Dynamics of a generalized Beverton-Holt competition model subject to Allee effects

We propose and study {a generalized Beverton-Holt competition model} subject to Allee effects to obtain insights on how the interplay of Allee effects and contest competition affects the persistence and the extinction of two competing species. By using {the theory of monotone dynamics} and the properties of critical curves for non-invertible maps, our analysis shows that our model has simple equilibrium dynamics for most conditions. The coexistence of two competing species occurs only if the system has four interior equilibria.} We provide an {approximation to} the basins of the boundary attractors (i.e., the extinction of one or both species) where our results suggests that {contest species are more prone to extinction than scramble ones are at low densities}. In addition, {in comparison to the} dynamics of two species scramble competition models subject to Allee effects, our study suggests that (i) Both contest and scramble competition models can have only three boundary attractors without the coexistence equilibria, or four attractors among which only one is the {persistent attractor}, {whereas} scramble competition {models} may have the extinction of both species as its only attractor under certain conditions, i.e., \emph{the essential extinction} of two species due to strong Allee effects; (ii) {Scramble competition models like Ricker type models can} have much more complicated dynamical structure of interior attractors than contest ones like Beverton-Holt type models have; and (iii) Scramble competition models {like Ricker type competition models} may be more likely to promote the coexistence of two species at low and high densities under certain conditions

Permanence of a general discrete-time two-species-interaction model with non-monotonic per capita growth rates

Combined with all density-dependent factors, the per capita growth rate of a species may be non-monotonic. One important consequence is that species may suffer from weak Allee effects or strong Allee effects. In this paper, we study the permanence of a discrete-time two-species-interaction model with non-monotonic per capita growth rates for the first time. By using the average Lyapunov functions and extending the ecological concept of the relative nonlinearity, we find a simple sufficient condition for guaranteeing the permanence of systems that can model complicated two-species interactions. The extended relative nonlinearity allows us to fully characterize the effects of nonlinearities in the per capita growth functions with non-monotonicity. These results are illustrated with specific two species competition and predator-prey models of generic forms with non-monotone per capita growth rates.Comment: 18 page

Quantitative estimates of the field excited by an emitter in a narrow region between two circular inclusions

A field excited by an emitter can be enhanced due to presence of closely located inclusions. In this paper we consider such field enhancement when inclusions are disks of the same radii, and the emitter is of dipole type and located in the narrow region between two inclusions. We derive quantitatively precise estimates of the field enhancement in the narrow region. The estimates reveal that the field is enhanced by a factor of $\epsilon^{-1/2}$ in most area, where $\epsilon$ is the distance between two inclusions. This factor is the same as that of gradient blow-up when there is a smooth back-ground field, not a field excited by an emitter. The method of deriving estimates shows clearly that enhancement is due to potential gap between two inclusions.Comment: 15 page

Noise and seasonal effects on the dynamics of plant-herbivore models with monotonic plant growth functions

We formulate general plant-herbivore interaction models with monotone plant growth functions (rates). We study the impact of monotone plant growth functions in general plant-herbivore models on their dynamics. Our study shows that all monotone plant growth models generate a unique interior equilibrium and they are uniform persistent under certain range of {parameters} values. However, if the attacking rate of herbivore is too small or the quantity of plant is not enough, then herbivore goes extinct. Moreover, these models lead to noise sensitive bursting which can be identified as a dynamical mechanism for almost periodic outbreaks of the herbivore infestation. Monotone and non-monotone plant growth models are contrasted with respect to bistability and crises of chaotic attractors.Comment: 20 pages, 8 figure

Global Dynamics of a Discrete Two-species Lottery-Ricker Competition Model

In this article, we study the global dynamics of a discrete two dimensional competition model. We give sufficient conditions on the persistence of one species and the existence of local asymptotically stable interior period-2 orbit for this system. Moreover, we show that for a certain parameter range, there exists a compact interior attractor that attracts all interior points except a Lebesgue measure zero set. This result gives a weaker form of coexistence which is referred to as relative permanence. This new concept of coexistence combined with numerical simulations strongly suggests that the basin of attraction of the locally asymptotically stable interior period-2 orbit is an infinite union of connected components. This idea may apply to many other ecological models. Finally, we discuss the generic dynamical structure that gives relative permanence.Comment: 20 pages, 10 figure

Optimal estimates of the field enhancement in presence of a bow-tie structure of perfectly conducting inclusions in two dimensions

This paper deals with the field enhancement, that is, the gradient blow-up, due to presence of a bow-tie structure of perfectly conducting inclusions in two dimensions. The bow-tie structure consists of two disjoint bounded domains which have corners with possibly different aperture angles. The domains are parts of cones near the vertices, and they are nearly touching to each other. We characterize the field enhancement using explicit functions and, as consequences, derive optimal estimates of the gradient in terms of the distance between two inclusions and aperture angles of the corners. The estimates show that the field is enhanced beyond the corner singularities due to the interaction between two inclusions.Comment: 27pages, 3 figure

Existence of global solutions for a Keller-Segel-fluid equations with nonlinear diffusion

We consider a coupled system consisting of the Navier-Stokes equations and a porous medium type of Keller-Segel system that model the motion of swimming bacteria living in fluid and consuming oxygen. We establish the global-in-time existence of weak solutions for the Cauchy problem of the system in dimension three. In addition, if the Stokes system, instead Navier-Stokes system, is considered for the fluid equation, we prove that bounded weak solutions exist globally in time.Comment: 24page

Co-evolutionary dynamics of a host-parasite interaction model: obligate versus facultative social parasitism

To examine the co-evolution of quantitative traits in hosts and parasites, we present and study a co-evolutionary model of a social parasite-host system that incorporates (1) ecological dynamics that feed back into their co-evolutionary outcomes; (2) variation in whether the parasite is obligate or facultative; and (3) Holling Type II functional responses between host and parasite, which are particularly suitable for social parasites that face time costs for host location and its social manipulation. We perform local and global analyses for the co-evolutionary model and the corresponding ecological model. In the absence of evolution, the facultative parasite system can have one, two, or three interior equilibria, while the obligate parasite system can have either one or three interior equilibria. Multiple interior equilibria result in rich dynamics with multiple attractors. The ecological system, in particular, can exhibit bi-stability between the facultative-parasite-only equilibrium and the interior coexistence equilibrium when it has two interior equilibria. Our findings suggest that: (a) The host and parasite can select different strategies that may result in local extinction of one species. These strategies can have convergence stability (CS), but may not be evolutionary stable strategies (ESS); (b) The host and its facultative (or obligate) parasite can have ESS that drive the host (or the obligate parasite) extinct locally; (c) Trait functions play an important role in the CS of both boundary and interior equilibria, as well as their ESS; and (d) A small variance in the trait difference that measures parasitism efficiency can destabilize the co-evolutionary system, and generate evolutionary arms-race dynamics with different host-parasite fluctuating patterns

Characterization of the electric field concentration between two adjacent spherical perfect conductors

When two perfectly conducting inclusions are located closely to each other, the electric field concentrates in a narrow region in between two inclusions, and becomes arbitrarily large as the distance between two inclusions tends to zero. The purpose of this paper is to derive an asymptotic formula of the concentration which completely characterizes the singular behavior of the electric field, when inclusions are balls of the same radii in three dimensions.Comment: 20page

The explicit formula of flat Lagrangian H-umbilical submanifolds in quaternion Euclidean spaces

Using the idea of special Legendre curves, the authors obtained the explicit description of flat Lagrangian H-umbilical submanifolds in quaternion Euclidean spaces