DSpace collection: 第47卷第1期

DYNAMICS OF THE PREDATOR-PREY MODELS ON THE TWO-PATCH FRAGMENTED HABITAT WITH DISPERSAL

title: DYNAMICS OF THE PREDATOR-PREY MODELS ON THE TWO-PATCH FRAGMENTED HABITAT WITH DISPERSAL abstract: In this work, we consider the population-dispersal dynamics for predator-prey interactions in a two-patch environment. On each fragmented patch, there is a two-species predator-prey ecological system. It is assumed that the predator species are mobile. The existence and local dynamics of boundary equilibria and interior equilibria with respect to parameters are completely classified. Moreover, global extinction results are established analytically. In particular, the phenomenon of over-exploitation is also found in these discrete patches models. Finally, some biological interpretations are discussed.

REGIME SHIFT IN A PHYTOPLANKTON–PHOSPHORUS MODEL WITH VERTICAL STRUCTURE AND SEASONALITY

title: REGIME SHIFT IN A PHYTOPLANKTON–PHOSPHORUS MODEL WITH VERTICAL STRUCTURE AND SEASONALITY abstract: Many ecological systems are influenced by positive feedbacks between organisms and abiotic environments, which generates multiple stable equilibria of a mathematical model with a hysteresis structure. In addition, discontinuous shifts of system at equilibrium is predicted, which is often called regime shift in ecosystem sciences. This hysteresis structure is unfavorable from environmental management point of view, because the reconstruction of original equilibrium state requests much lower levels of external forcing. Mathematical models proposed in previous works are simple and mathematically tractable ([7],[2]).However, it is difficult to extrapolate from such simple models the occurrence likelihood of regime shift in natural environments since temporally dynamic features in ecology and physico-chemical environments, and spatial dimension are less explored in those models. In this study, we construct a realistic but mathematically tractable model of interaction between phytoplankton and phosphorus, which incorporates (1) 1-dimensional vertical structure of lake ecosystem and (2) seasonal periodic cycle of water mixing. We aim to understand the impact of changes in seasonality in various types of lakes on the occurrence of multiple attractors (periodic solution) and hysteresis structure.

ON THE EVOLUTIONARY STABILITY OF MALE HARASSMENT IN A COERCIVE MATING GAME

title: ON THE EVOLUTIONARY STABILITY OF MALE HARASSMENT IN A COERCIVE MATING GAME abstract: In many animals, males employ coercive mating strategies to help them maximize their expected number of offspring. In such systems, selection will favor behavioral adaptations in females that help them mitigate harassment costs and maximize their reproductive fitness. Previously, Bokides et al. [1] presented a model showing how male harassment strategies can co-evolve with female habitat preferences in a mating game. Their results indicated that if females dispersed freely between habitats where males were present and where males were excluded, selection could favor males who strategically harassed at high (or low) levels, depending on the proximity of their phenotype to a threshold level h∗h∗. This article is a continuation of that work addressing the questions of stability at equilibria where males harass at the threshold level (i.e., h∗h∗). We show these states are both locally and globally asymptotically stable. Further, we argue based on these results that h∗h∗ is an evolutionary stable male harassment level at which females will be ideally distributed to match the resource quality and social environments of their alternative habitats.

A REACTION-DIFFUSION SYSTEM AND ITS SHADOW SYSTEM DESCRIBING HARMFUL ALGAL BLOOMS

title: A REACTION-DIFFUSION SYSTEM AND ITS SHADOW SYSTEM DESCRIBING HARMFUL ALGAL BLOOMS abstract: The occurrence of harmful algal blooms (HAB) in ecosystems is a worldwide environmental issue that currently needs to be addressed. An attempt to theoretically understand the mechanism behind the formation of HAB has led to the proposal of a reaction-diffusion model of the Lotka--Volterra type. In particular, a shadow system, as a limiting system of the model in which the diffusion rate tends to infinity, has been proposed to study whether or not stable nonconstant equilibrium solutions of the system exist, because these solutions are mathematically associated with HAB. In this paper, we discuss the convergence property between solutions of the full system and its shadow system from the point of view of an evolutional problem.

