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Neuron activation, which involves the state transitioning from inhibition to excitation between adjacent neuron units due to minor perturbations, can induce intricate dynamics such as synchronized oscillations and local excitation. These phenomena arising from coupling-induced changes are vital for information processing in Cellular Neural Networks. This study takes the coupled memristive neuron circuits based on the N-type Locally Active Memristor (LAM) as an example to elucidate the activation mechanism. We predict the oscillation conditions of neurons under static bias and investigate the factors affecting the system stability under dynamic parameters. Then, by analyzing global dynamics before and after coupling, we find that even slight initial differences in identical inactive neurons can lead to reactivation and periodic spiking. The dynamics map visually represents the parameter domain where the neuron activation occurs. Moreover, the impact of initial condition differences between the neurons on the dynamics of the coupled system is explored. Finally, according to the mathematical expression of N-type LAM, an equivalent emulator is designed, and the fourth-order coupled circuit is subsequently constructed. The hardware implementation demonstrates the correctness of theoretical analysis and simulation results. This work provides a systematic method for the coupling of neuron circuits, laying a stepstone for neuromorphic computing.

Seed dispersal mutualisms assume a crucial role in sustaining overall ecosystem functionality. This study extends the foundational framework of seed dispersal mutualism by incorporating animal cooperation and environmental variability. Through the integration of stability theory and phase analysis, we investigate the global dynamics of seed dispersal mutualism in deterministic environments. Theoretical findings reveal that the deterministic system exclusively exhibits equilibrium dynamics, ruling out periodic solutions and chaotic phenomena. Notably, in cases of multiple coexisting equilibria, adjacent equilibria display distinct stability properties. Furthermore, employing stochastic analysis and Lyapunov exponents, we ascertain the uniqueness of global positive solutions in the stochastic model, along with stochastic ultimate boundedness, and precise thresholds for the stochastic persistence and extinction. These results illuminate the theoretical mechanisms governing transitions in seed dispersal mutualism across various stable population levels and underscore the insights into how essential biological factors, including animal cooperation efforts and environmental variability, impact the long-term dynamics of seed dispersal mutualism.

Related studies indicate that the Rosenzweig–MacArthur predator–prey model with constant search rate exhibits the paradox of enrichment, while variable search rate may avoid this event. We investigate the effect of variable search rate on the dynamics of the predator–prey model, a complete and explicit classification of global dynamics is characterized using carrying capacity $K$ and half-saturation constant $g$ as parameters. First, this is a trichotomy result, the boundary equilibrium or the positive equilibrium is globally asymptotically stable; otherwise, there exists a globally stable limit cycle. Furthermore, we demonstrate that for the case where the limit cycle is stable, the predator can adjust search rate to restore the positive equilibrium to stable state, thereby avoiding the occurrence of the paradox of enrichment. Specifically, we adopt a new planar analysis method that differs from classical ways, proving that local stability of the positive equilibrium implies its global stability. Finally, numerical simulation is conducted to verify the results of theoretical analysis. Our results can help understand the evolutionary mechanisms through which organisms adapt to environmental changes in some sense.

Some observations are made on a class of one-predator, two-prey models via numerical analysis. The simulations are performed with the aid of an adaptive grid method for constructing bifurcation diagrams and cell-to-cell mapping for global analysis. A two-dimensional bifurcation diagram is constructed to show that regions of coexistence of all three species, which imply the balance of competitive and predatory forces, are surrounded by regions of extinction of one or two species. Two or three coexisting attractors which may have a chaotic member are found in some regions of the bifurcation diagram. Their separatrices are computed to show the domains of attraction. The bifurcation diagram also contains codimension-two bifurcation points including Bogdanov–Takens, Gavrilov–Guckenheimer, and Bautin bifurcations. The dynamics in the vicinity of these codimension-two bifurcation points are discussed. Some global bifurcations including homoclinic and heteroclinic bifurcations are investigated. They can account for the disappearance of chaotic attractors and limit cycles. Bifurcations of limit cycles such as transcritical and saddle-node bifurcations are also studied in this work. Finally, some relevant calculations of Lyapunov exponents and power spectra are included to support the chaotic properties.

