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This paper analytically investigates a nonlocal system of reaction-diffusion-advection equations modeling the competition of two phytoplankton species for a limiting nutrient and light in a water column, where dead phytoplankton species can get recycled back into the system as a resource for growth. The threshold dynamics of the single population model is first established. Then the utilization of abstract persistence theory enables us to show that two species population system admits a coexistence steady state and the system is uniformly persistent if the trivial steady state and two global attractors on the boundary are all weak repellers.

Urban crime such as residential burglary is a social problem in every major urban area. As such, many mathematical models have been proposed to study the collective behavior of these crimes. In [V. B. Pasour, G. E. Tita, P. J. Brantingham, A. L. Bertozzi, M. B. Short, M. R. D’Orsogna and L. B. Chayes, A statistical model of crime behavior, Math. Methods Appl. Sci 107 (2008) 1249–1267; M. B. Short, A. L. Bertozzi and P. J. Brantingham, Nonlinear patterns in urban crime: Hotspots, bifurcations, and suppression, SIAM J. Appl. Dyn. Syst. 9 (2010) 462–483], Short et al. proposed an agent-based statistical model of residential burglary to model the crime hotspot phenomena. From the point of view of reaction–diffusion systems, the model is a chemotactic system with cross diffusion that exhibit hotspot phenomena. In this paper, we first construct a radial hotspot solution of this system, then study the linear stability of this hotspot solution by studying a nonlocal eigenvalue problem. It turns out that the stability of the hotspot is completely different depending on which spatial dimension the system is on. The main mathematical difficulty of the system involves treating the steep change of diffusion near the core of the hotspot, because of the quasilinearity induced by the cross diffusion. We believe that the techniques used in this paper can be developed to treat many other chemotactic systems.

The Hopf bifurcation from spike solutions for the classical Gierer–Meinhardt system in a onedimensional interval is considered. The existence of time-periodic solution near the Hopf bifurcation parameter for a boundary spike is rigorously proved by the classical Crandall–Rabinowitz theory. The criteria for the stability of the limit cycle are determined, and it is shown that the limit cycle isunstable.

The Hopf bifurcations for the classical Gray–Scott system and the Schnakenberg system in an one-dimensional interval are considered. For each system, the existence of time-periodic solutions near the Hopf bifurcation parameter for a boundary spike is rigorously proved by the classical Crandall–Rabinowitz theory. The criteria for the stability of limit cycles are determined and it is shown that the Hopf bifurcation is supercritical for the Schnakenberg system, and hence the bifurcating periodic solutions are linearly stable. For the Gray–Scott system, there is a critical feeding rate, when the feeding rate of the system is greater than this critical feeding rate, the Hopf bifurcation is supercritical which implies the bifurcating periodic solutions are linearly stable, while when the feeding rate is smaller than this critical feeding rate, the Hopf bifurcation is subcritical, implying the bifurcating periodic solutions are linearly unstable.

Since tremendous resources are consumed in the architecture, engineering, and construction (AEC) industry, the sustainability and efficiency in this field have received increasing concern in the past few decades. With the advent and development of computational tools and information technologies, structural optimization based on mathematical computation has become one of the most commonly used methods for the sustainable and efficient design in the field of civil engineering. However, despite the wide attention of researchers, there has not been a critical review of the recent research progresses on structural optimization yet. Therefore, the main objective of this paper is to comprehensively review the previous research on structural optimization, provide a thorough analysis on the optimization objectives and their temporal and spatial trends, optimization process, and summarize the current research limitations and recommendations of future work. The paper first introduces the significance of sustainability and efficiency in the AEC industry as well as the background of this review work. Then, relevant articles are retrieved and selected, followed by a statistical analysis of the selected articles. Thereafter, the selected articles are analyzed regarding the optimization objectives and their temporal and spatial trends. The four major steps in the structural optimization process, including structural analysis and modelling, formulation of optimization problems, optimization techniques, and computational tools and design platforms, are also reviewed and discussed in detail based on the collected articles. Finally, research gaps of the current works and potential directions of future works are proposed. This paper critically reviews the achievements and limitations of the current research on structural optimization, which provide guidelines for future research on structural optimization in the field of civil engineering.

This paper study nonstationary homoclinic-type solutions for a fractional reaction-diffusion system with asymptotically linear and local super linear nonlinearity. The resulting problem engages two major difficulties: one is that the associated functional is strongly indefinite, the second lies in verifying the link geometry and showing the boundedness of Cerami sequences when the nonlinearity is not super quadratic at infinity globally. These enable us to develop a direct approach and new tricks to overcome the difficulties. We establish the existence of homoclinic orbit under some weak assumptions on nonlinearity.

