Conflict between the need to forage and the need to avoid competition: persistence of two-species model

We consider a model in which the need to forage and the need to avoid a competitor are in conflict. The model is composed of two Lotka-Volterra patches. The system has two competitors; one can diffuse between two patches, but the other is confined to one of the patches and cannot diffuse. It is proved that the system can be made persistent under appropriate diffusion conditions that ensure the instability of boundary equilibria, even if the competitive patch is not persistent without diffusion. Further it is shown that the system is globally stable for any diffusion rate if the competition between the two species is weak.     

扩散对污染斑块上Smith种群生存的影响

通过建立两斑块的单种群扩散系统,主要研究扩散对其中受污染的斑块上种群生存的影响．文中所考虑的是污染斑块上外界毒素的输入量存在极限值的情形,得到如下结论：无扩散时,若此极限值超过某一定值,则污染斑块上的种群趋于灭绝;扩散存在且两斑块上种群的扩散系数满足一定的条件时,则可使该系统的种群永久生存或灭绝．     

Diffusion-mediated persistence in three-species competition models with heteroclinic cycles.

We consider a model composed of two patches. One patch has three competing species forming a heteroclinic cycle within the path. The other is a refuge for one of the three species, which can diffuse between the two patches. The remaining two competitors are confined to the competitive patch and cannot diffuse. A new heteroclinic cycle can exist in the model, and the underlying cycle in the competitive patch cannot appear with a positive diffusion rate. It is proved that the model can be made persistent under appropriate diffusion conditions even if the underlying heteroclinic cycle is an attractor in the competitive patch and the patch is not persistent without the refuge. Further it is shown that the model with a specific structure is globally stable if the underlying cycle is a repeller.     

一类带有扩散和时滞的非自台Lotka-Volterra系统

本文讨论有时滞的扩散系统,此系统有两个种群两个斑块,其中一种种群可以在两斑块中自由扩散,另一种群被限定在斑块中不能扩散,当系数数满足一定的条件时,得到系统有持续生存和全局稳定的解。     

Persistence and periodic orbits of a three-competitor model with refuges.

We consider a model composed of four patches. One patch has three competing species forming a heteroclinic cycle within the patch. The remaining patches are refuges for the three competitors, and each species can diffuse between the competitive patch and its refuge. It is proved that the model can be made persistent by the introduction of the refuges for the competitors even if the isolated competitive patch has an attracting heteroclinic cycle. Further it is shown that Hopf bifurcation is possible when we change the value of the diffusion constant and periodic orbits may exist in a specific case.     

Effective dispersal rate is a function of habitat size and corridor shape: mechanistic formulation of a two‐patch compartment model for spatially continuous systems

Two‐patch compartment models have been explored to understand the spatial processes that promote species coexistence. However, a phenomenological definition of the inter‐patch ‘dispersal rate’ has limited the quantitative predictability of these models to community dynamics in spatially continuous habitats. Here, we mechanistically rederived a two‐patch Lotka–Volterra competition model for a spatially continuous reaction‐diffusion system where a narrow corridor connects two large habitats. We provide a mathematical formula of the dispersal rate appearing in the two‐patch compartment model as a function of habitat size, corridor shape (ratio of its width to its length), and organism diffusion coefficients. For most reasonable settings, the two‐patch compartment model successfully approximated not only the steady states, but also the transient dynamics of the reaction–diffusion model. Further numerical simulations indicated the general applicability of our formula to other types of community dynamics, e.g. driven by resource‐competition, in spatially homogeneous and heterogeneous environments. Our results suggest that the spatial configuration of habitats plays a central role in community dynamics in space. Furthermore, our new framework will help to improve experimental designs for quantitative test of metacommunity theories and reduce the gaps among modeling, empirical studies, and their application to landscape management.     

非自治Lotka-Volterra型捕食扩散系统中有害时滞对该系统持久性的影响

研究了时滞对一类非自治Lotka-Volterra型捕食扩散系统的影响,该系统由n个斑块组成,食饵种群可以在斑块间迁移,而摘食者限制在某一个斑块不能扩散．我们假设密度制约项系数并不总是严格正的．通过运用比较定理及时滞泛函微分方程的基本原理,分两种情况表明了在一定条件下系统是一致持久的．两种情况的结果表明时滞的引入和变化即可能是“有害”,也可能是”无害”．进一步还说明了系统在一致持久性的条件下至少存在一个正周期解．这些结果是对已知的非自治Lotka-Volterra系统的一些结果的推广与改进．     

Hydrodynamics of a permeable patch in the fluid membrane.

A patch of cross-linked proteins in the fluid membrane is considered for the case in which the patch is permeable (porous) for the lipid flow in the membrane. The Bretscher flow field is studied quantitatively and the distribution of Brownian particles over the surface of the cell is given. This leads to a simple quantitative criterion for cap formation. Finally, explicit expressions for the rotational and translational diffusion coefficients of a permeable patch, as calculated from hydrodynamics, are given.     

