Periodic Oscillatory Phenomenon in Fractional-Order Neural Networks Involving Different Types of Delays

This research is chiefly concerned with the stability and Hopf bifurcation for newly established fractional-order neural networks involving different types of delays. By means of an appropriate variable substitution, equivalent fractional-order neural network systems involving one delay are built. By discussing the distribution of roots of the characteristic equation of the established fractional-order neural network systems and selecting the delay as bifurcation parameter, a novel delay-independent bifurcation condition is derived. The investigation verifies that the delay is a significant parameter which has an important influence on stability nature and Hopf bifurcation behavior of neural network systems. The computer simulation plots and bifurcation graphs effectively illustrate the reasonableness of the theoretical fruits.

Hopf Bifurcation of an Age-Structured Epidemic Model with Quarantine and Temporary Immunity Effects

In this paper, we investigate an epidemic model with quarantine and recovery-age effects. Reformulating the model as an abstract nondensely defined Cauchy problem, we discuss the existence and uniqueness of solutions to the model and study the stability of the steady state based on the basic reproduction number. After analyzing the distribution of roots to a fourth degree exponential polynomial characteristic equation, we also derive the conditions of Hopf bifurcation. Numerical simulations are performed to illustrate the results.

Assembly history modulates vertical root distribution in a grassland experiment

The order of arrival of plant species during assembly can affect the structure and functioning of grassland communities. These so-called priority effects have been extensively studied aboveground, but we still do not know how they affect the vertical distribution of roots in the soil and the rooting depth of plant communities. To test this hypothesis, we manipulated the order of arrival of three plant functional groups (forbs, grasses and legumes) in a rhizobox experiment. Priority effects were created by sowing one functional group 10 days before the other two. Rhizoboxes in which all functional groups were sown simultaneously were used as controls. During the experiment, the mean rooting depth of plant communities was monitored using image analysis and a new methodological approach using deep learning (RootPainter) for root segmentation. At harvest, we measured aboveground (community and species level) and belowground (community level) biomass, and assessed the vertical distribution of the root biomass in different soil layers. At the community level, all scenarios where one functional group was sown before the other two had similar shoot and root productivity. At the species level, two forbs (Achillea millefolium and Centaurea jacea) benefited from arriving early, and one legume (Trifolium pratense) had a disadvantage when it was sown after the grasses. Priority effect treatments also affected the vertical distribution of roots. When grasses were sown first, plant communities rooted more shallowly than when forbs or legumes were sown first,. In addition, roots moved down the soil profile 24% more slowly in grasses-first communities. Our results highlight that plant functional group order of arrival in grassland communities can affect the vertical distribution of roots in the soil and this may have implications for species coexistence.

Spatial and temporal distribution of mung bean (Vigna radiata) and soybean (Glycine max) roots

Spatial and temporal distribution of roots of mung bean and soybean originated from different geographical backgrounds is an important scientific issue. The aim of this study was to research the spatial and temporal distribution of roots system of soybean cultivar ‘Hefeng55’ and mung bean cultivar ‘Jilv7’ which can elucidate differences between soybean roots and mung bean roots in the key spatial and temporal locations. The roots at V6, R2, R4, R5, R6, and R7 stages were collected to acquire data of root length, root surface area, root volume and root dry weight. 49.8%, 11.7%, 13.2%, 14.7% and 10.6% of soybean roots and 57.8%, 10.7%, 11.2%, 11.9% and 8.4% of mung bean roots were in 0-5, 5-10, 10-15, 15-20 and 20-25 cm horizontal soil layers, respectively; 79.2%, 11.5%, 4.3%, 1.8%, 1.1%, 1.0% and 1.1% of soybean roots and 70.0%, 12.3%, 8.0%, 3.0%, 1.6%, 1.7% and 3.4% of mung bean roots were in 0-20, 20-40, 40-60, 60-80, 80-100, 100-120 and 120-140 cm vertical soil layers, respectively. Compared with mung bean, soybean had a much larger root system during development. In horizontal direction, soybean root tended to be more laterally developed, but the distribution of mung bean root was more uniform in vertical direction. With a greater root surface area to weight ratio (AWR), mung bean had a finer root system than soybean. These findings can help to clarify the four-dimensional spatial and temporal distribution characteristics of legumes and may provide reference for production practice of soybean and mung bean in the future.