The use of inhomogeneous boundary conditions has become increasingly important in the study of pattern forming reaction diffusion systems. Such inhomogeneities can be used to describe, for example, the interaction between an individual cell with its environment. In this paper we consider arbitrary mixed inhomogeneous boundary conditions for the activator in the singularly perturbed Gierer-Meinhardt system. By using formal asymptotics we derive an algebraic system and nonlocal eigenvalue problem respectively describing the structure and linear stability of multi-spike solutions in the case of a one-dimensional domain, thereby extending previous results obtained in the case of inhomogeneous Neumann boundary conditions by Gomez and Wei (2021). We also rigorously prove partial stability results and provide detailed stability thresholds for two examples consisting of one- and two-spike solutions. In higher dimensions we restrict our attention to the shadow limit in which the inhibitor is well-mixed and for which we rigorously establish both the existence and stability of a boundary layer solution. In both the one- and higher-dimensional cases we find that when the boundary conditions are symmetric then a symmetric solution is stable only when the magnitude of the inhomogeneity exceeds some threshold. Below this threshold we demonstrate that a stable asymmetric two spike solution emerges in the case of a one-dimensional domain, while in the case of a two-dimensional domain we show numerical simulations illustrating the formation of a near-boundary spike.

Liouville type results for positive radial entire solutions (u,v)∈[C2(RN∖{0})∩C0(RN)]×[C2(RN∖{0})∩C0(RN)] (N≥3) of the weighted Lane–Emden system div(|x|θ1∇u)+|x|l1vp=0in RN∖{0},div(|x|θ2∇v)+|x|l2u...

We study the structure of radial solutions of the quasilinear elliptic problem (Q) where , , and . It is seen that (Q) admits a global continuum of radial solutions . Moreover, there is such that has infinitely many turning points for and does not admit any turning point for .

Bifurcation of non-radial solutions from radial solutions of a semilinear elliptic equation with negative exponent in expanding annuli of ℝ2 is studied. To obtain the main results, we use a blow-up argument via the Morse index of the regular entire solutions of the equationThe main results of this paper can be seen as applications of the results obtained recently for finite Morse index solutions of the equationwith N ⩾ 2 and p > 0.

New embeddings of some weighted Sobolev spaces with weights a(x) and b(x) are established. The weights a(x) and b(x) can be singular. Some applications of these embeddings to a class of degenerate elliptic problems of the form −div(a(x)∇u) = b(x)f(x, u) in Ω, u = 0 on ∂Ω, where Ω is a bounded or unbounded domain in R N , N > 2, are presented. The main results of this paper also give some generalizations of the well-known Caffarelli-Kohn-Nirenberg inequality.

In this paper, narrow region principle for cooperative systems involving fractional Laplacian is developed. As an application, some Liouville type results of the fractional Lane–Emden system(1) are obtained via the method of moving planes. Different to the conventional method by studying the equivalent integral system, we work directly on the differential system. Our method can also be employed to obtain Liouville type results for general cooperative systems involving fractional Laplacian.

Bifurcation of nonradial solutions from radial solutions of 0.1 \begin{document}$-Δ u=λ e^u$ \end{document} in expanding annuli of ${\mathbb{R}^N}$ with $3 ≤q N ≤q 9$ is studied. To obtain the main results, we use a blow-up argument via Morse indices of the regular entire solutions of (0.1).

A reaction-diffusion logistic population model with spatially nonhomogeneous harvesting is considered. It is shown that when the intrinsic growth rate is larger than the principal eigenvalue of the protection zone, then the population is always sustainable; while in the opposite case, there exists a maximum allowable catch to avoid the population extinction. The existence of steady state solutions is also studied for both cases. The existence of an optimal harvesting pattern is also shown, and theoretical results are complemented by some numerical simulations for one-dimensional domains.

Phytoplankton species in a water column compete for mineral nutrients and light, and the existing models usually neglect differences in the nutrient content and the amount of light absorbed of individuals. In this current paper, we examine a size-structured and nonlocal reaction–diffusion–advection system which describes the dynamics of a single phytoplankton species in a water column where the species depends simply on light for its growth. Our model is under the assumption that the amount of light absorbed by individuals is proportional to cell size, which varies for populations that reproduce by simple division into two equally-sized daughters. We first establish the existence of a critical death rate and our analysis indicates that the phytoplankton survives if and only if its death rate is less than the critical death rate. The critical death rate depends on a general reproductive rate, the characteristics of the water column (e.g., turbulent diffusion rate, sinking, depth), cell growth, cell division, and cell size.

In this paper, we analyze a system modeling the growth of single phytoplankton populations in a water column, where population growth increases monotonically with the nutrient quota stored within individuals. We establish a threshold result on the global extinction and persistence of phytoplankton. Condition for persistence is shown to depend on the principal eigenvalue of a boundary value problem, which is related to the physical transport properties of the water column (i.e. the diffusivity and the sinking speed), nutrient uptake rate, and growth rate.

