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Abstract
We consider the Fisher equation and its generalization for an asocial population in a linear, hostile environment. The method of center manifold analysis is used to obtain the time-dependent solution of the former, nonlinear equation. The correct critical habitat size is obtained; in addition, the result for the steady state central density compares favorably with the exact result for relatively large population sizes (up to one half of the carrying capacity). For a model of asocial growth we obtain the expanded criteria for survival. This includes the habitat size, the population size at which positive growth begins, and also the minimum initial central density.
The spatial critical shelter sizes above which populations would survive are investigated for the infection of hantavirus among rodent populations surrounded by a deadly environment. We show that the critical shelter sizes for the infected population and the susceptible population are different due to symmetry breaking in the reproduction and the transmission processes. Therefore, there exists a shelter size gap within which the infected population becomes extinct while only the susceptible population survives. With the field data reported in the literature, we estimate that, if one confines the rodent population within a stripe region surrounded by a deadly environment with the shorter dimension between 335.5±27.2m and 547.9±78.3m, the infected population would become extinct. In addition, we introduce two factors that influence the movement of rodents, namely, the spatial asymmetry of the landscape and the sociality of rodents, to study their effects on the shelter size gap. The effects on the critical size due to environmental bias are twofold: it enhances the overall competition among rodents which increases the critical size, but on the other hand it promotes the spread of the hantavirus which reduces the critical size for the infected population. On the contrary, the sociality of rodents gives rise to a more localized population profile which promotes the spread of the hantavirus and reduces the shelter size gap. The results shed light on a possible strategy of eliminating hantavirus while preserving the integrity of food webs in ecosystems.
The talk consists of two parts. In the first, we have demonstrated enhancement of nonlinear signals and suggested a new approach that extends Raman spectroscopy to media which has a high level of scattering. The stimulated Raman scattering has an advantage of relaxing phase-matching condition which is difficult or even impossible to meet under condition of strong scattering. The second topic is related to enhanced optico-mechanical coupling that occurs when the group velocity of light is ultra-slow. We show theoretically that, due to dragging of the light by the mechanical motion of the medium, it is possible to control phase-matching condition.
We theoretically predict strong coherent scattering in the backward direction from forward propagating fields only. It is achieved by applying laser fields with proper chosen detunings and an approprate spatial phase to excite a molecular vibrational coherence that corresponds to a wave propagating backwards. The applications of the technique to coherent scattering and remote sensing are discussed.
The article illustrates recent experiments in which light is slowed, frozen, reversed, and stored in hot atomic vapors via electromagnetically induced transparency.
The problem of nonlinear Hamiltonian oscillators is one of the classical questions in physics. When an analytic solution is not possible, one can resort to obtaining a numerical solution or using perturbation theory around the linear problem. We apply the Fourier series expansion to find approximate solutions to the oscillator position as a function of time as well as the period-amplitude relationship. We compare our results with other recent approaches such as variational methods or heuristic approximations, in particular the Ren-He's method. Based on its application to the Duffing oscillator, the nonlinear pendulum and the eardrum equation, it is shown that the Fourier series expansion method is the most accurate.
We present an approach to determining the speed of wave-front solutions to reaction-transport processes. This method is more accurate than previous ones. This is explicitly shown for several cases of practical interest: (i) the anomalous diffusion reaction, (ii) reaction diffusion in an advective field, and (iii) time-delayed reaction diffusion. There is good agreement with the results of numerical simulations.
We extend the previous results, describing the population dispersal that occurs in some insects and small animal populations when this process is not strictly random, by including both the downgradient diffusion and the full Pearl-Verhulst logistic growth term in the equation of evolution. Motivated by the increasing fragmentation of natural habitats that is the result of human activities, we consider a finite habitat surrounded by a hostile environment. Previous work [Phys Rev. E 62, 4032 (2000)] considered only the case of an unbounded habitat, obviating issues concerned with the critical habitat size and the adoption of strategies best suited to achieve lower densities by dispersal through downgradient diffusion.
