Bistable Case Research Articles

Traveling wave solutions in a half-space for boundary reactions

We prove the existence and uniqueness of a traveling front and of its speed for the homogeneous heat equation in the half-plane with a Neumann boundary reaction term of non-balanced bistable type or of combustion type. We also establish the monotonicity of the front and, in the bistable case, its behavior at infinity. In contrast with the classical bistable interior reaction model, its behavior at the side of the invading state is of power type, while at the side of the invaded state its decay is exponential. These decay results rely on the construction of a family of explicit bistable traveling fronts. Our existence results are obtained via a variational method, while the uniqueness of the speed and of the front rely on a comparison principle and the sliding method.

Multi-type Entire Solutions in a Nonlocal Dispersal Epidemic Model

This paper deals with entire solutions of a nonlocal dispersal epidemic model. Unlike local (random) dispersal problems, a nonlocal dispersal operator is not compact and the solutions of nonlocal dispersal system studied here lack regularity in suitable spaces, which affects the uniform convergence of the solution sequences and the technique details in constructing the entire solutions. In the monostable case, some new types of entire solutions are constructed by combining leftward and rightward traveling fronts with different speeds and a spatially independent solution. In the bistable case, the existence of many different entire solutions with merging fronts are proved by constructing different sub- and super-solutions. Various qualitative features of the entire solutions are also investigated. A key idea is to characterize the asymptotic behaviors of the traveling wave solutions at infinite in terms of appropriate sub- and super-solutions. Finally, we also obtain the smoothness of the entire solutions in space, i.e., the solutions established in our paper are global Lipschitz continuous in space.

A multi-stable energy harvester: Dynamic modeling and bifurcation analysis

A theoretical investigation is conducted on the dynamic and energetic characteristics of a multi-stable bimorph cantilever energy harvester that uses magnetic attraction effect. The multi-stable energy harvester under study is composed of a bimorph cantilever beam with soft magnetic tip and two externally fixed permanent magnets that are arranged in series. With this configuration, the magnetic force and the moment that are exerted on the cantilever tip tend to be highly dependent on the magnetic field induced by the external magnets. Such an energy harvester can possess multi-stable potential functions, ranging from mono-stable to penta-stable. The mechanism that governs the formation of this multi-stability is thoroughly identified and examined thorough a bifurcation analysis performed on the system׳s equilibrium solutions. From this analysis, it is found that the transitions between these multi-stable states occur through very complicated bifurcation scenarios that include degenerate pitchfork bifurcations and mergers of pitchfork bifurcations or saddle-node bifurcations. Bifurcation set diagram is obtained, which is composed of five separate parametric regions, from mono- to penta-stability. The resulting stability map satisfactorily describes the multi-stable characteristics of the present energy harvester. In addition, the dynamic and energetic characteristics of the present multi-stable energy harvester are more thoroughly examined using its potential energy diagrams and a series of numerical simulations, and the obtained results are compared with those for the equivalent bi-stable cases.

Traveling Wave Solutions of Reaction-Diffusion Equations Arising in Atherosclerosis Models

In this short review article, two atherosclerosis models are presented, one as a scalar equation and the other one as a system of two equations. They are given in terms of reaction-diffusion equations in an infinite strip with nonlinear boundary conditions. The existence of traveling wave solutions is studied for these models. The monostable and bistable cases are introduced and analyzed.

On the basins of attraction for a class of delay differential equations with non-monotone bistable nonlinearities

We consider the delay differential equation (DDE) x˙(t)=−g(x(t))+f(x(t−τ)) which shares the same equilibria with the corresponding ordinary differential equation (ODE) x˙(t)=−g(x(t))+f(x(t)). For the bistable case, both the DDE and ODE share three equilibria x0=0<x1<x2 with x0 and x2 being stable and x1 being unstable for the ODE. We are concerned with stability of these equilibria for the DDE and the basins of attraction of x0 and x2 when they are asymptotically stable for the DDE. Combining the idea of relating the dynamics of a map to the dynamics of a DDE and invariance arguments for the solution semiflow, we are able to characterize some subsets of basins of attraction of these equilibria for the DDE. In addition, existence of heteroclinic orbits is also explored. The general results are applied to a particular model equation describing the matured population of some species demonstrating the Allee effect.

