Pattern dynamics in a Gierer–Meinhardt model with a saturating term

Gierer–Meinhardt system is a typical mathematical model to describe chemical and biological phenomena. In this paper, the Gierer–Meinhardt model with a saturating term is considered. By the linear stability analysis, we find the parameter area where possible Turing instability can occur. Then the multiple scales method is applied to obtain the amplitude equations at the critical value of Turing bifurcation, which help us to derive parameter space more specific where certain patterns such as spot-like pattern, stripe-like pattern and the coexistence pattern will emerge. Furthermore, the numerical simulations provide an indication of the wealth of patterns that the system can exhibit and the two-dimensional Fourier transform by the image processing interface of GUI from Matlab enables us to gain intensive understanding of these patterns. Besides, the interesting patterns including labyrinthine-like patterns are also numerically observed. All the results obtained reveal the mechanism of morphogenetic processes in adult mesenchymal cells. The patterns obtained corresponding to the patterns induced by morphogens in the vascular mesenchymal cells may play a role in atherosclerotic vascular calcification.     

Dynamics of pulse solutions in Gierer–Meinhardt model with time dependent diffusivity

Dispersive processes with a time dependent diffusivity appear in a plethora of physical systems. Most often a solution is attained for a predefined form of diffusion coefficient $D(t)$. Here existence of pulse solutions with an arbitrary time dependence thereof is proved for the Gierer–Meinhardt model with three types of transport: regular diffusion, sub-diffusion and Lévy flights. Admission of a solution of the classical pulse shape, but for an unencumbered form of $D(t)$ is a valuable property that allows to study phenomena of the ilk observed in various ostensibly unrelated applications. Closed form solutions are obtained for some pulse constellations. Transitions between periods of nearly constant diffusivities trigger respective cross-over between counterpart solutions known for a constant diffusivity, thereupon exhibiting otherwise unattainable behaviour, qualitatively reconstructing observable evolution peculiarities of tagged molecular structures, such as essential slowing down or speeding up during various stages of motion, inexplicable with a single constant diffusion coefficient.     

Bifurcation analysis of the Gierer–Meinhardt system with a saturation in the activator production

The reaction–diffusion Gierer–Meinhardt system with a saturation in the activator production is considered. Stability of the unique positive constant steady state solution is analysed, and associated Hopf bifurcations and steady state bifurcations are obtained. A global bifurcation diagram of non-trivial periodic orbits and steady state solutions with respect to key system parameters is obtained, which improves the understanding of dynamics of Gierer–Meinhardt system with a saturation in different parameter regimes.     

Spikes for the Gierer–Meinhardt System with Discontinuous Diffusion Coefficients

We rigorously prove results on spiky patterns for the Gierer–Meinhardt system (Kybernetik (Berlin) 12:30–39, 1972) with a jump discontinuity in the diffusion coefficient of the inhibitor. Using numerical computations in combination with a Turing-type instability analysis, this system has been investigated by Benson, Maini, and Sherratt (Math. Comput. Model. 17:29–34, 1993a; Bull. Math. Biol. 55:365–384, 1993b; IMA J. Math. Appl. Med. Biol. 9:197–213, 1992). Firstly, we show the existence of an interior spike located away from the jump discontinuity, deriving a necessary condition for the position of the spike. In particular, we show that the spike is located in one-and-only-one of the two subintervals created by the jump discontinuity of the inhibitor diffusivity. This localization principle for a spike is a new effect which does not occur for homogeneous diffusion coefficients. Further, we show that this interior spike is stable. Secondly, we establish the existence of a spike whose distance from the jump discontinuity is of the same order as its spatial extent. The existence of such a spike near the jump discontinuity is the second new effect presented in this paper. To derive these new effects in a mathematically rigorous way, we use analytical tools like Liapunov–Schmidt reduction and nonlocal eigenvalue problems which have been developed in our previous work (J. Nonlinear Sci. 11:415–458, 2001). Finally, we confirm our results by numerical computations for the dynamical behavior of the system. We observe a moving spike which converges to a stationary spike located in the interior of one of the subintervals or near the jump discontinuity.      

