Narrow Results

This paper is concerned with a classical two-species prey–predator reaction–diffusion system with ratio-dependent functional response and subject to homogeneous Neumann boundary condition in a two-dimensional rectangle domain. By analyzing the associated eigenvalue problem, the spatially homogeneous Hopf bifurcation curve and Turing bifurcation curve of system at the constant coexistence equilibrium are established. Then when the bifurcation parameter is in the interior of range for Turing instability and near Turing bifurcation curve, the amplitude equations of the original system near the constant coexistence equilibrium are obtained by multiple-scale time perturbation analysis. On the basis of the obtained amplitude equations, the stability and classifications of spatiotemporal patterns of the original system at the constant coexistence equilibrium are discussed. Finally, to verify the validity of the obtained theoretical results, numerical simulations are also carried out.

From the perspective of mathematical epidemiology, COVID-19 epidemics emerge due to instabilities in epidemiological systems. It is shown that the COVID-19 outbreaks follow highly specified paths in epidemiological state spaces. These paths are described by phase portraits that can be readily computed from epidemiological models defined in terms of nonlinear dynamical systems. The paths are predicted by order parameters and amplitude equations that are well known in nonlinear physics and synergetics to exist at instability points. The approach is illustrated for SIR, SEIR and SEIAR models and epidemic outbreaks in China, Italy and West Africa. Identifying such COVID-19 order parameters may help in forecasting COVID-19 epidemics and predicting the impacts of intervention measures.

The dynamics of an interface connecting a stationary stripe pattern with a homogeneous state is studied. The conventional approach which describes this interface, Newell–Whitehead–Segel amplitude equation, does not account for the rich dynamics exhibited by these interfaces. By amending this amplitude equation with a nonresonate term, we can describe this interface and its dynamics in a unified manner. This model exhibits a rich and complex transversal dynamics at the interface, including front propagations, transversal patterns, locking phenomenon, and transversal localized structures.

We study the bifurcation and stability of constant stationary solutions (u₀, v₀) of a particular system of parabolic partial differential equations as amplitude equations on a bounded domain (0, L) with Neumann boundary conditions. In this paper, the asymptotic behavior of the solutions (u₀, v₀) of the amplitude equations are considered. With the length L of the domain regarded as bifurcation parameter, branches of nontrivial solutions are shown by using the perturbation method. Moreover, local behavior of these branches are studied. We also analyze the stability of the bifurcated solutions.

The amplitude equation describes a reduced form of a reaction–diffusion system, yet still retains its essential dynamical features. By approximating the analytic solution, the amplitude equation allows the examination of mode instability when the system is near a bifurcation point. Multiple-scale expansion (MSE) offers a straightforward way to systematically derive the amplitude equations. The method expresses the single independent variable as an asymptotic power series consisting of newly introduced independent variables with differing time and space scales. The amplitude equations are then formulated under the solvability conditions which remove secular terms. To our knowledge, there is little information in the research literature that explains how the exhaustive workflow of MSE is applied to a reaction–diffusion system. In this paper, detailed mathematical operations underpinning the MSE are elucidated, and the practical ways of encoding these operations using MAPLE are discussed. A semi-automated MSE computer algorithm Amp_solving is presented for deriving the amplitude equations in this research. Amp_solving has been applied to the classical Brusselator model for the derivation of amplitude equations when the system is in the vicinity of a Turing codimension-1 and a Turing–Hopf codimension-2 bifurcation points. Full open-source Amp_solving codes for the derivation are comprehensively demonstrated and available to the public domain.

A Lotka–Volterra model for an arbitrary number of competitors is studied for different ratios of self-inhibition versus cross-inhibition. It is shown that winner-takes-all fixed points (states of single surviving species) are the only stable fixed points of the model when cross-inhibition exceeds self-inhibition. Secondary bifurcations in terms of bifurcations between winner-takes-all fixed points induced by changes in the exponential growth rates of competitors are studied and the critical control parameters are identified. A selection principle is derived that states that evolution proceeds in such a way that exponential growth rates of surviving competitors are magnified in evolutionary bifurcation steps. The interacting competitor model is applied as an amplitude equation model for interacting patterns of self-organizing pattern formation systems with an eye on action selection and compulsive behaviors in humans. The possibility is discussed that human behavior is subjected to the selection principle of "faster growth rates".