Entire solutions in periodic lattice dynamical systems

This paper deals with entire solutions of periodic lattice dynamical systems. Unlike homogeneous problems, the periodic equation studied here lacks symmetry between increasing and decreasing pulsating traveling fronts, which affects the construction of entire solutions. In the bistable case, the existence, uniqueness and Liapunov stability of entire solutions are proved by constructing different sub- and supersolutions. In the monostable case, the existence and asymptotic behavior of spatially periodic solutions connecting two steady states are first established. Some new types of entire solutions are then constructed by combining leftward and rightward pulsating traveling fronts with different speeds and a spatially periodic solution. Various qualitative features of the entire solutions are also investigated.

Dynamical behaviors determined by the Lyapunov function in competitive Lotka-Volterra systems

Dynamical behaviors of the competitive Lotka-Volterra system even for 3 species are not fully understood. In this paper, we study this problem from the perspective of the Lyapunov function. We construct explicitly the Lyapunov function using three examples of the competitive Lotka-Volterra system for the whole state space: (1) the general 2-species case, (2) a 3-species model, and (3) the model of May-Leonard. The basins of attraction for these examples are demonstrated, including cases with bistability and cyclical behavior. The first two examples are the generalized gradient system, where the energy dissipation may not follow the gradient of the Lyapunov function. In addition, under a new type of stochastic interpretation, the Lyapunov function also leads to the Boltzmann-Gibbs distribution on the final steady state when multiplicative noise is added.

Generation of hysteresis cycles with two and four jumps in a shape memory oscillator

A numerical investigation of the generation of hysteresis cycles with two and four jumps in a shape memory oscillator is presented. The fast subsystem is examined using local stability and bifurcations. When the upper and lower attractors disappear, the fast subsystem can be stable or bi-stable. This leads to a classification of the bifurcation behaviors of the fast subsystem, i.e., the stable case and the bi-stable case, which are associated with different parameter conditions. For the stable case, the hysteresis cycles with two and four jumps are obtained by properly modulating the slow external forcing, and the associated dynamical mechanisms are analyzed. While for the bi-stable case, the transition mechanisms underlying the appearance of hysteresis cycles with two and four jumps are revealed by computing attraction domains of the attractors.

Reaction-diffusion waves with nonlinear boundary conditions

A reaction-diffusion equation with nonlinear boundary condition is considered in a two-dimensional infinite strip. Existence of waves in the bistable case is proved by the Leray-Schauder method.

Entire Solutions in Lattice Delayed Differential Equations with Nonlocal Interaction: Bistable Cases

This paper is concerned with entire solutions of a class of bistable delayed lattice differential equations with nonlocal interaction. Here an entire solution is meant by a solution defined for all (n, t) is an element of Z x R. Assuming that the equation has an increasing traveling wave front with nonzero wave speed and using a comparison argument, we obtain a two-dimensional manifold of entire solutions. In particular, it is shown that the traveling wave fronts are on the boundary of the manifold. Furthermore, uniqueness and stability of such entire solutions are studied.

Existence of Reaction-Diffusion Waves with Nonlinear Boundary Conditions

The paper is devoted to a reaction-diffusion equation in an infinite two-dimensional strip with nonlinear boundary conditions. The existence of travelling waves is proved in the bistable case by the Leray-Schauder method. It is based on a topological degree for elliptic problems in unbounded domains and on a priori estimates of solutions.

Recent developments on wave propagation in 2-species competition systems

In this paper, we shall survey some recent results on the wavepropagation in 2-species competition systems with Lotka-Volterratype nonlinearity. This includes systems with continuous anddiscrete diffusion (or migration). We are interested in bothmonostable case and bistable with strong competition case. Questionson minimal speed for the monostable case, uniqueness of wave speedand propagation failure in the bistable case, monotonicity anduniqueness of wave profile for both cases are addressed. Finally, wegive some open problems on wave propagation in 2-species competitionsystems.