Stability and bifurcations of equilibria in a delayed Kirschner–Panetta model

In this paper, a delay differential equation model of immunotherapy for tumor-immune response is presented. The dynamics that interplays between the three model factors, namely, effector cells, tumor cells and interleukin-2 is studied and the quantitative analysis is performed. We estimate the length of delay to preserve the stability of an equilibrium state of biological significance. The impact of delay in the immunotherapy with interleukin-2, especially, at different antigenicity levels, is discussed, along with the scenarios under which the tumor remission can be prolonged.     

Period-doubling and Neimark–Sacker bifurcations in a larch budmoth population model

We investigate a discrete consumer-resource system based on a model originally proposed for studying the cyclic dynamics of the larch budmoth population in the Swiss Alps. It is shown that the moth population can persist indefinitely for all of the biologically feasible parameter values. Using intrinsic growth rate of the consumer population as a bifurcation parameter, we prove that the system can either undergo a period-doubling or a Neimark–Sacker bifurcation when the unique interior steady state loses its stability.     

Global attractivity of equilibrium in Gierer–Meinhardt system with activator production saturation and gene expression time delays

In this work we investigate a diffusive Gierer–Meinhardt system with gene expression time delays in the production of activators and inhibitors, and also a saturation in the activator production, which was proposed by Seirin Lee et al. (2010) [10]. We rigorously consider the basic kinetic dynamics of the Gierer–Meinhardt system with saturation. By using an upper and lower solution method, we show that when the saturation effect is strong, the unique constant steady state solution is globally attractive despite the time delays. This result limits the parameter space for which spatiotemporal pattern formation is possible.     

Hopf bifurcations of a ratio-dependent predator–prey model involving two discrete maturation time delays

In this paper we give a detailed Hopf bifurcation analysis of a ratio-dependent predator–prey system involving two different discrete delays. By analyzing the characteristic equation associated with the model, its linear stability is investigated. Choosing delay terms as bifurcation parameters the existence of Hopf bifurcations is demonstrated. Stability of the bifurcating periodic solutions is determined by using the center manifold theorem and the normal form theory introduced by Hassard et al. Furthermore, some of the bifurcation properties including direction, stability and period are given. Finally, theoretical results are supported by some numerical simulations.     

Stable spike clusters for the precursor Gierer–Meinhardt system in $$\mathbb {R}^2$$

We consider the Gierer–Meinhardt system with small inhibitor diffusivity, very small activator diffusivity and a precursor inhomogeneity. For any given positive integer k we construct a spike cluster consisting of k spikes which all approach the same nondegenerate local minimum point of the precursor inhomogeneity. We show that this spike cluster can be linearly stable. In particular, we show the existence of spike clusters for spikes located at the vertices of a polygon with or without centre. Further, the cluster without centre is stable for up to three spikes, whereas the cluster with centre is stable for up to six spikes. The main idea underpinning these stable spike clusters is the following: due to the small inhibitor diffusivity the interaction between spikes is repulsive, and the spikes are attracted towards the local minimum point of the precursor inhomogeneity. Combining these two effects can lead to an equilibrium of spike positions within the cluster such that the cluster is linearly stable.     

Hopf bifurcations in a predator–prey system with a discrete delay and a distributed delay

A delayed Lotka–Volterra two-species predator–prey system with discrete hunting delay and distributed maturation delay for the predator population described by an integral with a strong delay kernel is considered. By linearizing the system at the positive equilibrium and analyzing the associated characteristic equation, the asymptotic stability of the positive equilibrium is investigated and Hopf bifurcations are demonstrated. It is found that under suitable conditions on the parameters the positive equilibrium is asymptotically stable when the hunting delay is less than a certain critical value and unstable when the hunting delay is greater than this critical value. Meanwhile, according to the Hopf bifurcation theorem for functional differential equations (FDEs), we find that the system can also undergo a Hopf bifurcation of nonconstant periodic solution at the positive equilibrium when the hunting delay crosses through a sequence of critical values. In particular, by applying the normal form theory and the center manifold reduction for FDEs, an explicit algorithm determining the direction of Hopf bifurcations and the stability of bifurcating periodic solutions occurring through Hopf bifurcations is given. Finally, to verify our theoretical predictions, some numerical simulations are also included at the end of this paper.     

