Recent results demonstrate the dynamical peculiarities of the quantum chaos within the hybrid systems by chaotic parameters and probe the pattern formation under the influence of condensation. The complex dynamic behavior of the considered systems was determined with numerical simulation and presented an efficient technique that studied fractional systems comprising chaos-coherence fractions. The findings divulge the peculiar association between the coherence structure and the correlations at finite relative momenta. Thus the present study helps to explore the partially chaos hybrid systems in order to stimulate the experimental applications of nonlinear phenomena. The coherent-chaotic parameters can be measured by examining the chaos peculiarities that possess explicit relations with the condensations to demonstrate the environs of the physical systems. We investigate the influence of the multiplicities, chaos, momentum and temperature of the nonlinear system on the coherent-chaotic normalized correlations. The chaotic parameters are suppressed considerably with the coherence fraction and it appears numerically zero at maximum condensation and one at ideal chaos emissions. We procure that the meaningful parameters decrease significantly with the multiplicity of the nonlinear systems and increase with the momentum in the specified regimes. The identical multiplicity leads to contemplating the coherence and thus the normalized chaotic parameters within its spectacular influences exhibit significance worth contemplating in earnest. The findings underscore the significance of cogitating correlations in deciphering the nonlinear system characteristics and bestowing extraordinary perceptiveness into the convoluted essence of complex systems. The contemplated methodology can be applied to evaluating and analyzing the nonlinear systems and such an innovative approach computes the problems of celestial mechanics, heartbeats and chemical reactions in engineering and medical fields.

In biology, evolutionary game-theoretical models often arise in which players\' strategies impact the state of the environment, driving feedback between strategy and the surroundings. In this case, cooperative interactions can be applied to studying ecological systems, animal or microorganism populations, and cells producing or actively extracting a growth resource from their environment. We consider the framework of eco-evolutionary game theory with replicator dynamics and growth-limiting public goods extracted by population members from some external source. It is known that the two sub-populations of cooperators and defectors can develop spatio-temporal patterns that enable long-term coexistence in the shared environment. To investigate this phenomenon and unveil the mechanisms that sustain cooperation, we analyze two eco-evolutionary models: a well-mixed environment and a heterogeneous model with spatial diffusion. In the latter, we integrate spatial diffusion into replicator dynamics. Our findings reveal rich strategy dynamics, including bistability and bifurcations, in the temporal system and spatial stability, as well as Turing instability, Turing-Hopf bifurcations, and chaos in the diffusion system. The results indicate that effective mechanisms to promote cooperation include increasing the player density, decreasing the relative timescale, controlling the density of initial cooperators, improving the diffusion rate of the public goods, lowering the diffusion rate of the cooperators, and enhancing the payoffs to the cooperators. We provide the conditions for the existence, stability, and occurrence of bifurcations in both systems. Our analysis can be applied to dynamic phenomena in fields as diverse as human decision-making, microorganism growth factors secretion, and group hunting.

A vasculature network supplies blood to feather buds in the developing skin. Does the vasculature network during early skin development form by sequential sprouting from the central vasculature or does local vasculogenesis occur first that then connect with the central vascular tree? Using transgenic Japanese quail Tg(TIE1p.H2B-eYFP), we observe that vascular progenitor cells appear after feather primordia formation. The vasculature then radiates out from each bud and connects with primordial vessels from neighboring buds. Later they connect with the central vasculature. Epithelial-mesenchymal recombination shows local vasculature is patterned by the epithelium, which expresses FGF2 and VEGF. Perturbing noggin expression leads to abnormal vascularization. To study endothelial origin, we compare transcriptomes of TIE1p.H2B-eYFP+ cells collected from the skin and aorta. Endothelial cells from the skin more closely resemble skin dermal cells than those from the aorta. The results show developing chicken skin vasculature is assembled by (1) physiological vasculogenesis from the peripheral tissue, and (2) subsequently connects with the central vasculature. The work implies mesenchymal plasticity and convergent differentiation play significant roles in development, and such processes may be re-activated during adult regeneration. SUMMARY STATEMENT: We show the vasculature network in the chicken skin is assembled using existing feather buds as the template, and endothelia are derived from local bud dermis and central vasculature.

This paper investigates the Hamiltonian energy of a modified Hindmarsh-Rose (HR) model to observe its effect on short-term memory. A Hamiltonian energy function and its variable function are given in the reduced system with a single node according to Helmholtz\'s theorem. We consider the role of the coupling strength and the links between neurons in the pattern formation to show that the coupling and cooperative neurons are necessary for generating the fire or a clear short-term memory when all the neurons are in sync. Then, we consider the effect of the degree and external stimulus from other neurons on the emergence and disappearance of short-term memory, which illustrates that generating short-term memory requires much energy, and the coupling strength could further reduce energy consumption. Finally, the dynamical mechanisms of the generation of short-term memory are concluded.

The milestone of compound leaf development is the generation of separate leaflet primordia during the early stages, which involves two linked but distinct morphogenetic events: leaflet initiation and boundary establishment for leaflet separation. Although some progress in understanding the regulatory pathways for each event have been made, it is unclear how they are intrinsically coordinated. Here, we identify the PINNATE-LIKE PENTAFOLIATA2 (PINNA2) gene encoding a newly identified GRAS transcription factor in Medicago truncatula. PINNA2 transcripts are preferentially detected at organ boundaries. Its loss-of-function mutations convert trifoliate leaves into a pinnate pentafoliate pattern. PINNA2 directly binds to the promoter region of the LEAFY orthologue SINGLE LEAFLET1 (SGL1), which encodes a key positive regulator of leaflet initiation, and downregulates its expression. Further analysis revealed that PINNA2 synergizes with two other repressors of SGL1 expression, the BEL1-like homeodomain protein PINNA1 and the C2H2 zinc finger protein PALMATE-LIKE PENTAFOLIATA1 (PALM1), to precisely define the spatiotemporal expression of SGL1 in compound leaf primordia, thereby maintaining a proper pattern of leaflet initiation. Moreover, we showed that the enriched expression of PINNA2 at the leaflet-to-leaflet boundaries is positively regulated by the boundary-specific gene MtNAM, which is essential for leaflet boundary formation. Together, these results unveil a pivotal role of the boundary-expressed transcription factor PINNA2 in regulating leaflet initiation, providing molecular insights into the coordination of intricate developmental processes underlying compound leaf pattern formation.