This paper studies an eigenvalue problem associated with a linear parabolic equation and a coupled ordinary differential equation. The existence and the uniqueness of the principal eigenvalue for this eigenvalue problem is first established. Then, the qualitative dependence of the principal eigenvalue with respect to the several parameters involved in the system is analyzed. Finally, these results are applied to a system in flowing habitats with a hydraulic storage zone and light limitation.

We study a nonlocal reaction-diffusion-advection equation arising from the study of a single phytoplankton species competing for light in a poorly mixed water column. When the diffusion coefficient is very small, the phytoplankton population concentrates around certain zeros of the advection function. The corresponding phytoplankton distribution approaches a $\delta$-like function centered at those zeros.

We obtain non-radial bifurcation from radial solutions of a semilinear elliptic equation in expanding annuli of \(\mathbb {R}^N\). To obtain the main results, we use a blow-up argument via the Morse index of the regular entire solutions of the equation $$\begin{aligned} -\Delta u=\lambda u^p \quad \text {in}\quad \mathbb {R}^N. \end{aligned}$$ (0.1) The main results of this paper can be seen as applications of the properties of regular entire solutions of (0.1).

We investigate the steady state solutions of a phytoplankton-nutrient system proposed by Huisman et al. in [14] that models the dynamics of a single phytoplankton species whose growth is limited by light and nutrients in a vertical water column. We first study the existence and nonexistence problem of the model and prove there is at least one positive solution to the system when the parameters involved are in a suitable range. We then analyze the limiting profiles of the positive solutions as the specific phytoplankton loss rate approaches zero and as the diffusion coefficient of the system tends to zero, respectively. In the small diffusion case, we show that the phytoplankton species all die out regardless of how large the nutrient supply is, and the nutrients distribution approaches a linear function determined by the parameters of the system. This phenomenon is in sharp contrast to that of the model studied by Du and Hsu [3, 4], where the phytoplankton species can concentrate at the bottom, at the surface or at a specific depth of the water column decided by the amplitude of the nutrient supply. We also study the asymptotic profile of the positive solution when the diffusion coefficient is very large. Our results reveal that in such a case the phytoplankton and the nutrients distribute evenly in the water column. Our concentration results also reveal that passive diffusion and active movement (sinking or floating) should be in proportion to the oscillation phenomena showed in [14, 24] to occur.

We investigate a reaction–diffusion–advection equation that models the dynamics of a single phytoplankton species in a eutrophic vertical water column. First, we extend the results of Du and Hsu (2010 SIAM J. Math. Anal.42 1305–33) to show that even with variable diffusion and sinking rates, the global dynamics of the model is completely determined by its unique steady-state solution. This implies that the bistable behaviour observed through numerical simulation in Ryabov et al (2010 J. Theor. Biol. 263 120–33) for the phytoplankton dynamics can only occur when one assumes limitation of nutrients in the model. Second, we examine the asymptotic profiles of the positive steady-state solution for small diffusion, large diffusion and deep water column, respectively. Our results reveal that for small diffusion, the phytoplankton population concentrates at the bottom of the water column, while for large diffusion, the population tends to distribute evenly in the water column, and when all the other factors are the same, in a water column with positive background turbidity, the total biomass is bigger in the large diffusion case than in the small diffusion case, and in a water column with zero (or negligible) background turbidity, the total biomass tends to the same limit in both cases; when the water column depth goes to infinity, the population distribution approaches that obtained in Ishii and Takagi (1982 J. Math. Biol. 16 1–24) with infinite water depth, and it reaches a unique maximum at a certain finite water level. We also give a complete answer to a question left open in Hsu and Lou (2010 SIAM J. Appl. Math. 70 245–54) regarding the behaviour of the critical death rate for deep water column, which plays a key role in determining whether a critical water depth exists.

We study positive solutions of the equation − Δ u = λ u − a ( x ) u p , x ∈ R N ( N ≥ 3 ) , where p > 1 is a constant, λ is a parameter and a ( x ) changes sign in a bounded domain. Under suitable conditions on the behavior of a ( x ) near its zero set and near | x | = ∞ , we obtain multiple positive solutions to this problem. Our approach is by approximating the problem with corresponding problems on enlarging balls. Our results improve and extend those of Du (2004) [15] .

This paper considers positive solutions to problem−Δu=λu+h(x)eu,x∈R2, where λ is a parameter, h(x)∈C2(R2,R) changes sign. Uniform a priori bounds are obtained using less restrictive conditions from previous works.

