nature. On the other hand, the analysis of the model with adult-stage predation shows that the system admits a fold-Hopf bifurcation with the adult growth rate and the predator mortality rate as the two bifurcation parameters. We also demonstrate the existence of a unique limit cycle arising from this codimension-2 bifurcation. These results reveal far richer dynamics compared to models without stage-structure. Numerical simulations are done to support analytical results.     

A MICROFOUNDATION OF PREDATOR‐PREY DYNAMICS

ABSTRACT. Predator‐prey relationships account for an important part of all interactions betweenspecies. In this paper we provide a microfoundation for such predator‐prey relations in afood chain. Basic entities of our analysis are representative organisms of species modeled similar to economic households. With prices as indicators of scarcity, organisms are assumed to behave as if they maximize their net biomass subject to constraints which express the organisms' risk of being preyed upon during predation. Like consumers, organisms face a ‘budget constraint’ requiring their expenditure on prey biomass not to exceed their revenue from supplying own biomass. Short‐run ecosystem equilibria are defined and derived. The net biomass acquired by the representative organism in the short term determines the positive or negative population growth. Moving short‐run equilibria constitute the dynamics of the predator‐prey relations that are characterized in numerical analysis. The population dynamics derived here turn out to differ significantly from those assumed in the standard Lotka‐Volterra model.     

MODELING THE EFFECTS OF PREDATOR EXCLOSURES ON A WESTERN SNOWY PLOVER POPULATION

Abstract In Humboldt County, California, nest exclosures were used between 2001 and 2006 to reduce the predation of eggs aiding in the recovery of a threatened population segment of western snowy plovers (Charadrius nivosus). Due to a sudden increase in adult mortality in 2006, field biologists abandoned the use of nest exclosures. This paper describes a discrete‐time stochastic model designed to compare two different management strategies (with and without exclosures) to predict the change in the plover population. The model uses beta distributions to model demographic parameters, and whenever possible, these distributions were fit to survey data of these populations. The model shows nest exclosures to be effective in increasing chick fledging rates. However, the model also shows that an increase in adult mortality potentially caused by the nest exclosures would counteract this increase in fledging rates. The model predicts that there will be a net negative effect on the population if these exclosures reduce adult survival to 90% of its unexclosed rate. The model also demonstrates that the population is dependent on immigration.     

Modelling the depletion of forestry resources by population and population pressure augmented industrialization

In this paper, a nonlinear mathematical model is proposed and analysed to study the depletion of forestry resources caused by population and population pressure augmented industrialization. It is shown that the equilibrium density of resource biomass decreases as the equilibrium densities of population and industrialization increase. It is found that even if the growth of population (whether intrinsic or by migration) is only partially dependent on resource, still the resource biomass is doomed to extinction due to large population pressure augmented industrialization. It is noted that for sustained industrialization, control measures on its growth are required to maintain the ecological stability.     

On the influence of aggressive driving behavior on the dynamics of traffic flow

The car following model under discussion is a system of ordinary differential equations describing N cars moving around a circular road. We extend the stability analysis of the model in two directions: variable reaction time (depending on the headway) and an additional part describing the aggressive driving tendency. We study the global bifurcation dynamics and in particular its dependence on these two new ingredients. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Lotka-Volterra's model and migrations: Breaking of the well-known center

This paper is devoted to the study of the effect of individual behavior on the Lotka-Volterra predation. We assume that the individuals have many activities in a day for example. Each population is subdivided into subpopulations corresponding to different activities. In order to be clear, I have chosen the case of two activities for each population. We assume that the activities change is faster than the other processes (reproduction, mortality, predation…). This means that we consider population in which the individuals change their activities many times in a day while the reproduction and the predation effects are sensible after about ten days, for example. We use the aggregation method developed in [1] to obtain the global dynamics. Indeed, we start with a micro-model governing the micro-variables, which are the subpopulation densities; the aggregation method permits us to obtain a simpler system governing the macro-variables, which are the global population densities. Furthermore, this method allows us to observe emergence of the dynamics. Indeed, the method implies that the dynamics of the micro-system is close to an invariant manifold after a short time. We show that the dynamics on this manifold is a perturbation of the well-known center of the Lotka-Volterra model. Finally, we prove that a weak change of behavior can lead to a subcritical Hopf bifurcation in the global dynamics.     

VOLE POPULATION DYNAMICS UNDER THE INFLUENCE OF SPECIALIST AND GENERALIST PREDATION

Abstract To understand the impact of predation by different types of predators on the vole population dynamics, we formulate a three differential equation model describing the population dynamics of voles, the “specialist predator” and the “generalist predator.” First we perform a local stability study of the different steady states of the basic model and deduce that the predation rates of the “specialist” as well as the “generalist” predator are the main parameters controlling the existence/extinction criteria of the concerned populations. Next we analyze the model from a thermodynamic perspective and study the thermodynamic stability of the different equilibria. Finally using stochastic driving forces, we incorporate the exogenous factor of environmental forcing and investigate the stochastic stability of the system. We compare the stability criteria of the different steady states under deterministic, thermodynamic and stochastic situations. The analysis reveals that when the “specialist” and the “generalist” predator are modeled separately, the system exhibits rich dynamics and the predation rates of both types of predators play a major role in controlling vole oscillation and/or stability. These findings are also seen to resemble closely with the observed behavior of voles in the natural setting. Numerical simulations are carried out to illustrate analytical findings.     

A Multivariate CLT for Decomposable Random Vectors with Finite Second Moments

Stein's method is used to derive a CLT for dependent random vectors possessing the dependence structure from Barbour et al. J. Combin. Theory Ser. B 47, 125–145, but under the assumption of second moments only. This allows us to derive Lindeberg–Feller type theorems for sums of random vectors with certain dependence structures. We apply the main theorem to the study of three problems: local dependence, random graph degree statistics and finite population statistics. In particular, we consider U-statistics of independent observations as well as of observations drawn without replacement.     

JOINTLY‐DETERMINED ECOLOGICAL THRESHOLDS AND ECONOMIC TRADE‐OFFS IN WILDLIFE DISEASE MANAGEMENT

ABSTRACT. We investigate wildlife disease management, in a bioeconomic framework, when the wildlife host is valuable and disease transmission is density‐dependent. Disease prevalence is reduced in density‐dependent models whenever the population is harvested below a host‐density threshold a threshold population density below which disease prevalence declines and above which a disease becomes epidemic. In conventional models, the threshold is an exogenous function of disease parameters. We consider this case and find a steady state with positive disease prevalence to be optimal. Next, we consider a case in which disease dynamics are affected by both population controls and changes in human‐environmental interactions. The host‐density threshold is endogenous in this case. That is, the manager does not simply manage the population relative to the threshold, but rather manages both the population and the threshold. The optimal threshold depends on the economic and ecological trade‐offs arising from the jointly‐determined system. Accounting for this endogene‐ity can lead to reduced disease prevalence rates and higher population levels. Additionally, we show that ecological parameters that may be unimportant in conventional models that do not account for the endogeneity of the host‐density threshold are potentially important when host density threshold is recognized as endogenous.