Persistence and Periodic Solutions of a System of Two Competing Species with Functional Response

1IntroductionOneofthemOStnit~tingquestionsinrnathernaticalbiologyconcernsthesurvivalofspeCiesinecologiCalmodels.Perslstenceisanimportantconceptindabingwiththeseproblems.Therearemanyliteraturesaboutthedy'ndricsofdiffuSivecompetingspeCies,butthefunctionalresPOnseofthisfOITnhasnotbeenst'Udiedtoomuchyet.Inthispaper,weconsiderthepersistenceproblemforanonautonomoussystemoftwOcompetingspecieswithfunctionalreSPOnse,themodelweconsiderinthispaperishereallri(t),ail(t),D,(t)anda(t)areassumedtobecon…     

An SIS patch model with variable transmission coefficients

In this paper, an SIS patch model with non-constant transmission coefficients is formulated to investigate the effect of media coverage and human movement on the spread of infectious diseases among patches. The basic reproduction number R₀ is determined. It is shown that the disease-free equilibrium is globally asymptotically stable if R₀?1, and the disease is uniformly persistent and there exists at least one endemic equilibrium if R₀>1. In particular, when the disease is non-fatal and the travel rates of susceptible and infectious individuals in each patch are the same, the endemic equilibrium is unique and is globally asymptotically stable as R₀>1. Numerical calculations are performed to illustrate some results for the case with two patches.     

Analysis of Microscopic Parameters of Single-Particle Trajectories in Neurons

We performed a comparative study of the statistical uncertainties that arise when calculating the velocity and diffusion coefficients from single-particle trajectories. We show that a method where particle mean displacement is used to calculate velocity and mean square fluctuation is used to calculate diffusion coefficient offers greater accuracy than analysis of time-dependent mean square displacement. Our assessment of the performance of the two analysis strategies is conducted in two ways. First, we apply each of the methods to simulated trajectories where each parameter term is known. Second, we analyze the motion of previously uncharacterized EphB2 receptors in the membrane of hippocampal neurons. We find that EphB2 receptors display different types of motion mode and transition between these modes. We present our data as a distribution of microscopic diffusion coefficients for each particle trajectory, which we refer to as partial distributions. Partial distributions are summed to form a cumulative distribution of diffusion coefficients for EphB2 receptors in hippocampal neurons. The structure and interpretation of the EphB2 cumulative distribution are discussed.     

Homogenization techniques for population dynamics in strongly heterogeneous landscapes

An important problem in spatial ecology is to understand how population-scale patterns emerge from individual-level birth, death, and movement processes. These processes, which depend on local landscape characteristics, vary spatially and may exhibit sharp transitions through behavioural responses to habitat edges, leading to discontinuous population densities. Such systems can be modelled using reaction–diffusion equations with interface conditions that capture local behaviour at patch boundaries. In this work we develop a novel homogenization technique to approximate the large-scale dynamics of the system. We illustrate our approach, which also generalizes to multiple species, with an example of logistic growth within a periodic environment. We find that population persistence and the large-scale population carrying capacity is influenced by patch residence times that depend on patch preference, as well as movement rates in adjacent patches. The forms of the homogenized coefficients yield key theoretical insights into how large-scale dynamics arise from the small-scale features.     

具有分离扩散的两种群Lotka—Volterra模型的持久性

本文考虑具有分离扩散的捕食-被捕食系统的持续性。此模型由两种群组成，其中被捕食种群可在两个生态环境中生存，而捕食种群仅能在一个生态环境中生存，两种群的动态行为都用Lotka－Volterra模型来描述。得到了系统强持续的充分必要条件，并证明了无论无扩散时系统是共存的，还是主导的都可以适当选择分离扩散系数使整个系统强持续。     

Remarks on the number of persistent states to a fragmented predator–prey model system

We consider a predator–prey model system for spatially distributed species over patches. Each predator species has a unique preferred patch (shelter and reproduction site) and travel for chasing prey. Its individuals are split into resident from the preferred patch and travelers. Further there is at most one resident predator species per patch. Depending on the availability of local anthropized resources not related to local prey on the preferred patch, one distinguishes between well-fed and starving predators. We assume prey species do not disperse at the predator scale.In this study we are interested in the number of persistent stationary states for the resulting ordinary differential equations model system. There exists at most one persistent predator–prey stationary state when there is exactly one starving resident predators per patch provided all functional responses to predation are Lotka–Volterra like or when a single starving resident predators is available. Else multiple persistent predator–prey stationary state are likely to exist. A specific emphasis is put on toy-model systems with 2 or 3 patches. Slow–fast dynamical methodology is also used for locally asymptotically stable purposes.Numerical experiments suggest that several scalings may govern the dynamics at stabilization.     

Lipid and water diffusion in bicontinuous cubic phases measured by NMR.