We have exploited the positive entropy of the twin boundary energy to refine twin structure in YBa 2 Cu 3 O 7− x (Y123) which is the matrix phase in melt-textured grown (MTG) Y–Ba–Cu–O (YBCO) pellets. Refinement of twin microstructure in MTG pellets is achieved by annealing at higher temperatures. Decrease of twin spacing with increasing temperature is confirmed by electron microscopy. The critical current density ( J c ), the pinning force and the magnetic field at maximum pinning-force all increase with increasing annealing temperature of oxygenation. This thermal method provides a viable way to improve these superconducting properties that are evidently extrinsic and strong functions of the microstructure. For the MTG YBCO samples with initial composition of 28.6 wt% Y 2 BaCuO 5 /71.4 wt% Y123/0.5 wt% PtO 2 , twin spacing reduces from 1900 to 540 A as the oxygenation temperature increases from 450 to 680 °C. Concomitantly, the J c (77 K, 1 T) is improved by 22,120–31,920 A/cm 2 and the maximum flux-pinning force nearly triples from 1.2 × 10 8 to 3.4 × 10 8 N/m 3 .

Potassium fluoroboratoberyllate KBe2BO3F2 (KBBF) has been revealed theoretically and experimentally as a novel ultraviolet nonlinear optical crystal, but it is found to be very difficult to grow in a large size, because of the weak binding interaction between the (Be2BO3)∞ units, which leads to an apparent layer habit in the growth. By using a molecular engineering approach, oxygen bridges when brought in to strengthen the binding between the infinite units are found to be useful to overcome the above shortcoming of KBBF, and in the light of it another new ultraviolet nonlinear optical crystal—strontium boratoberyllate Sr2Be2B2O7 (SBBO) has been discovered. The linear optical properties of SBBO are similar to KBBF’s, but its nonlinear optical properties are better than that of the latter. d22(SBBO)≂d22(β‐BaB2O4), which is two times higher than d11 of KBBF. SBBO has very good mechanical properties, and it is also not deliquescent. So SBBO is expected to have great potential for the application in ultraviol...

An important use of nonlinear optical (NLO) materials is in vacuum ultraviolet (VUV) applications, and no crystal except KBe2BO3F2 (KBBF) can be used in the VUV region thus far. In this paper, the crystal structure of a new monoclinic VUV NLO crystal, sodium beryllium borate fluoride, NaBe2BO3F2 (NBBF), is reported. The crystal structure has been determined by single crystal X-ray diffraction, and the final R value for 612 unique reflections with I>3.00σ (I) is 5.6%. The compound has the same polyanions as KBe2BO3F2. All the B atoms are nearly triangularly coordinated in the plane, while all the Be atoms are tetrahedrally coordinated. The structure is characterized by infinite sheets [Be2BO3F2], which may be described in terms of six-membered rings of [Be2BO6F2] composed of one [BO3] and two [BeO3F] with edges in common. Its large, nonlinear optical (NLO) coefficients, short absorption edge, and moderate birefringence show that it will also be a promising VUV NLO material.

Twin boundary energy (γtw) and its temperature dependence are important factors to be evaluated for engineering twin morphology for strong flux pinning and high critical current density in superconducting YBa2Cu3O7−x. Here, it has been measured as a function of oxygenation temperature from 450 to 680 °C using both the twin spacing method and a twin tip method. By comparing the twin energies obtained by the two methods, the geometrical factor α used in twin spacing method is determined. This geometrical factor is directly related to the sharpness of the twin tips. From the temperature dependence of twin boundary energy per unit area, twin boundary entropy (Δs) and enthalpy (Δh) for twin boundaries are projected to be 0.24mJ∕Km2 and 246mJ∕m2, respectively.

MBe2BO3F2 (M=Na,K) have been discovered to be nonlinear optical (NLO) materials for UV applications. They have relatively large NLO coefficients: deff is about 2.5 times as large as that of KH2PO4 (λ=1079 nm, Nd:YAP laser), the UV cutoff wavelengths are as short as 155 nm, and they have a moderate birefringence of nearly 0.10. Their structures are characterized by an infinite sheet which is formed by six‐membered rings [Be2BO6F2] with Be‐O and B‐O edges in common. Such structures are favorable for good NLO properties. These NLO properties make MBe2BO3F2 promising candidates for vacuum UV NLO materials.

KBe 2 BO 3 F 2 (KBBF) crystals of up to 20X10X1.0 mm were grown for the first time by the flux method, both by top-seeded solution growth and by spontaneous nucleation.

In this paper we study a nonlocal reaction–diffusion–advection system modeling the growth of multiple competitive phytoplankton species in a vertical water column with incomplete mixing. We find that when the diffusion of the system is large, there is no positive steady states, and when the diffusion is not large, there exists at least one positive steady states under certain conditions. The main tools we use are the fixed point index theory, a refined comparison theorem and fine properties of the principal eigenvalues.