Inside dynamics of pulled and pushed fronts

We investigate the inside structure of one-dimensional reaction–diffusion traveling fronts. The reaction terms are of the monostable, bistable or ignition types. Assuming that the fronts are made of several components with identical diffusion and growth rates, we analyze the spreading properties of each component. In the monostable case, the fronts are classified as pulled or pushed ones, depending on the propagation speed. We prove that any localized component of a pulled front converges locally to 0 at large times in the moving frame of the front, while any component of a pushed front converges to a well determined positive proportion of the front in the moving frame. These results give a new and more complete interpretation of the pulled/pushed terminology which extends the previous definitions to the case of general transition waves. In particular, in the bistable and ignition cases, the fronts are proved to be pushed as they share the same inside structure as the pushed monostable critical fronts. Uniform convergence results and precise estimates of the left and right spreading speeds of the components of pulled and pushed fronts are also established.

Survivor’s dilemma: Defend the group or flee?

We consider a survival game of gregarious individuals, in which the aim of the players is survival to reproductive age under predator attacks. The survivor’s dilemma (shortly: SVD) game consists in the following: a group member either surely survives alone by fleeing, while its defensive mate may be killed; or tries to save its mate’s life, risking to get killed. The dilemma is that, in every single attack, fleeing ensures maximal survival probability, but if its mate survives by fighting both, and they remain together, its risk to be killed at the next attack will be lower. We show that, if defense is successful enough, then the one-attack game is a prisoner’s dilemma (PD), where fleeing is the strict ESS. We have additively decomposed the SVD game, according to the survival of the group mate of the focal prey, into two games: the aim of the “ collective game” is survival of the group of prey. Counter-wise, the aim of the “ hostile game” is survival alone (focal prey survives and its mate is killed by the predator). We obtain the following results: if the attack number is large enough, the multi-attack SVD game is dominated by the “collective game” in the sense that each individual can ensure its own maximal survival probability by maximizing the group survival probability in each attack. In the hostile game, the only strict ESS is the fleeing strategy. In the collective game there are two different cases: either defense is a unique strict ESS, or the collective game is bistable, i.e. fleeing and defense are local strict ESS’s. If defense is the only strict ESS in the collective game, and the attack number is large enough, defense replaces fleeing strategy in the multi-attack SVD game. However, in the bistable case, defense cannot invade into the fleeing population. It is shown that, if the interaction between relatives is frequent enough, than defense can replace fleeing strategy, in spite of the fact that in the well-mixed population the collective game is bistable.

Control-based continuation for investigating nonlinear experiments

In this paper we present a systematic experimental study of two one-degree-of-freedom nonlinear devices using the newly introduced control-based continuation method of Sieber and Krauskopf. By considering hardening, softening and bistable spring characteristics, we demonstrate the versatility and power of the control-based continuation method for investigating nonlinear experiments. We show that, using this method, it is possible to track the stable orbits of the devices through a saddle-node bifurcation (fold) where they lose stability and continue them up to the resonance peak where they undergo a second saddle-node bifurcation. For the bistable case, a bifurcation diagram is produced that is strongly reminiscent of the bifurcation diagram produced using the classical harmonic balance solution. A detailed introduction to general continuation methods is included to enable implementation by other experimentalists.

Bistability and bacterial infections.

Bacterial infections occur when the natural host defenses are overwhelmed by invading bacteria. The main component of the host defense is impaired when neutrophil count or function is too low, putting the host at great risk of developing an acute infection. In people with intact immune systems, neutrophil count increases during bacterial infection. However, there are two important clinical cases in which they remain constant: a) in patients with neutropenic-associated conditions, such as those undergoing chemotherapy at the nadir (the minimum clinically observable neutrophil level); b) in ex vivo examination of the patient's neutrophil bactericidal activity. Here we study bacterial population dynamics under fixed neutrophil levels by mathematical modelling. We show that under reasonable biological assumptions, there are only two possible scenarios: 1) Bacterial behavior is monostable: it always converges to a stable equilibrium of bacterial concentration which only depends, in a gradual manner, on the neutrophil level (and not on the initial bacterial level). We call such a behavior type I dynamics. 2) The bacterial dynamics is bistable for some range of neutrophil levels. We call such a behavior type II dynamics. In the bistable case (type II), one equilibrium corresponds to a healthy state whereas the other corresponds to a fulminant bacterial infection. We demonstrate that published data of in vitro Staphylococcus epidermidis bactericidal experiments are inconsistent with both the type I dynamics and the commonly used linear model and are consistent with type II dynamics. We argue that type II dynamics is a plausible mechanism for the development of a fulminant infection.

