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We hypothesize that dynamical systems concepts used to study the transition to turbulence in shear flows are applicable to other transition phenomena in fluid mechanics. In this paper, we consider a finite air bubble that propagates within a Hele-Shaw channel containing a depth-perturbation. Recent experiments revealed that the bubble shape becomes more complex, quantified by an increasing number of transient bubble tips, with increasing flow rate. Eventually, the bubble changes topology, breaking into multiple distinct entities with non-trivial dynamics. We demonstrate that qualitatively similar behaviour to the experiments is exhibited by a previously established, depth-averaged mathematical model and arises from the model’s intricate solution structure. For the bubble volumes studied, a stable asymmetric bubble exists for all flow rates of interest, while a second stable solution branch develops above a critical flow rate and transitions between symmetric and asymmetric shapes. The region of bistability is bounded by two Hopf bifurcations on the second branch. By developing a method for a numerical weakly nonlinear stability analysis we show that unstable periodic orbits (UPOs) emanate from the first Hopf bifurcation. Moreover, as has been found in shear flows, the UPOs are edge states that influence the transient behaviour of the system.

Mechanical instabilities can be exploited to design innovative structures, able to change their shape in the presence of external stimuli. In this work, we derive a mathematical model of an elastic beam subjected to an axial force and constrained to smoothly slide along a rigid support, where the distance between the rod midline and the constraint is fixed and finite. Using both theoretical and computational techniques, we characterize the bifurcations of such a mechanical system, in which the axial force and the natural curvature of the beam are used as control parameters. We show that, in the presence of a straight support, the rod can deform into shapes exhibiting helices and perversions, namely transition zones connecting together two helices with opposite chirality. The mathematical predictions of the proposed model are also compared with some experiments, showing a good quantitative agreement. In particular, we find that the buckled configurations may exhibit multiple perversions and we propose a possible explanation for this phenomenon based on the energy landscape of the mechanical system.

In this work, we have studied a spatiotemporal prey–predator model with Allee effect in prey and hunting cooperation in predator. In available literature, a prey-dependent functional response is mostly considered to model the prey–predator interaction. But empirical data show that functional response can depend on both prey and predator populations. Here, we have introduced the cooperative hunting in a Holling type III functional response for the predator population and extended the model spatially. Both Turing and non-Turing patterns produced by the diffusion added prey–predator model have been studied in detail. Emphasis is given to the analytical study of the spiral and target patterns applying the amplitude equation through weakly nonlinear analysis. The analytical results are verified with extensive numerical simulations.

To investigate the formation mechanism of vegetation patterns in dry-land ecosystems, this paper delves into the impact of non-local grazing on the stability and spatiotemporal dynamics of a plant-water model. We first establish the conditions for the occurrence of codimension-1 bifurcations: Turing bifurcations, Hopf bifurcations, as well as codimension-2 bifurcations: Turing–Turing bifurcations, Turing-Hopf bifurcations, and determine the stable and unstable regions of the positive equilibrium. Regarding Turing bifurcations, utilizing weakly nonlinear analysis methods to derive amplitude equations, we conclude that under the influence of non-local grazing, the system exhibits complex patch patterns, including spot, mixed, and stripe patterns. The main analytical challenges arise from non-local interactions, which increase the difficulty of deriving the amplitude equations. From a biological perspective, besides water and nutrients, herbivores also play a significant role in the self-organization of patch patterns in dry-land ecosystems.

This paper studies the influence of orienting external fields on pattern formation, particularly mesoscale turbulence, in microswimmer suspensions. To this end, we apply a hydrodynamic theory that can be derived from a microscopic microswimmer model (Reinken et al 2018 Phys. Rev. E 97, 022613). The theory combines a dynamic equation for the polar order parameter with a modified Stokes equation for the solvent flow. Here, we extend the model by including an external field that exerts an aligning torque on the swimmers (mimicking the situation in chemo-, photo-, magneto- or gravitaxis). Compared to the field-free case, the external field breaks the rotational symmetry of the vortex dynamics and leads instead to strongly asymmetric, traveling stripe patterns, as demonstrated by numerical solution and linear stability analysis. We further analyze the emerging structures using a reduced model which involves only an (effective) microswimmer velocity field. This model is significantly easier to handle analytically, but still preserves the main features of the anisotropic pattern formation. We observe an underlying transition between a square vortex lattice and a traveling stripe pattern. These structures can be well described in the framework of weakly nonlinear analysis, provided the strength of nonlinear advection is sufficiently weak.