We study the chaotic dynamics of one-degree-of-freedom nonlinear oscillator representing a density perturbation in plasma device model excited by parametric and external driven forces. Critical parameters for the onset of chaotic motions are specified using Melnikov method. The analytical results are confirmed by numerical simulations. The global dynamical changes of the system have been examined by evaluating parametric changes of the bifurcation diagrams, maximum Lyapunov exponent, Poincaré map and the basin boundaries of attraction. The transitions to chaos caused by the cascade bifurcation and intermittency are clearly shown by graphical methods.

In this paper, we examine the global dynamics of the complete Owen–Sherratt model describing the tumor–macrophage interactions. We show for this dynamics that there is a positively invariant polytope. We give upper and lower ultimate bounds for densities of cell populations involved in this model. Besides, we derive sufficient conditions under which each trajectory in tends to the mutant cells-free equilibrium point or to the equilibrium point of macrophages in isolation or to the coordinate plane corresponding to the absence of normal tissue cells depending on initial conditions. The biological sense of our results is discussed as well.

This paper proposes a Filippov epidemic model with piecewise continuous function to represent the enhanced vaccination strategy being triggered once the proportion of the susceptible individuals exceeds a threshold level. The sliding bifurcation and global dynamics for the proposed system are investigated. It is shown that as the threshold value varies, the proposed system can exhibit variable sliding mode domains and local sliding bifurcations including boundary node (focus) bifurcation, double tangency bifurcation and other sliding mode bifurcations. Model solutions ultimately approach either one of two endemic states for two structures or the pseudo-equilibrium on the switching surface, depending on the threshold level. The findings indicate that proper combinations of threshold level and enhanced vaccination rate based on threshold policy can lead disease prevalence to a previously chosen level if eradication of disease is impossible.

In this paper we study some features of global behavior of a seven-dimensional tumor growth model under immunotherapy described by Joshi et al. [2009]. We find the upper bounds for ultimate dynamics of all types of cell populations involved in this model. A few lower bounds are found as well. Further, we prove the existence of the bounded positively invariant polytope. Finally, we show that if the parameter modeling the flow of antigen presenting cells is very large then the tumor-free equilibrium point attracts all points in the positive orthant.

The paper is concerned with the effect of a nonlinear incidence rate S^p I^q on dynamical behaviors of a parasite-host model. It is shown that the global attractor of the parasite-host model is an equilibrium if q = 1, which is similar to that of the parasite-host model with a nonlinear incidence rate of the fractional function . However, when q is greater than one, more positive equilibria appear and limit cycles arise from Hopf bifurcations at the positive equilibria for the model with the incidence rate S^p I^q. It reveals that the nonlinear incidence rate of the exponential function S^p I^q for generic p and q can lead to more complicated and richer dynamics than the bilinear incidence rate or the fractional incidence rate for this model.

In this paper, we present a novel 4-dimensional (4D) smooth quadratic autonomous hyperchaotic system with complex dynamics. In order to investigate the dynamics evolution of the system, the Lyapunov exponent spectrum, bifurcation diagram and various phase portraits are provided. The local dynamics of this hyperchaotic system, such as the stability, pitchfork bifurcation, and Hopf bifurcation of equilibrium point, are analyzed by using the center manifold theorem and bifurcation theory. About the global dynamics, the ultimate bound sets of the system are found by combining the Lyapunov function method and appropriate optimization method. Numerical simulations are given to demonstrate the emergence of the two bifurcations and show the ultimate boundary regions.

A multiscale system for environmentally-driven infectious disease is proposed, in which control measures at three different scales are implemented when the number of infected hosts exceeds a certain threshold. Our coupled model successfully describes the feedback mechanisms of between-host dynamics on within-host dynamics by employing one-scale variable guided enhancement of interventions on other scales. The modeling approach provides a novel idea of how to link the large-scale dynamics to small-scale dynamics. The dynamic behaviors of the multiscale system on two time-scales, i.e. fast system and slow system, are investigated. The slow system is further simplified to a two-dimensional Filippov system. For the Filippov system, we study the dynamics of its two subsystems (i.e. free-system and control-system), the sliding mode dynamics, the boundary equilibrium bifurcations, as well as the global behaviors. We prove that both subsystems may undergo backward bifurcations and the sliding domain exists. Meanwhile, it is possible that the pseudo-equilibrium exists and is globally stable, or the pseudo-equilibrium, the disease-free equilibrium and the real equilibrium are tri-stable, or the pseudo-equilibrium and the real equilibrium are bi-stable, or the pseudo-equilibrium and disease-free equilibrium are bi-stable, which depends on the threshold value and other parameter values. The global stability of the pseudo-equilibrium reveals that we may maintain the number of infected hosts at a previously given value. Moreover, the bi-stability and tri-stability indicate that whether the number of infected individuals tends to zero or a previously given value or other positive values depends on the parameter values and the initial states of the system. These results highlight the challenges in the control of environmentally-driven infectious disease.