The Lotka–Volterra–Haken equations have been frequently used in ecology and pattern formation. Recently, the equations have been proposed by several research groups as amplitude equations for task-related patterns of brain activity. In this theoretical study, the focus is on the circular causality aspect of pattern formation systems as formulated within the framework of synergetics. Accordingly, the stable modes of a pattern formation system inhibit the unstable modes, whereas the unstable modes excite the stable modes. Using this circular causality principle it is shown that under certain conditions the Lotka–Volterra–Haken amplitude equations can be derived from a general model of brain activity akin to the Wilson–Cowan model. The model captures the amplitude dynamics for brain activity patterns in experiments involving several consecutively performed multiple-choice tasks. This is explicitly demonstrated for two-choice tasks involving grasping and walking. A comment on the relevance of the theoretical framework for clinical psychology and schizophrenia is given as well.

The spatiotemporal pattern induced by cross-diffusion of a toxic-phytoplankton–zooplankton model with nonmonotonic functional response is investigated in this paper. The linear stability analysis shows that cross-diffusion is the key mechanism for the formation of spatial patterns. By taking cross-diffusion rate as bifurcation parameter, we derive amplitude equations near the Turing bifurcation point for the excited modes in the framework of a weakly nonlinear theory, and the stability analysis of the amplitude equations interprets the structural transitions and stability of various forms of Turing patterns. Furthermore, we illustrate the theoretical results via numerical simulations. It is shown that the spatiotemporal distribution of the plankton is homogeneous in the absence of cross-diffusion. However, when the cross-diffusivity is greater than the critical value, the spatiotemporal distribution of all the plankton species becomes inhomogeneous in spaces and results in different kinds of patterns: spot, stripe, and the mixture of spot and stripe patterns depending on the cross-diffusivity. Simultaneously, the impact of toxin-producing rate of toxic-phytoplankton (TPP) species and natural death rate of zooplankton species on pattern selection is also explored.

We explore the impact of time-periodic forcing on pattern transitions in a plankton system of reaction–diffusion type. Here, we mainly focus on the forced states near the Turing–Hopf bifurcation. A normal form analysis leads to the finding that weak forcing exhibits a destabilizing effect on the dynamics by exciting the transitions from a spatially homogeneous stationary state to a periodic oscillation in time. The results are obtained by studying the amplitude equations derived using weakly nonlinear analysis in the presence of forcing, which enables us to calculate the changes of states. Examples are given to confirm the theoretical results.

In this paper, we have presented Turing pattern selection in a spatial epidemic model with zero-flux boundary conditions, for which we have given a general survey of Hopf and Turing bifurcations, and have derived amplitude equations for the excited modes. Furthermore, we present novel numerical evidence of typical Turing patterns, and find that the model dynamics exhibits complex pattern replication: on increasing the control parameter r, the sequence "H₀-hexagons → H₀-hexagon-stripe mixtures → stripes → H_π-hexagon-stripe mixtures → H_π-hexagons" is observed. This may enrich the research of the pattern formation in diffusive epidemic models.

Spatial patterns through diffusion-driven instability are stationary structures that appear spontaneously upon breaking the symmetry of the spatial domain, which results only from the coupling between the reaction and the diffusion processes. This paper is concerned with a modified Leslie–Gower-type model with cross-diffusion and indirect predation effect. We first prove the global existence, non-negativity and uniform boundedness for the considered model. Then the linear stability analysis shows that the cross-diffusion is the key mechanism of spatiotemporal pattern formation. Amplitude equations are derived near Turing bifurcation point under nonlinear cross-diffusion to interpret pattern selection among spot pattern, stripe pattern and the mixture of spot and stripe patterns, which reflects the species’s spatially inhomogeneous distribution, and it is also found that the fear factor has great influence on spatially inhomogeneous distribution of the two species under certain cross-diffusivity, that is, high level of fear can induce striped inhomogeneous distribution, low level of fear can induce spotted inhomogeneous distribution, and the intermediate level of fear can induce the mixture of spotted and striped inhomogeneous distribution. Finally, numerical simulations illustrate the effectiveness of all theoretical results.

This paper considers a diffusive prey–predator system with fear and group defense in the prey population. Also, we consider that the mortality of predators is linear and quadratic. By using local stability analysis, we get the prerequisite of Turing instability. Using comprehensive numerical computations, we get non-Turing pattern formation in the system with linear death of predator. Turing patterns are obtained for the system with the quadratic death of the predator. The modeling technique of multiple scale analysis is used to determine amplitude equations near the Turing bifurcation origin for the model with the predator’s quadratic mortality rate. The amplitude equations stability leads to various Turing patterns such as spots, stripes, and mixed. The result focuses on changing the mortality rate linear to quadratic of a predator in the prey–predator system. The derived results support us in a more immeasurable understanding of prey–predator interaction dynamics in the actual world.