We study the conditions for extinction and survival of populations living in a patch surrounded by a hostile environment. We find analytic expressions for the steady states when population dynamics is described by diffusion and reaction is driven by compensation, depensation, or critical depensation growths. The role of initial population density is studied, and the complete bifurcation diagrams are constructed and validated numerically for the three cases studied.
We consider the Dirichlet problem for semilinear parabolic equations of the form ut = Δu + h(u, ∇u), t > 0, x ∈ Ω, (1) on a smooth bounded domain Ω ⊂ ℝN, N ≥ 2. The nonlinearity h : ℝ × ℝN → ℝ is assumed continuously difierentiable and spatially homogeneous (that is, independent of x). Applying the method of realization of vector fields, we show that (1) can generate very complicated dynamics. For example, choosing h and Ω suitably, one achieves that (1) has an invariant manifold W the flow on which has a transverse homoclinic orbit to a periodic orbit. Another choice of h and Ω yields a transitive Anosov flow on W. The invariant manifold W can have any dimension n ≥ 3 while N = dimΩ is fixed (greater then 1). In particular, the solutions of (1) can have ω-limit sets of arbitrarily high dimensions even though Ω has low dimension. We describe the method of realization of vector fields in detail, first in an abstract context and then for the above specific class of equations. A large piece of our analysis is devoted to the study of perturbations of multiple eigenvalues and corresponding eigenfunctions of the Laplacian under perturbation of the domain.
The transient stage of the random dispersal of logistic populations is investigated, using a Sturm-Liouville series leading
to an infinite system of non-linear integral equations. These equations are then solved via a successive approximation scheme.
R. A. Fisher's (steady-state) velocity of advance paradox is discussed. An illustrative example is worked to the second order
of approximation.
An approximation method using a sine function is used to solve the second degree growth equation for the case in which an
organism may simultaneously become dispersed throughout a uniform region. The resulting expression for a special case is compared
with the expression obtained by R. Barakat (1959,Bull. Math. Biophysics,21, 141–51), giving the first two terms, by an iterative, procedure. The agreement is satisfactory.
Asocial populations, for which the population growth function at small densities is negative (due either to paucity of reproductive opportunity or to other causes) are fundamentally different in character from logistic populations. This paper establishes the spatial distributions which are steady under the combined influence of birth, death, and one-dimensional dispersion, together with a general analysis of their stability.The distributions possible for a given growth function F(N) (F is population density growth rate at normalized density N) depend on whether ∫01FdN⋛0 The principal ones of interest are spatially periodic (e.g. in a finite region with reflecting boundaries) or aperiodic (e.g. in a finite region with absorbing boundaries).The theory of stability to small disturbances is established with the aid of Sturm-Liouville comparison theorems. It is proved for arbitrary F(N) (not only for asocial populations) that all periodic and all “intersecting” aperiodic distributions are unstable, and that all “non-intersecting” aperiodic distributions are stable. For asocial populations with ∫01FdN>0, the aperiodic distribution with central density N0 = N0† and minimum range is neutrally stable. Distributions with N0 >N0† are stable, and all others (apart from the trivial case N0 = 0) are unstable.This result that the population distribution can be stable only if the habitat and the central density exceed certain critical sizes has important and timely ecological implications.
The random-walk problem is adopted as a starting point for the analytical study of dispersal in living organisms. The solution is used as a basis for the study of the expanson of a growing population, and illustrative examples are given. The law of diffusion is deduced and applied to the understanding of the spatial distribution of population density in both linear and two-dimensional habitats on various assumptions as to the mode of population growth or decline. For the numerical solution of certain cases an iterative process is described and a short table of a new function is given. The equilibrium states of the various analytical models are considered in relation to the size of the habitat, and questions of stability are investigated. A mode of population growth resulting from the random scattering of the reproductive units in a population discrete in time, is deduced and used as a basis for study on interspecific competition. The extent to which the present analytical formulation is applicable to biological situations, and some of the more important biological implications are briefly considered.