Spatial periodic signal for cell differentiation in some multicellular organisms is generated according to Turing\'s principle for pattern formation. How a dividing cell responds to the signal of differentiation is addressed with the filamentous cyanobacterium Nostoc sp. PCC 7120, which forms the patterned distribution of heterocysts. We show that differentiation of a dividing cell was delayed until its division was completed and only one daughter cell became heterocyst. A mutant of patU3, which encodes an inhibitor of heterocyst formation, showed no such delay and formed heterocyst pairs from the daughter cells of cell division or dumbbell-shaped heterocysts from the cells undergoing cytokinesis. The patA mutant, which forms heterocysts only at the filament ends, restored intercalary heterocysts by a single nucleotide mutation of patU3, and double mutants of patU3/patA and patU3/hetF had the phenotypes of the patU3 mutant. We provide evidence that HetF, which can degrade PatU3, is recruited to cell divisome through its C-terminal domain. A HetF mutant with its N-terminal peptidase domain but lacking the C-terminal domain could not prevent the formation of heterocyst pairs, suggesting that the divisome recruitment of HetF is needed to sequester HetF for the delay of differentiation in dividing cells. Our study demonstrates that PatU3 plays a key role in cell-division coupled control of differentiation.

Interaction between active materials and the boundaries of geometrical confinement is key to many emergent phenomena in active systems. For living active matter consisting of animal cells or motile bacteria, the confinement boundary is often a deformable interface, and it has been unclear how activity-induced interface dynamics might lead to morphogenesis and pattern formation. Here, we studied the evolution of bacterial active matter confined by a deformable boundary. We found that an ordered morphological pattern emerged at the interface characterized by periodically spaced interfacial protrusions; behind the interfacial protrusions, bacterial swimmers self-organized into multicellular clusters displaying +1/2 nematic defects. Subsequently, a hierarchical sequence of transitions from interfacial protrusions to creeping branches allowed the bacterial active drop to rapidly invade surrounding space with a striking self-similar branch pattern. We found that this interface patterning is geometrically controlled by the local curvature of the interface, a phenomenon we denote as collective curvature sensing. Using a continuum active model, we revealed that the collective curvature sensing arises from enhanced active stresses near high-curvature regions, with the active length scale setting the characteristic distance between the interfacial protrusions. Our findings reveal a protrusion-to-branch transition as a unique mode of active matter invasion and suggest a strategy to engineer pattern formation of active materials.

In an arid or semi-arid environment, precipitation plays a vital role in vegetation growth. Recent researches reveal that the response of vegetation growth to precipitation has a lag effect. To explore the mechanism behind the lag phenomenon, we propose and investigate a water-vegetation model with spatiotemporal nonlocal effects. It is shown that the temporal kernel function does not affect Turing bifurcation. For better understanding the influences of lag effect and nonlocal competition on the vegetation pattern formation, we choose some special kernel functions and obtain some insightful results: (i) Time delay does not trigger the vegetation pattern formation, but can postpone the evolution of vegetation. In addition, in the absence of diffusion, time delay can induce the occurrence of stability switches, while in the presence of diffusion, spatially nonhomogeneous time-periodic solutions may emerge, but there are no stability switches; (ii) The spatial nonlocal interaction may trigger the pattern onset for small diffusion ratio of water and vegetation, and can change the number and size of isolated vegetation patches for large diffusion ratio. (iii) The interaction between time delay and spatial nonlocal competition may induce the emergence of traveling wave patterns, so that the vegetation remains periodic in space, but is oscillating in time. These results demonstrate that precipitation can significantly affect the growth and spatial distribution of vegetation.

Formation building for multi-small-AUV systems with on-board cameras is crucial under the limited communication underwater environment. A hybrid coordination strategy is proposed for the rapid convergence to a leader-follower pattern. The strategy consists of two parts: a time-optimal local-position-based controller (TOLC) and a distributed asynchronous discrete weighted consensus controller (ADWCC). The TOLC controller is designed to optimize the assignation of AUVs\' destinations in the given pattern and guide each AUV to its destination by the shortest feasible distance. The ADWCC controller is developed to direct the AUVs blocked by obstacles to reach their destinations with the information from the perceived neighbors by on-board cameras. The rapidity of the proposed strategy is theoretically discussed. The effectiveness of the proposed algorithm has been verified in the simulation environments in both MATLAB and Blender.

Reputation and reciprocity are key mechanisms for cooperation in human societies, often going hand in hand to favor prosocial behavior over selfish actions. Here we review recent researches at the interface of physics and evolutionary game theory that explored these two mechanisms. We focus on image scoring as the bearer of reputation, as well as on various types of reciprocity, including direct, indirect, and network reciprocity. We review different definitions of reputation and reciprocity dynamics, and we show how these affect the evolution of cooperation in social dilemmas. We consider first-order, second-order, as well as higher-order models in well-mixed and structured populations, and we review experimental works that support and inform the results of mathematical modeling and simulations. We also provide a synthesis of the reviewed researches along with an outlook in terms of six directions that seem particularly promising to explore in the future.