EDGE PERMEABILITY AND POPULATION PERSISTENCE IN ISOLATED HABITAT PATCHES

ABSTRACT. Population persistence in isolated habitat fragments is investigated using integrodifference equations. The propensity of individual dispersers encountering the boundary of the patch to emigrate is defined by edge permeability. A dispersal model incorporating movement, settlement and edge permeability defines dispersal success as a function of a disperser's starting location. This dispersal model is used to generate dispersal kernels for integrodifference equation models, analysis of which gives a condition for population persistence in terms of edge permeability, patch size and average dispersal distance. An approximation reduces the spatial problem to a simple nonspatial model that can be easily analyzed.     

Numerical simulation of reaction‐diffusion telegraph systems in unbounded domains

We present a boundary element method for computing numerical solutions of the reaction‐diffusion telegraph equation in unbounded domains. This technique does not need artificial boundary conditions at the computational domain and uses a new algorithm to compute the Fourier transform, the convolution theorem, and the fact that the exact solution of the telegraph equation can be written as an integral transform in terms of the fundamental solution. We use the logistic growth model to find how the population of an organism evolves according to its growth rate. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 326–335, 2015     

Population model with diffusion and supplementary forest resource in a two-patch habitat

A mathematical model is proposed to study the role of supplementary self-renewable resource on species population in a two-patch habitat. It is assumed that the density of forest resource biomass is governed by the logistic equation in both the regions but with the different intrinsic growth rate but the same carrying capacity in the entire habitat. It is further assumed that the densities of species population is also governed by the generalized logistic equations in both the regions but with different growth rates and carrying capacities. It is shown that the steady state solutions are positive, monotonic and continuous under both reservoir and no-flux boundary conditions. The linear and non-linear asymptotic stability conditions of non-uniform steady state are compared with the case of the model with and without diffusion in a homogeneous habitat.     

A stationary population model with an interior interface-type boundary

We propose a stationary system that might be regarded as a migration model of some population abandoning their original place of abode and becoming part of another population, once they reach the interface boundary. To do so, we show a model where each population follows a logistic equation in their own environment while assuming spatial heterogeneities. Moreover, both populations are coupled through the common boundary, which acts as a permeable membrane on which their flow moves in and out. The main goal we face in this work will be to describe the precise interplay between the stationary solutions with respect to the parameters involved in the problem, in particular the growth rate of the populations and the coupling parameter involved on the boundary where the interchange of flux is taking place.     

A semigroup approach to age-dependent population dynamics with time delay

We study the McKendrick type models of population dynamics with instantaneous time delay in the birth rate. The models involve first order partial differential equations with nonlocal and delayed boundary conditions. We show that a semigroup can be associated

to it and identify the infinistimal generator. Its spectral properties are analyzed yielding large time behaviour. An interesting result is that if the total population converges to an equilibrium it will converge to it in an oscillatory fashion. Further, we consider a logistic ara age-dependent model with delay. A nonlinear semigroup is constructed to describe the evolution of the population. Existence and uniqueness of the nonlinear equation are proved.     

Nonclassical boundary value problem for the transport equation

We consider a nonclassical boundary value problem for the transport equation. The particle transport process is described by a stationary linear integro-differential equation, and the outgoing particle flux density on some part of the boundary is specified as the boundary condition. We find the particle flux density for a given outgoing flux and known coefficients of the equation. We show that, under some restrictions on the medium, there exists a unique solution of the problem, which can be represented by an infinite convergent series.     

Incorporating boundary conditions in a stochastic volatility model for the numerical approximation of bond prices

In this paper, we consider a two-factor interest rate model with stochastic volatility, and we assume that the instantaneous interest rate follows a jump-diffusion process. In this kind of problems, a two-dimensional partial integro-differential equation is derived for the values of zero-coupon bonds. To apply standard numerical methods to this equation, it is customary to consider a bounded domain and incorporate suitable boundary conditions. However, for these two-dimensional interest rate models, there are not well-known boundary conditions, in general. Here, in order to approximate bond prices, we propose new boundary conditions, which maintain the discount function property of the zero-coupon bond price. Then, we illustrate the numerical approximation of the corresponding boundary value problem by means of an alternative direction implicit method, which has been already applied for pricing options. We test these boundary conditions with several interest rate pricing models.     

Approximating the ideal free distribution via reaction-diffusion-advection equations

We consider reaction-diffusion-advection models for spatially distributed populations that have a tendency to disperse up the gradient of fitness, where fitness is defined as a logistic local population growth rate. We show that in temporally constant but spatially varying environments such populations have equilibrium distributions that can approximate those that would be predicted by a version of the ideal free distribution incorporating population dynamics. The modeling approach shows that a dispersal mechanism based on local information about the environment and population density can approximate the ideal free distribution. The analysis suggests that such a dispersal mechanism may sometimes be advantageous because it allows populations to approximately track resource availability. The models are quasilinear parabolic equations with nonlinear boundary conditions.     