Lipid and water diffusion coefficients in bicontinuous cubic liquid crystalline phases have been determined with the NMR pulsed magnetic field gradient technique. In the monoolein-water system, a discontinuity in the variation of the water diffusion coefficient with water content is observed, which coincides with the two-phase region between the two cubic phases in this system. The degree of water association to the lipid has been determined, considering the obstruction factor for diffusion in the cubic phases. The lipid diffusion coefficient increases with increased unsaturation of the lipid, and decreases when larger amphiphile molecules like cholesterol, gramicidin-A, and lyso-oleoyl-phosphatidylcholine are solubilized in the cubic phase. In a cubic liquid crystal of monoolein (MO), dioleoylphosphatidylcholine (DOPC), and water, the individual lipid diffusion coefficients have been determined simultaneously in the same sample. The diffusion coefficients of MO and DOPC differ by a factor of two, and both decrease with increasing DOPC content. The results are discussed in relation to probe techniques for measurements of lipid diffusion.     

Effect of media-induced social distancing on disease transmission in a two patch setting

We formulate an SIS epidemic model on two patches. In each patch, media coverage about the cases present in the local population leads individuals to limit the number of contacts they have with others, inducing a reduction in the rate of transmission of the infection. A global qualitative analysis is carried out, showing that the typical threshold behavior holds, with solutions either tending to an equilibrium without disease, or the system being persistent and solutions converging to an endemic equilibrium. Numerical analysis is employed to gain insight in both the analytically tractable and intractable cases; these simulations indicate that media coverage can reduce the burden of the epidemic and shorten the duration of the disease outbreak.     

A delay equation model for oviposition habitat selection by mosquitoes

We propose a patch type model for mosquitoes that have aquatic larvae inhabiting ponds. Partial differential equations (PDEs) model the larvae on each of several disconnected patches representing the ponds, with conditions varying in each patch, coupled via the adults in the air. From the PDEs a scalar delay differential equation, with multiple delays, for the total adult mosquito population is derived. The various delays represent the larval development times in the patches. The coefficients contain all the relevant information about the sizes and geometry of the individual patches inhabited by the larvae, the boundary conditions applicable to those patches and the diffusivity of the larvae in each patch. For patches of general shapes and sizes, and without the need to specify the criteria by which an adult mosquito selects an oviposition patch, the modern theory of monotone dynamical systems and persistence theory enables a complete determination of the conditions for the mosquito population to go extinct or to persist. More detailed biological insights are obtained for the case when the patches are squares of various sizes, which allows a detailed discussion of the effects of scale, and for two particular criteria by which mosquitoes might select patches for oviposition, being (i) selection based solely on patch area, and (ii) selection based both on area and expected larval survival probability for each patch. In some parameter regimes, counterintuitive phenomena are predicted.     

Fixation probabilities in a spatially heterogeneous environment

 We consider a simple model of a one-locus, two-allele population inhibiting a two-patch system and experiencing spatially heterogeneous viability selection. The populaton size is finite. We use a diffusion approximation and singular perturbation techniques to find the probability of fixation of a mutant allele. We focus on situations in which each allele is advantageous in one patch and deleterious in the other patch. Our theoretical results support the previous conclusions that, under certain conditions, small populations respond faster to selection than do large populations. We emphasize that knowledge of the dependence of migration rates on population size is crucial in evaluating the effects of population size on the rate of evolution.     

Clonal growth strategies in simultaneously persistent and expanding <Emphasis Type="Italic">Trifolium repens</Emphasis> patches

Plants with clonal growth can generate patches dominated by a single species. In time, patches can change and may fragment, form a ring, dissolve or both persist and expand. For patches to maintain their original habitat and simultaneously increase in size, ramets or clonal fragments must both promote local persistence inside the patch and grow out of the patch into new habitats. This study analyses simultaneously expanding and persistent Trifolium patches in a nutrient-poor lawn that is frequently cut, and where the Trifolium is competitively superior to the grass species. Trifolium primary stolon growth strategies were analyzed in relation to their location (border, middle, and center) inside the patch, and according to patch size (small, medium, and large). It was hypothesized that different growth strategies inside a patch can explain both persistent and expanding patch of Trifolium, and that growth strategies were different between patch sizes. Primary Trifolium stolons had two different growth strategies inside and at the border of patches: (i) stolons at the border were long, grew fast, had few lateral stolons, and grew out of the patch, while (ii) stolons inside the patch were smaller, grew slowly, and had more lateral stolons and a wide range of growth directions. Growth strategies were not different between patch sizes. The directional growth and the high growth rate at the border will increase the patch size with time, while the growth strategy near the center consolidates the patch in space and time, by placing ramets inside the patch. Different growth strategies near the center and on the border result in Trifolium patches that are simultaneously persistent and on the increase. The results also indicate a division of labor among primary Trifolium stolons in a patch.