A weakly nonlinear analysis provides a system constituting amplitude equations and its related analysis is capable of predicting parameter regimes with different patterns expected to co-exist in dynamical circumstances that exhibit complex fractional-order system characteristics. The Turing mechanism of pattern formation as a result of diffusion-induced instability of the homogeneous steady state is concerned with unpredictable conditions. The Turing instability caused by fractional diffusion in a Human Immunodeficiency Virus model has been addressed in this study. It is important that the effect of the Human Immunodeficiency Virus to the immune system can be modeled by the interaction of uninfected cells, unhealthy cells, virus particles and antigen-specific. Initially, all potential equilibrium points are defined and the stability of the interior equilibrium point is then evaluated using the Routh–Hurwitz criteria. The conditions for Turing instability are obtained by local equilibrium points with stability analysis. In the neighborhood of the Turing bifurcation point, weakly nonlinear analysis is employed to deduce the amplitude equations. After applying amplitude equations, it is observed that this system has a very rich dynamical behavior. The constraints for the formation of the patterns like a hexagon, spot, mixed and stripe patterns are identified for the amplitude equations by dynamic analysis. Furthermore, by using the numerical simulations, the theoretical results are verified. Within this framework, this study through the dynamical behavior of the complex system perspective and bifurcation point based on the viral death rate can provide the basis for several researchers working on Human Immunodeficiency Virus model through various aspects. Accordingly, the Turing bifurcation point and weakly nonlinear analysis employed within the complex fractional-order dynamics addressed herein are highly relevant experimentally since the related effects can be studied and applied concerning different mathematical, physical, engineering and biological models.

Density distributions of populations with self-diffusion and interaction in a spatial domain are dynamically visualized with coupled nonlinear reaction–diffusion equations. Incorporating self-diffusion terms creates a more pragmatic modeling paradigm and provides meaningful descriptions of influences on spatiotemporal pattern formation phenomena. This paper examines the effect of self-diffusion in a food chain system with a Holling type-IV functional response and the type of spatial structures forms on a geographical scale due to the random movement of species. We discussed the existence and uniqueness of a positive equilibrium solution and obtained the Turing instability conditions for the self-diffusive food chain model. Moreover, weakly nonlinear analysis close to the Turing bifurcation boundary is used to derive the amplitude equations. The stability of the amplitude equations and sufficient conditions for the emanation of spatiotemporal patterns (such as spots, stripes, and blended patterns) are investigated. The analytical results are verified with numerical simulations. The results are applicable to all environments and can be used to understand the effects of self-diffusion in other food chain models both qualitatively and quantitatively.

We study stochastic partial differential equations (SPDEs) with potentially very rough fractional noise with Hurst parameter $H\in (0,1)$. Close to a change of stability measured with a small parameter $𝜀$, we rely on the natural separation of time-scales and establish a simplified description of the essential dynamics. Up to an error term bounded by a power of $𝜀$ depending on the Hurst parameter we can approximate the solution of the SPDE in first order by an SDE, the so-called amplitude equation, which describes the amplitude of the dominating pattern changing stability. In second order the approximation is given by a fast infinite-dimensional Ornstein–Uhlenbeck process. To this aim, we need to establish an explicit averaging result for stochastic integrals driven by rough fractional noise for small Hurst parameters.

Several recent modeling studies have attempted to understand the human immune reaction to SARS-CoV-2 infections. Such a model-based understanding provides a sound basis for fighting COVID-19 on the level of individual patients. However, in this context, a worked-out nonlinear physics analysis providing insights into underlying amplitude equations and potential COVID-19 order parameters has not been conducted so far. In order to conduct such an analysis, a three-variable virus dynamics model is considered that can account for the human immune reaction to SARS-CoV-2 infections. The model amplitudes equations are derived and the relevant order parameter is determined. In line with theoretical reasoning, it is demonstrated that the order parameter and its amplitude determine the initial stage of SARS-CoV-2 infections, in general, and the initial dynamics of immune reactions, in particular. Explicitly, this finding is demonstrated for data from four COVID-19 patients. For those patients it is also demonstrated that the remnant of the order parameter determines the final disease decline phase. In this context, a time-resolved eigenvalue analysis is conducted that reveals that the transition from the initial stage to the decline stage which is associated with a switch of the leading eigenvalue from a positive to a negative value. It is argued that the immune reaction essentially contributes to this switch. From a medical-physics point of view, this observation suggests that the immune reaction of COVID-19 patients can stabilize the virus-free fixed point of affected sites.