LACK OF SYMMETRY IN LINEAR DETERMINACY DUE TO CONVECTIVE EFFECTS IN REACTION-DIFFUSION-CONVECTION PROBLEMS

title: LACK OF SYMMETRY IN LINEAR DETERMINACY DUE TO CONVECTIVE EFFECTS IN REACTION-DIFFUSION-CONVECTION PROBLEMS abstract: This paper is concerned with linear determinacy in monostable reaction- diffusion-convection equations and co-operative systems. We present sufficient conditions for minimal travelling-wave speeds (equivalent to spreading speeds) to equal values obtained from linearisations of the travelling-wave problem about the unstable equilibrium. These conditions involve both reaction and convection terms. We present separate conditions for non-increasing and non-decreasing travelling waves, called `right' and `left' conditions respectively, because of the asymmetry in propagation caused by the convection terms. We also give a necessary condition on the reaction term for the existence of convection terms such that both the right and left conditions can be satisfied simultaneously. Examples show that our sufficient conditions for linear determinacy are not necessary and compare these conditions in the scalar case with alternative conditions observed in Malaguti-Marcelli [15] and Benguria-Depassier-Mendez [3]. We also illustrate, for both an equation and a system, the existence of reaction and (non-trivial) convection terms for which the right and left linear determinacy conditions are simultaneously satisfied. An example is given of an equation which is right but not left linearly determinate.

ON THE EXISTENCE OF TWO STATIONARY SOLUTIONS FOR A FREE BOUNDARY PROBLEM DESCRIBING CELL MOTILITY

title: ON THE EXISTENCE OF TWO STATIONARY SOLUTIONS FOR A FREE BOUNDARY PROBLEM DESCRIBING CELL MOTILITY abstract: This paper is concerned with the existence of stationary solutions for a free boundary problem related to cell motility. In recent years, the author and Ninomiya \cite{monobe_ninomiya} showed that there exist at least two stationary solutions with disk-shaped domains in isotropic boundary conditions. In this paper, it will be shown that there exist exactly two stationary solutions for the free boundary problem under the same boundary conditions. The proof is based on the weak maximum principle and the mean-valued theorem.

THE OPTIMAL DISPERSAL STRATEGY: A TWO-PATCH MODEL WITH TRAVEL LOSS

title: THE OPTIMAL DISPERSAL STRATEGY: A TWO-PATCH MODEL WITH TRAVEL LOSS abstract: The dispersal of organisms plays an important role in determining the dynamics of ecological models. Ecologically, it is of interest in understanding how dispersal strategy influences the distribution of populations. An ideal free distribution (IFD) of populations has been used to predict the distribution of organisms among patches, where a key assumption is to assume that species can move freely between patches without paying any cost. If instead one assumes that there are losses when species moves from one patch to another, then ideal free distributions may not appear. In this note, we examine a two-patch resident-mutant model with travel loss and predict the optimal dispersal strategy for resident and mutant. Moreover, such strategy which produces a non-IFD is evolutionarily stable. Some same and different features of patch models with travel loss are discussed.

ASYMPTOTIC ANALYSIS OF A MONOSTABLE EQUATION IN PERIODIC MEDIA

title: ASYMPTOTIC ANALYSIS OF A MONOSTABLE EQUATION IN PERIODIC MEDIA abstract: We consider a multidimensional monostable reaction-diffusion equation whose nonlinearity involves periodic heterogeneity. This serves as a model of invasion for a population facing spatial heterogeneities. As a rescaling parameter tends to zero, we prove the convergence to a limit interface, whose motion is governed by the minimal speed (in each direction) of the underlying pulsating fronts. This dependance of the speed on the (moving) normal direction is in contrast with the homogeneous case and makes the analysis quite involved. Key ingredients are the recent improvement \cite{A-Gil} %[4]of the well-known spreading properties \cite{Wein02}, %[32], \cite{Ber-Ham-02}, %[9],and the solution of a Hamilton-Jacobi equation.