Predator foraging facilitation or cooperative hunting increases per predator consumption rate as predator density increases. This affects predator extinction in a prey–predator interaction model when the predator density is low. This is an indication of Allee effect in predator’s growth rate. Here, we take a Gause type model with a generalized type II functional response which depends on both prey and predator densities. We also assume that prey’s growth is subjected to Allee effect. Strong Allee effect in prey’s growth rate enhances the stability of the coexisting steady state. A region is found in a two-parameter plane where the coexisting steady state is a global attractor when the prey’s growth is subjected to weak Allee effect. In addition, codimension two bifurcation points (cusp and Bogdanov–Takens points) have also been found in the bifurcation diagram.

Considering the effectiveness of introducing the change rate of viral loads into the threshold setting policy for triggering interventions, we propose an immune-virus Filippov system with a nonlinear threshold. By developing new analytical and numerical methods, we systematically studied the rich dynamical behaviors and bifurcations of the proposed system, including the existence of three sliding segments and three pseudo-equilibria, boundary-center bifurcation, boundary-saddle bifurcation, pseudo-saddle-node bifurcation and tangency bifurcation. We further showed that the proposed system can exhibit virous structures in the coexistence of multiple steady states. Phenomena include bistability of two pseudo-equilibria, tristability and multiplestability of two pseudo-equilibria with regular equilibria or touching cycles. The modeling methods, as well as the analytical and numerical methods, can be widely applied to many other fields.

In this article, we consider a SIV infectious disease control system with two-threshold guidance, in which infection rate and vaccination rate are represented by a piecewise threshold function. We analyze the global dynamics of the discontinuous system using the theory of differential equations with discontinuous right-hand sides. We find some dynamical behaviors, including the disease-free equilibrium and endemic equilibria of three subsystems, a globally asymptotically stable pseudo-equilibrium and two endemic equilibria bistable, one of the two pseudo-equilibria or pseudo-attractor is stable. Conclusions can be used to guide the selection of the most appropriate threshold and parameters to achieve the best control effect under different conditions. We hope to minimize the scale of the infection so that the system can eventually stabilize at the disease-free equilibrium, pseudo-equilibrium or pseudo-attractor, corresponding to the disease disappearing or becoming endemic to a minimum extent, respectively.

This paper investigates a two-species amensalism model that includes the fear effect on the first species and the Beddington–DeAngelis functional response. The existence and stability of possible equilibria are investigated. Under different parameters, there exist two stable equilibria which means that this model is not always globally asymptotically stable. Together with the existence of all possible equilibria and their stability, saddle connection and close orbits, we derive some conditions for transcritical bifurcation and saddle-node bifurcation. Furthermore, global dynamics analysis of the model is performed. It is observed that under certain parameter conditions, when the intensity of the fear effect is below a certain threshold value, as the fear effect increases it will only reduce the density of the first species population and will have no influence the extinction or existence of the first species population. However, when the fear effect exceeds this threshold, the increase of the fear effect will accelerate the extinction of the first species population. Finally, numerical simulations are performed to validate theoretical results.