Stability and Hopf bifurcation for a delayed predator–prey model with disease in the prey

This paper is concerned with a mathematical model dealing with a predator–prey system with disease in the prey. Mathematical analysis of the model regarding stability has been performed. The effect of delay on the above system is studied. By regarding the time delay as the bifurcation parameter, the stability of the positive equilibrium and Hopf bifurcations are investigated. Furthermore, the direction of Hopf bifurcations and the stability of bifurcated periodic solutions are determined by applying the normal form theory and the center manifold reduction for functional differential equations. Finally, to verify our theoretical predictions, some numerical simulations are also included.     

Controlling bifurcations and chaos in TCP–UDP–RED

We study the bifurcation and chaotic behavior of the Transmission Control Protocol (TCP) and User Datagram Protocol (UDP) network with Random Early Detection (RED) queue management. These bifurcation and chaotic behaviors may cause heavy oscillation of an average queue length and induce network instability. We propose an impulsive control method for controlling bifurcations and chaos in the internet congestion control system. The theoretical analysis and the simulation experiments show that this method can obtain the stable average queue length without sacrificing the other advantages of RED.     

Phase portraits,Hopf bifurcations and limit cycles of the Holling–Tanner models for predator–prey interactions

The phase portraits, existence and uniqueness of stable limit cycles and Hopf bifurcations of the well-known Holling–Tanner models for predator–prey interactions are studied. The ranges of the parameters involved are provided under which the unique interior equilibrium can be determined to be a stable (or an unstable) node or focus. The Hopf bifurcations and the existence and uniqueness of stable limit cycles of the models are obtained by computing the Lyapunov number involved. Our results confirm some previous results observed and suggested from the real ecological systems.     

Hopf bifurcation in a conceptual climate model with ice–albedo and precipitation–temperature feedbacks

In this paper we analyse a dynamical system based on the so-called KCG (Källén, Crafoord, Ghil) conceptual climate model. This model describes an evolution of the globally averaged temperature and the average extent of the ice sheets. In the nondimensional form the equations can be simplified facilitating the subsequent analysis. We consider the limiting case of a stationary snow line for which the phase plane can be completely analysed and the type of each critical point can be determined. One of them can exhibit the Hopf bifurcation and we find sufficient conditions for its existence. Those, in turn, have a straightforward physical meaning and indicate that the model predicts internal oscillations of the climate. Using the typical real-world values of appearing parameters we conclude that the obtained results are in the same ballpark as the conditions on our planet during the quaternary ice ages. Our analysis is a rigorous justification of a generalization of some previous results by KCG and other authors.     

Hopf bifurcation and bistability of a nutrient–phytoplankton–zooplankton model

In this paper we consider a nutrient–phytoplankton–zooplankton model in aquatic environment and study its global dynamics. The existence and stability of equilibria are analyzed. It is shown that the system is permanent as long as the coexisting equilibrium exists. The discontinuous Hopf and classical Hopf bifurcations of the model are analytically verified. It is shown that phytoplankton bloom may occur even if the input rate of nutrient is low. Numerical simulations reveal the existence of saddle-node bifurcation of nonhyperbolic periodic orbit and subcritical discontinuous Hopf bifurcation, which presents a bistable phenomenon (a stable equilibrium and a stable limit cycle).     

Hopf bifurcation at the nonzero foci in 1∶4 resonance

The differential equation

Hopf bifurcation in a delayed reaction–diffusion–advection population model

In this paper, we investigate a reaction–diffusion–advection model with time delay effect. The stability/instability of the spatially nonhomogeneous positive steady state and the associated Hopf bifurcation are investigated when the given parameter of the model is near the principle eigenvalue of an elliptic operator. Our results imply that time delay can make the spatially nonhomogeneous positive steady state unstable for a reaction–diffusion–advection model, and the model can exhibit oscillatory pattern through Hopf bifurcation. The effect of advection on Hopf bifurcation values is also considered, and our results suggest that Hopf bifurcation is more likely to occur when the advection rate increases.