Plant–herbivore interactions mediated by plant toxicity

We explore the impact of plant toxicity on the dynamics of a plant–herbivore interaction, such as that of a mammalian browser and its plant forage species, by studying a mathematical model that includes a toxin-determined functional response. In this functional response, the traditional Holling Type 2 response is modified to include the negative effect of toxin on herbivore growth, which can overwhelm the positive effect of biomass ingestion at sufficiently high plant toxicant concentrations. Two types of consumption decisions of the herbivore are considered. One of these (Case 1) incorporates the adaptation of the herbivore to control its rate of consumption of plant items when that is likely to lead to levels of toxicity that more than offset the marginal gain to the herbivore of consuming more plant biomass, while the other (Case 2) simply assumes that, although the herbivore’s rate of ingestion of plant biomass is negatively affected by increasing ingestion of toxicant relative to the load it can safely deal with, the herbivore is not able to prevent detrimental or even lethal levels of toxicant intake. A primary result of this work is that these differences in behavior lead to dramatically different outcomes, summarized in bifurcation diagrams. In Case 2, a wide variety of dynamics may occur due to the interplay of Holling Type 2 dynamics and the effect of the plant toxicant. These dynamics include the occurrence of bistability, in which both a periodic solution and the herbivore-extinction equilibrium are attractors, as well the possibility of a homoclinic bifurcation. Whether the herbivore goes to extinction in the bistable case depends on initial conditions of herbivore and plant biomasses. For relatively low herbivore resource acquisition rates, the toxicant effect increases the likelihood of ‘paradox of enrichment’ type limit cycle oscillations, but at higher resource acquisition rates, the toxicant may decrease the likelihood of these cycles.

Measurement of variable magnetic reversal paths in electrically contacted pseudo-spin-valve rings

In this work we show that the measurement of single magnetic reversal events is of critical importance in order to correctly characterize the switching of magnetic microstructures. Magnetoresistance measurements are performed on two pseudo-spin-valve ring structures with high enough signal to noise to allow the probing of single reversal events. Using this technique we acquire ‘switching spectra’ which demonstrate that the rings exhibit a range of variable reversal paths, including a bistable reversal mechanism of the hard layer, where the two switching routes have substantially different switching fields. The signature of the variable reversal paths would have been obscured in field cycle averaged data and in the bistable case would cause a fundamental misinterpretation of the reversal behaviour.

Temporally Interruptive Interaction Allows Mutual Invasion of Two Competing Species Dispersing in Space

With a reaction-difiusion system, we consider the dispersing two-species Lotka- Volterra model with a temporally periodic interruption of the interspeciflc competitive re- lationship. We assume that the competition coe-cient becomes a given positive constant and zero by turns periodically in time. We investigate the condition for the coexistence of two competing species in space, especially in the bistable case for the population dynamics without dispersion. We could flnd that the spatial coexistence, that is, the spatially mutual invasion of two competing species appears with two opposite-directed travelling waves if a condition for the temporal interruption of the interspeciflc relationship is satisfled. Further, we give a suggested mathematical expression of the velocity of travelling waves.

Optical bullets and double bullet complexes in dissipative systems

We show that optical light bullets can coexist with double bullet complexes in nonlinear dissipative systems. Coexistence occurs for a relatively large range of the system parameters, and is associated with either marginal stability or bistable existence of the two dissipative soliton species. In the case of marginal stability, spontaneous transformations of single bullets into double bullet complexes are observed. Among the bistable cases, we show how both clockwise and anticlockwise rotating double bullet complexes can be formed out of the phase-controlled interaction of two single bullets. The internal dynamics of pulsating double bullet complexes, with oscillations in both the spatial separation between the two bullets and the bullet shape in time domain is also detailed.