Static stability analysis of a thin plate with a fixed trailing edge in axial subsonic flow: Possio integral equation approach

In this work, the static stability of a thin plate in axial subsonic airflow is studied using the framework of Possio integral equation. Specifically, we consider the cases when the plate’s leading edge is free and the plate’s trailing edge is either pinned or clamped. We formulate the problem under consideration using a partial differential equations (PDE) model and then linearize the model about the free stream velocity, density, and pressure, to enable analytical treatment. Based on the linearized model, we introduce a new derivation of a Possio integral equation that relates the pressure jump along the thin plate to the plate’s downwash. The steady state solution to the Possio equation is then used to account for the aerodynamic loads in the plate steady state governing equation resulting in a singular differential-integral equation which is transformed to a singular integral equation that represents the static aeroelastic equation of the plate. We verify the solvability of the static aeroelastic equation based on the Fredholm alternative for compact operators in Banach spaces and the contraction mapping theorem. By constructing solutions to the static aeroelastic equation and matching the nonzero boundary conditions at the trailing edge with the zero boundary conditions at the leading edge, we obtain characteristic equations for the free-clamped and free-pinned plates. The minimum solutions to the characteristic equations are the divergence speeds which indicate when static instabilities start to occur. We show analytically that free-pinned plates are statically unstable. We also construct, analytically, flow speed intervals that correspond to static stability regions for free-clamped plates. Furthermore, we resort to numerical computations to obtain an explicit formula for the divergence speed of free-clamped plates. Finally, we apply the obtained results on piezoelectric plates and we show that free-clamped piezoelectric plates are statically more stable than conventional free-clamped plates due to the piezoelectric coupling.     

On the Convergence Rate of the Boundary Penalty Method

The convergence rate of the boundary penalty finite element method is discussed for a model Poisson equation with inhomogeneous Dirichlet boundary conditions and a sufficiently smooth solution. It is proved that an optimal convergence rate can be achieved which agrees with the rate obtained recently in the numerical experiments by Utku and Carey.     

Diffusive logistic equation with constant yield harvesting and negative density dependent emigration on the boundary

The structure of positive steady state solutions of a diffusive logistic population model with constant yield harvesting and negative density dependent emigration on the boundary is examined. In particular, a class of nonlinear boundary conditions that depends both on the population density and the diffusion coefficient is used to model the effects of negative density dependent emigration on the boundary. Our existence results are established via the well-known sub-super solution method.     

Nonlinear dynamics of open marine population with space-limited recruitment: The case of mortality control

In this paper we develop a nonlinear extension for the open marine population model which has been proposed by Roughgarden et al. [Ecology 66 (1985) 54-67]. To avoid the negative population density, which is a drawback of the original model, we introduce a nonlinear mechanism that the mortality rate depends on the size of area occupied by the adult population. Then we give a rigorous mathematical framework to analyse the model equation, and we show sufficient conditions for stability and instability of the steady state. Our instability result suggests, as was proposed by Roughgarden et al., that there exists a sustained oscillation of the population density.     

Uncertainty quantification for random parabolic equations with nonhomogeneous boundary conditions on a bounded domain via the approximation of the probability density function

This paper deals with the randomized heat equation defined on a general bounded interval [L₁, L₂] and with nonhomogeneous boundary conditions. The solution is a stochastic process that can be related, via changes of variable, with the solution stochastic process of the random heat equation defined on [0,1] with homogeneous boundary conditions. Results in the extant literature establish conditions under which the probability density function of the solution process to the random heat equation on [0,1] with homogeneous boundary conditions can be approximated. Via the changes of variable and the Random Variable Transformation technique, we set mild conditions under which the probability density function of the solution process to the random heat equation on a general bounded interval [L₁, L₂] and with nonhomogeneous boundary conditions can be approximated uniformly or pointwise. Furthermore, we provide sufficient conditions in order that the expectation and the variance of the solution stochastic process can be computed from the proposed approximations of the probability density function. Numerical examples are performed in the case that the initial condition process has a certain Karhunen‐Loève expansion, being Gaussian and non‐Gaussian.     

An explicit staggered-grid method for numerical simulation of large-scale natural gas pipeline networks

We present an explicit second order staggered finite difference (FD) discretization scheme for forward simulation of natural gas transport in pipeline networks. By construction, this discretization approach guarantees that the conservation of mass condition is satisfied exactly. The mathematical model is formulated in terms of density, pressure, and mass flux variables, and as a result permits the use of a general equation of state to define the relation between the gas density and pressure for a given temperature. In a single pipe, the model represents the dynamics of the density by propagation of a non-linear wave according to a variable wave speed. We derive compatibility conditions for linking domain boundary values to enable efficient, explicit simulation of gas flows propagating through a network with pressure changes created by gas compressors. We compare our staggered grid method with an explicit operator splitting method and a lumped element scheme, and perform numerical experiments to validate the convergence order of the new discretization approach. In addition, we perform several computations to investigate the influence of non-ideal equation of state models and temperature effects on pipeline simulations with boundary conditions on various time and space scales.     

