Forced Traveling Waves in a Reaction-Diffusion Equation with Strong Allee Effect and Shifting Habitat

We study a reaction-diffusion equation that describes the growth of a population with a strong Allee effect in a bounded habitat which shifts at a speed [Formula: see text]. We demonstrate that the existence of forced positive traveling waves depends on habitat size L, and [Formula: see text], the speed of traveling wave for the corresponding reaction-diffusion equation with the same growth function all over the entire unbounded spatial domain. It is shown that for [Formula: see text] there exists a positive number [Formula: see text] such that for [Formula: see text] there are two positive traveling waves and for [Formula: see text] there is no positive traveling wave. It is also shown if [Formula: see text] for any [Formula: see text] there is no positive traveling wave. The dynamics of the equation are further explored through numerical simulations.

Local Well-Posedness to the Cauchy Problem for an Equation of the Nagumo Type

In this paper, we show the local well-posedness for the Cauchy problem for the equation of the Nagumo type in this equation (1) in the Sobolev spaces H^s(ℝ). If D > 0, the local well-posedness is given for s > 1/2 and for s > 3/2 if D=0.

On a new modified fractional analysis of Nagumo equation

In this work, a new modified fractional form of the Nagumo equation has been presented and deeply analyzed. Using the Caputo–Fabrizio and Atangana–Baleanu time-fractional derivatives, classical Nagumo model is transformed to a new fractional version. The modified equation has been solved by using the homotopy analysis transform method. The convergence analysis has been also examined with the help of the so-called [Formula: see text]-curves and average residual error. Comparing the obtained approximate solution with the exact solution leaves no doubt believing that the proposed technique is very efficient and converges toward the exact solution very rapidly.

Parameters analysis of FitzHugh-Nagumo model for a reliable simulation

Derived from the pioneer ionic Hodgkin-Huxley model and due to its simplicity and richness from a point view of nonlinear dynamics, the FitzHugh-Nagumo model (FHN) is one of the most successful simplified neuron / cardiac cell model. There exist many variations of the original FHN model. Though these FHN type models help to enrich the dynamics of the FHN model, the parameters used in these models are often in biased conditions. The related results would be questionable. So, in this study, the aim is to find the parameter thresholds for one of the commonly used FHN model in order to provide a better simulation environment. The results showed at first that inappropriate time step and integration tolerance in numerical solution of FHN model can give some biased results which would make some publications questionable. Then the thresholds of parameters α, γ and ε are presented. α controls the global dynamics of FHN. α > 0, the cell is in refractory mode; α <; 0, the cell is excitable. ε controls the main morphology of the action potential generated and has a relation with the period (P = 3.065 × αα,γ(-0.8275)+ 4.397). To show oscillations of relaxation with FHN, ε should be smaller than 0.0085. 7 influences barely action potential, it showed linear relationship with the period and duration of action potential. Even though α <; 0.1, ε <; 0.0085, there is no definite threshold for γ, smaller values are recommended.

Spatial waves of advance with bistable dynamics: cytoplasmic and genetic analogues of Allee effects

Unlike unconditionally advantageous "Fisherian" variants that tend to spread throughout a species range once introduced anywhere, "bistable" variants, such as chromosome translocations, have two alternative stable frequencies, absence and (near) fixation. Analogous to populations with Allee effects, bistable variants tend to increase locally only once they become sufficiently common, and their spread depends on their rate of increase averaged over all frequencies. Several proposed manipulations of insect populations, such as using Wolbachia or "engineered underdominance" to suppress vector-borne diseases, produce bistable rather than Fisherian dynamics. We synthesize and extend theoretical analyses concerning three features of their spatial behavior: rate of spread, conditions to initiate spread from a localized introduction, and wave stopping caused by variation in population densities or dispersal rates. Unlike Fisherian variants, bistable variants tend to spread spatially only for particular parameter combinations and initial conditions. Wave initiation requires introduction over an extended region, while subsequent spatial spread is slower than for Fisherian waves and can easily be halted by local spatial inhomogeneities. We present several new results, including robust sufficient conditions to initiate (and stop) spread, using a one-parameter cubic approximation applicable to several models. The results have both basic and applied implications.

Reentrant excitation in an analog-digital hybrid circuit model of cardiac tissue

We propose an analog-digital hybrid circuit model of one-dimensional cardiac tissue with hardware implementation that allows us to perform real-time simulations of spatially conducting cardiac action potentials. Each active nodal compartment of the tissue model is designed using analog circuits and a dsPIC microcontroller, by which the time-dependent and time-independent nonlinear current-voltage relationships of six types of ion channel currents employed in the Luo-Rudy phase I (LR-I) model for a single mammalian cardiac ventricular cell can be reproduced quantitatively. Here, we perform real-time simulations of reentrant excitation conduction in a ring-shaped tissue model that includes eighty nodal compartments. In particular, we show that the hybrid tissue model can exhibit real-time dynamics for initiation of reentries induced by uni-directional block, as well as those for phase resetting that leads to annihilation of the reentry in response to impulsive current stimulations at appropriate nodes and timings. The dynamics of the hybrid model are comparable to those of a spatially distributed tissue model with LR-I compartments. Thus, it is conceivable that the hybrid model might be a useful tool for large scale simulations of cardiac tissue dynamics, as an alternative to numerical simulations, leading toward further understanding of the reentrant mechanisms.

Propagation in cardiac tissue adjacent to connective tissue: two-dimensional modeling studies

The conditions for activation transmission across a region of extracellular space was demonstrated in two-dimensional preparations with results consistent with those previously seen in the one-dimensional fiber studies. In addition, one sees changes in action potential morphology which occur in the tissue nearest the connective-tissue border as well as changes in conduction velocity along the border. These results hinge on an adequate representation of the connective-tissue region achieved by careful implementation of the boundary conditions in the intracellular and interstitial spaces and the expansion of the connective-tissue discretization to a "double-tier network" description. Through a series of simulations, a clear dependence on fiber orientation is illustrated in the efficacy to transmit activation. The collision of a front with an embedded connective-tissue region was also examined. The results revealed that fibers aligned normal to a planar stimulus would more greatly disrupt the advancement of a planar front. Such pronounced disruptions have been shown to be proarrhythmic in the literature. The increasing evidence of the ability of connective tissue to transmit activation has implications in understanding spread of activation through infarcted tissues and through the healthy ventricular wall in the presence of connective-tissue sheets.