In this paper, the impact of the strong Allee effect and ratio-dependent Holling–Tanner functional response on the dynamical behaviors of a predator–prey system is investigated. First, the positivity and boundedness of solutions of the system are proved. Then, stability and bifurcation analysis on equilibria is provided, with explicit conditions obtained for Hopf bifurcation. Moreover, global dynamics of the system is discussed. In particular, the degenerate singular point at the origin is proved to be globally asymptotically stable under various conditions. Further, a detailed bifurcation analysis is presented to show that the system undergoes a codimension-$1$ Hopf bifurcation and a codimension-$2$ cusp Bogdanov–Takens bifurcation. Simulations are given to illustrate the theoretical predictions. The results obtained in this paper indicate that the strong Allee effect and proportional dependence coefficient have significant impact on the fundamental change of predator–prey dynamics and the species persistence.

In this study, we discuss the global dynamics of the Holling-II amensalism model for a strong Allee effect of harmful species. We discuss the existence and stabilization of the extinction equilibria, exclusion equilibria, coexistence equilibria, and infinite singularities by analyzing the presence and stabilization of the system characteristics in terms of the possibilities and correspondences in the model when the death rate of the injured species is used as a threshold value. Also, we find that the two equilibrium points in the first quadrant are effective in proving that the model does not have globally stabilizing features and obtain two critical conditions and their corresponding global phase diagrams. Finally, we explore the weak Allee effect of the victim species, and using the analysis from numerical simulations, we recapitulate the analysis and dynamics of the model in equilibrium.

We have previously demonstrated intra- and extra-cellular factors that govern somatic evolution of the malignant phenotype can be modeled through evolutionary game theory, a mathematical approach that analyzes phenotypic adaptation to in-vivo environmental selection forces. Here we examine the global evolutionary dynamics that control evolutionary dynamics explicitly addressing conflicting data and hypothesis regarding the relative importance of the mutator phenotype and microenvironment controls.

We find evolution of invasive cancer follows a biphasic pattern. The first phase occurs within normal tissue, which possesses a remarkable adaptive landscape that permits non-competitive coexistence of multiple cellular populations but renders it vulnerable to invasion. When random genetic mutations produce a fitter phenotype, self-limited clonal expansion is observed — equivalent to a polyp or nevus. This step corresponds to tumor initiation in classical skin carcinogenesis experiments because the mutant population deforms the adaptive landscape resulting in the emergence of unoccupied fitness peaks — a premalignant configuration because, over time, extant cellular populations will tend to evolve toward available fitness maxima forming an invasive cancer. We demonstrate that this phase is governed by intracellular processes, such as the mutation rate, that promote phenotypic diversity and environmental factors that control cellular selection and population growth. These results provide an integrative model of carcinogenesis that incorporates cell-centric approaches such as the mutator phenotype hypothesis with the critical role of the environmental demonstrated by Bissell and others. The biphasic dynamics of carcinogenesis give a quantitative framework of understanding for the empirically observed initiation and promotion/progression stages in skin carcinogenesis experimental models.

This paper investigates which smooth manifolds arise as quotients (orbit spaces) of flows of vector fields. Such quotient maps were already known to be surjective on fundamental groups, but this paper shows that every epimorphism of countably presented groups is induced by the quotient map of some flow, and that higher homology can also be controlled. Manifolds of fixed dimension arising as quotients of flows on Euclidean space realize all even (and some odd) intersection pairings, and all homotopy spheres of dimension at least two arise in this manner. Most Euclidean spaces of dimensions five and higher have families of topologically equivalent but smoothly inequivalent flows with quotient homeomorphic to a manifold with flexibly chosen homology. For $m\ge 2r>2$, there is a topological flow on (ℝ^2r+1 − 8 points) × ℝ^m that is unsmoothable, although smoothable near each orbit, with quotient an unsmoothable topological manifold.

In this paper, we investigate two predator–prey models which take into consideration hunting cooperation (i.e., mutualism) between two different predators and within one predator species, respectively. Local and global dynamics are obtained for the model systems. By a detailed bifurcation analysis, we investigate the dependence of predation dynamics on mutualism (cooperative predation). From our study, we prove that mutualism may enhance the survival of mutualist predators in a severe condition and break the competitive exclusion principle. We further provide quantitative information about how the cooperative predation (mutualism) may (i) establish multiple stability switches on the positive equilibrium; (ii) generate backward bifurcation on equilibria; (iii) induce supercritical or subcritical Hopf bifurcations; and (iv) establish bi-stability phenomenon between the predator-free equilibrium and a positive equilibrium (or a limit cycle).