The asymptotic behaviour of the heat equation in a twisted Dirichlet-Neumann waveguide

We consider the heat equation in a straight strip, subject to a combination of Dirichlet and Neumann boundary conditions. We show that a switch of the respective boundary conditions leads to an improvement of the decay rate of the heat semigroup of the order of t−1/2. The proof employs similarity variables that lead to a non-autonomous parabolic equation in a thin strip contracting to the real line, that can be analysed on weighted Sobolev spaces in which the operators under consideration have discrete spectra. A careful analysis of its asymptotic behaviour shows that an added Dirichlet boundary condition emerges asymptotically at the switching point, breaking the real line in two half-lines, which leads asymptotically to the 1/2 gain on the spectral lower bound, and the t−1/2 gain on the decay rate in the original physical variables.This result is an adaptation to the case of strips with twisted boundary conditions of previous results by the authors on geometrically twisted Dirichlet tubes.     

Global existence of solutions to one-dimensional viscous quantum hydrodynamic equations

The existence of global-in-time weak solutions to the one-dimensional viscous quantum hydrodynamic equations is proved. The model consists of the conservation laws for the particle density and particle current density, including quantum corrections from the Bohm potential and viscous stabilizations arising from quantum Fokker-Planck interaction terms in the Wigner equation. The model equations are coupled self-consistently to the Poisson equation for the electric potential and are supplemented with periodic boundary and initial conditions. When a diffusion term linearly proportional to the velocity is introduced in the momentum equation, the positivity of the particle density is proved. This term, which introduces a strong regularizing effect, may be viewed as a classical conservative friction term due to particle interactions with the background temperature. Without this regularizing viscous term, only the nonnegativity of the density can be shown. The existence proof relies on the Faedo-Galerkin method together with a priori estimates from the energy functional.     

Singular limit of a competition–diffusion system with large interspecific interaction

We consider a competition–diffusion system for two competing species; the density of the first species satisfies a parabolic equation together with an inhomogeneous Dirichlet boundary condition whereas the second one either satisfies a parabolic equation with a homogeneous Neumann boundary condition, or an ordinary differential equation. Under the situation where the two species spatially segregate as the interspecific competition rate becomes large, we show that the resulting limit problem turns out to be a free boundary problem. We focus on the singular limit of the interspecific reaction term, which involves a measure located on the free boundary.     

A NUMERICAL STUDY OF THE SPRUCE BUDWORM REACTION‐DIFFUSION EQUATION WITH HOSTILE BOUNDARIES

ABSTRACT. One of the interesting single species reaction diffusion problems is the spruce budworm model describing insect dispersal behavior. In an earlier study, Singh et al. [7] considered the two‐dimensional spruce budworm model with density dependent diffusion balanced by an artificial wind equal to the population gradient. Here we extend the model by considering more realistic density dependent diffusion and advection with hostile boundaries. We solve this model using a splitting method in which advection, diffusion and reaction processes are separated. Various hostility conditions have been used at the boundary. The numerical results show that the population moves quickly to a steady state outbreak situation when the advective components due to the density dependent diffusion are included.     

INCORPORATION AND INFLUENCE OF VARIABILITY IN AN AGGREGATED FOREST MODEL

ABSTRACT. Variability influences ecological processes at various scales and is incorporated in different ways in forest models. The forest model Dis CFor M scales an individual based, stochastic forest patch model up to a height structured tree population model. To describe the variability arising from stochastic processes in the patch model, Dis CFor M uses theoretical random dispersions of trees in each height class over all patches. This yields a spatial distribution of light and consequently of light dependent process rates. Three major influences of variability on simulations are examined: site condition, patch to patch, and temporal environmental variability. Simulation studies and comparison with forest compositions from the Swiss National Forest Inventory reveal that these influences affect simulated forest dynamics, species composition, and biodiversity, depending on climatic boundary conditions and hence have to be taken into account in modeling.     

一个与硅氧化有关的粘性可压缩流体的自由边界问题

本文研究一个描述硅的氧化过程的自由边界问题．它的数学模型是一个可压缩的Ｎａｖｉｏｒ－Ｓｔｏｋｅｓ方程与一个抛物方程以及一个双曲方程的耦合，其中在自由边界上存在表面张力并且密度方程是非齐次的．本文将证明只要已知数据满足相容性条件，则上述问题有唯一局部强解．