In order to stop and reverse land degradation and curb the loss of biodiversity, the United Nations 2030 Agenda for Sustainable Development proposes to combat desertification. In this paper, a fractional vegetation–water model in an arid flat environment is studied. The pattern behavior of the fractional model is much more complex than that of the integer order. We study the stability and Turing instability of the system, as well as the Hopf bifurcation of fractional order α, and obtain the Turing region in the parameter space. According to the amplitude equation, different types of stationary mode discoveries can be obtained, including point patterns and strip patterns. Finally, the results of the numerical simulation and theoretical analysis are consistent. We find some novel fractal patterns of the fractional vegetation–water model in an arid flat environment. When the diffusion coefficient, d, changes and other parameters remain unchanged, the pattern structure changes from stripes to spots. When the fractional order parameter, β, changes, and other parameters remain unchanged, the pattern structure becomes more stable and is not easy to destroy. The research results can provide new ideas for the prevention and control of desertification vegetation patterns.

Due to factors such as climate change, natural disasters, and deforestation, most measurement processes and initial data may have errors. Therefore, models with imprecise parameters are more realistic. This paper constructed a new predator-prey model with an interval biological coefficient by using the interval number as the model parameter. First, the stability of the solution of the fractional order model without a diffusion term and the Hopf bifurcation of the fractional order$ \\alpha $were analyzed theoretically. Then, taking the diffusion coefficient of prey as the key parameter, the Turing stability at the equilibrium point was discussed. The amplitude equation near the threshold of the Turing instability was given by using the weak nonlinear analysis method, and different mode selections were classified by using the amplitude equation. Finally, we numerically proved that the dispersal rate of the prey population suppressed the spatiotemporal chaos of the model.

In this paper, we study a modification of the mathematical model describing inflammation and demyelination patterns in the brain caused by Multiple Sclerosis proposed in Lombardo et al. (J Math Biol 75:373–417, 2017). In particular, we hypothesize a minimal amount of macrophages to be able to start and sustain the inflammatory response. Thus, the model function for macrophage activation includes an Allee effect. We investigate the emergence of Turing patterns by combining linearised and weakly nonlinear analysis, bifurcation diagrams and numerical simulations, focusing on the comparison with the previous model.

This paper presents a novel high-order numerical method. The proposed scheme utilizes polynomial generating functions to achieve p order accuracy in time for the Grünwald–Letnikov fractional derivatives, while maintaining second-order spatial accuracy. By incorporating a short-memory principle, the method remains computationally efficient for long-time simulations. The authors rigorously analyze the stability of equilibrium points for the fractional vegetation–water model and perform a weakly nonlinear analysis to derive amplitude equations. Convergence analysis confirms the scheme’s consistency, stability, and convergence. Numerical simulations demonstrate the method’s effectiveness in exploring how different fractional derivative orders influence system dynamics and pattern formation, providing a robust tool for studying complex fractional systems in theoretical ecology.

In this paper, the pattern formation of a volume-filling chemotaxis model with bistable source terms was studied. First, it was shown that self-diffusion does not induce Turing patterns, but chemotaxis-driven instability occurs. Then, the asymptotic behavior of the chemotaxis model was analyzed by weakly nonlinear analysis with the method of multiple scales. When the chemotaxis coefficient exceeded a threshold value and there was a single unstable mode, the supercritical and subcritical bifurcation of the model was discussed. The amplitude equations and the asymptotic expressions of the patterns were obtained. When the chemotaxis coefficient was large enough, the two-mode competition behavior of the model with two unstable modes was analyzed, and the corresponding amplitude equations and the asymptotic expressions of the patterns were obtained. Finally, numerical simulations were provided to further illuminate the above analytical results.