Exact solution of a cluster model with next-nearest-neighbor interaction

A 1D cluster model with next-nearest-neighbor interactions and two additional composite interactions is solved; the free energy is obtained and a correlation function is derived exactly. The model is diagonalized by a transformation obtained automatically from its interactions, which is an algebraic generalization of the Jordan–Wigner transformation. The gapless condition is expressed as a condition on the roots of a cubic equation, and the phase diagram is obtained exactly. We find that the distribution of roots for this algebraic equation determines the existence of long-range order, and we again obtain the ground-state phase diagram. We also derive the central charges of the corresponding conformal field theory. Finally, we note that our results are universally valid for an infinite number of solvable spin chains whose interactions obey the same algebraic relations.

Notes on the Distribution of Roots Modulo a Prime of a Polynomial III

Let f (x) bea monicpolynomialwith integer coefficients and integers r1,..., rn with 0 ≤ r1 ≤··· ≤ rn <p the n roots of f (x) ≡ 0mod p for a prime p. We proposed conjectures on the distribution of the point (r1/p,...,rn/p) in the previous papers. One aim of this paper is to revise them for a reducible polynomial f (x), and the other is to show that they imply the one-dimensional equidistribution of r1/p,...,rn/p for an irreducible polynomial f (x) by a geometric way.

Trickle irrigation systems affect spatial distribution of roots of banana crop

ABSTRACT Trickle irrigation has been largely used for banana in Brazil, mainly due to the increase in water and fertilizer use efficiency. These irrigation systems have different options concerning number, type and flow rate of emitters as well as for hydraulics, number and location of lateral lines. The small area of soil wetted by these systems limits root spatial distribution of crops. This study aimed to evaluate the effect of different trickle irrigation systems on the root spatial growth and root spatial distribution of banana cv. Prata Gorutuba. Root length density and root length were evaluated in soil profiles of three micro-sprinkler systems, with emitter flow rates of 35, 53 and 70 L h-1 and of two drip irrigation systems, with one and two lateral lines per crop row. Trickle irrigation systems influence root spatial distribution, favoring a greater or smaller distribution of roots at different depth and distance from the plant according to micro-sprinkler flow rate and to the number of lateral lines per crop row. The effect on root spatial distribution is more accentuated for micro-sprinkler systems than for drip systems. The majority of the total root length (80%) was observed in the soil profiles from 0.33 to 0.57 m depth and at distances from the plants of 0.75 to 0.83 m.

The impact of the rhizobia–legume symbiosis on host root system architecture

Legumes form symbioses with rhizobia to fix N2 in root nodules to supplement their nitrogen (N) requirements. Many studies have shown how symbioses affect the shoot, but far less is understood about how they modify root development and root system architecture (RSA). RSA is the distribution of roots in space and over time. RSA reflects host resource allocation into below-ground organs and patterns of host resource foraging underpinning its resource acquisition capacity. Recent studies have revealed a more comprehensive relationship between hosts and symbionts: the latter can affect host resource acquisition for phosphate and iron, and the symbiont’s production of plant growth regulators can enhance host resource flux and abundance. We review the current understanding of the effects of rhizobia–legume symbioses on legume root systems. We focus on resource acquisition and allocation within the host to conceptualize the effect of symbioses on RSA, and highlight opportunities for new directions of research.

Hopf bifurcation of a delayed worm model with two latent periods

We investigate a delayed epidemic model for the propagation of worm in wireless sensor network with two latent periods. We derive sufficient conditions for local stability of the worm-induced equilibrium of the system and the existence of Hopf bifurcation by regarding different combination of two latent time delays as the bifurcation parameter and analyzing the distribution of roots of the associated characteristic equation. In particular, we investigate the direction and stability of the Hopf bifurcation by means of the normal form theory and center manifold theorem. To verify analytical results, we present numerical simulations. Also, the effect of some influential parameters of sensor network is properly executed so that the oscillations can be reduced and removed from the network.