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In this paper, the dynamical behavior of a mathematical model of cancer including tumor cells, immune cells, and normal cells is investigated when a delay term is induced. Though the model was originally proposed by De Pillis et al. (Math. Comput. Model. 37:1221–1244, 2003), to make the model more realistic, we have added a delay term into the model, and it has incorporated novelty in our present work. The stability of existing equilibrium points in the delay-induced system is studied in detail. Global stability conditions of the tumor-free equilibrium point have been found. It is shown that due to this delay effect, the coexisting equilibrium point may lose its stability through a Hopf bifurcation. The implicit function theorem is applied to characterize a complex function in a neighborhood of delay terms. Additionally, the presence of Hopf bifurcation is demonstrated when the transversality conditions are satisfied. The length of delay for which the solutions preserve the stability of the limit cycle is estimated. Finally, through a series of numerical simulations, the theoretical results are formally examined. [ABSTRACT FROM AUTHOR]

In this paper, we consider a fractional-order model of a brushless DC motor. To develop a mathematical model, we use the concept of the Liouville–Caputo noninteger derivative with the Mittag-Lefler kernel. We find that the fractional-order brushless DC motor system exhibits the character of chaos. For the proposed system, we show the largest exponent to be 0.711625. We calculate the equilibrium points of the model and discuss their local stability. We apply an iterative scheme by using the Laplace transform to find a special solution in this case. By taking into account the rule of trapezoidal product integration we develop two iterative methods to find an approximate solution of the system. We also study the existence and uniqueness of solutions. We take into account the numerical solutions for Caputo Liouville product integration and Atangana–Baleanu Caputo product integration. This scheme has an implicit structure. The numerical simulations indicate that the obtained approximate solutions are in excellent agreement with the expected theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, our aim is mathematical analysis and numerical simulation of a prey-predator model to describe the effect of predation between prey and predator with nonlinear functional response. First, we develop results concerning the boundedness, the existence and uniqueness of the solution. Furthermore, the Lyapunov principle and the Routh–Hurwitz criterion are applied to study respectively the local and global stability results. We also establish the Hopf-bifurcation to show the existence of a branch of nontrivial periodic solutions. Finally, numerical simulations have been accomplished to validate our analytical findings. [ABSTRACT FROM AUTHOR]

A novel stochastic turbidostat model is investigated in this paper. The stochasticity in the model comes from the maximal growth rate influenced by white noise. Firstly, the existence and uniqueness of the positive solution for the system are demonstrated. Secondly, we analyze the persistence in mean and stochastic persistence of the system, respectively. Sufficient conditions about the extinction of the microorganism are obtained. Finally, numerical simulation results are given to support the theoretical conclusions. [ABSTRACT FROM AUTHOR]

We investigate the collective dynamics of multi-quasi-synchronization of coupled fractional-order neural networks with delays. Using the pinning impulsive strategy, we design a novel controller to pin the coupled networks to realize the multi-quasi-synchronization. Based on the comparison principle and mathematical analysis, we derive some novel criteria of the multi-quasi-synchronization. Moreover, we discuss the effects of coupling strength and pinning control matrix. Finally, some simulation examples show the effectiveness of the presented results. [ABSTRACT FROM AUTHOR]

In this study, we discuss a cancer model considering discrete time-delay in tumor-immune interaction and stimulation processes. This study aims to analyze and observe the dynamics of the model along with variation of vital parameters and the delay effect on anti-tumor immune responses. We obtain sufficient conditions for the existence of equilibrium points and their stability. Existence of Hopf bifurcation at co-axial equilibrium is investigated. The stability of bifurcating periodic solutions is discussed, and the time length for which the solutions preserve the stability is estimated. Furthermore, we have derived the conditions for the direction of bifurcating periodic solutions. Theoretically, it was observed that the system undergoes different states if we vary the system's parameters. Some numerical simulations are presented to verify the obtained mathematical results. [ABSTRACT FROM AUTHOR]

We consider a susceptible-exposed-infected-removed (SEIR) epidemic model with discontinuous treatment strategies. The treatment rate has at most a finite number of jump discontinuities in every compact interval. By using Lyapunov theory for discontinuous differential equations and other techniques on non-smooth analysis, the basic reproductive number $\mathcal{R}_{0}$ is proved to be a sharp threshold value which completely determines the dynamics of the model. If $\mathcal{R}_{0}\leq 1$, then there only exists a disease-free equilibrium which is globally stable. If $\mathcal {R}_{0}>1$, the disease-free equilibrium becomes unstable and there exists a unique endemic equilibrium which is globally stable. The numerical simulations indicate that strengthening treatment measures after infective individuals reach some level is beneficial to disease control. Furthermore, we discuss that the disease will die out in a finite time, which is impossible for the corresponding SEIR model with continuous treatment. [ABSTRACT FROM AUTHOR]

Discrete N-fold Darboux transformation (DT) is used to derive new bright and dark multi-soliton solutions of two higher-order Toda lattice equations. Propagation and elastic interaction structures of such soliton solutions are shown graphically. The details of their evolutions are studied via numerical simulations. Numerical results show the accuracy of our numerical scheme and the stable evolutions of such bright and dark multi-solitons without a noise. To compare the numerical evolution results with the classical Toda lattice equation, we also investigate the dynamical behaviors of the multi-soliton solutions for Toda lattice equation via numerical simulations, and we find that the multi-soliton solutions of Toda lattice equation have better stability and are more robust against a big noise than its two corresponding higher-order equations. The same small noise has different effect on the evolutions of the multi-soliton solutions for three different equations in the same hierarchy. The possible reason is that the higher-order nonlinear terms of the higher-order equation cause the instability of the wave propagation. The discrete generalized (n,N−n)-fold DTs are constructed and used to derive some discrete rational solutions of three equations, and a few mathematical features for such rational solutions are also discussed. Results might be helpful for understanding the propagation of nonlinear waves in soliton theory. [ABSTRACT FROM AUTHOR]