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Abstract Let G be a connected graph. The edge k -partiteness of G is the minimum number of edges whose deletion from G yields a k -partite graph. The Kirchhoff index Kf (G) of G is the sum of the resistance distance between all unordered pairs of vertices. In this paper, we study the problem to determine the minimum Kirchhoff index of graphs with a given edge k -partiteness. First, we theoretically characterize the graphs with minimum Kirchhoff index in this graph family when the edge k -partiteness is not big compared to the number of vertices of G. Then we propose an exhaustive search algorithm to find the optimal graphs. At last, three strategies are used to reduce the computation complexity of our algorithm and several performance comparisons of these strategies are given. [ABSTRACT FROM AUTHOR]

Highlights • A time delay model of the complex interaction between Tumor, CD8+T cells and T - helper cells. • The model undergoes Hopf bifurcation in which time delay plays an important role to destabilize the system. • Extensive numerical simulations are performed to validate our analytical findings. Abstract In this paper, we consider a three-dimensional nonlinear delay differential system (tumor cells, cytotoxic-T lymphocytes, T-helper cells) with single interaction delay. We perform linear stability of the equilibria and the existence of Hopf bifurcation in which the discrete time delay is used as a bifurcation parameter. We estimate the length of delay to preserve the stability of period-1 limit cycle. We also investigate the direction, period, and the stability of bifurcated periodic solutions by applying normal form method and center manifold theory. We observe that the discrete time delay plays an important role in stability switching. Numerical simulations are presented to illustrate the rich dynamical behavior of the model with different values for the time delay τ including the existence of periodic oscillations, which demonstrate the phenomena of long-term tumor relapse. [ABSTRACT FROM AUTHOR]

This paper is dedicated to the qualitative analysis as well as numerical simulations of a one dimensional open channel hydraulics system which is commonly used in hydraulic engineering to model the unsteady flow dynamics in a river. First, an output feedback control is proposed. Next, the closed-loop system is proved to possess a unique solution in a functional space. Furthermore, the spectrum and resolvent sets of the system operator are characterized. Then, stability results are stated and proved according to a smallness assumption on the feedback gain. The proof invokes Lyapunov direct method. Last but not least, we adopt the Chebychev collocation method, that uses backward Euler method and the Gauss-Lobatto points, to provide numerical simulations in order to ascertain the correctness of the theoretical outcomes. [ABSTRACT FROM AUTHOR]

This paper systematically investigates the optimal harvesting of a stochastic delay competitive Lotka–Volterra model with Lévy jumps. Under some simple assumptions, the sufficient conditions for extinction and stable in the time average of each species are established. The stability in distribution of this model is proved under our assumptions. Finally, the sufficient and necessary criteria for the existence of optimal harvesting policy are established and the explicit expression of the optimal harvesting effort and the maximum of sustainable yield are also obtained. And some numerical simulations are introduced to demonstrate the theoretical results. [ABSTRACT FROM AUTHOR]

This paper deals with a parabolic equation with variable coefficient modeling MEMS technology, and we will prove the solution of the problem quenches in finite time in the case without stationary solutions. Numerical simulations are presented to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

Based on biological meaning, a kind of diffusive intraguild predation (IGP: resource, IG prey and IG predator) model with delay effect is investigated in this paper. The model has Holling-Type I functional response between resource-IG prey and resource-IG predator; Holling-Type II functional response between IG prey and IG predator. We first give sufficient conditions on the stability of possible nonnegative constant steady-state solutions for the proposed model, which give us a complete picture of the global dynamics. Then we investigate Hopf bifurcation near the unique positive constant steady-state solution by taking delay as bifurcation parameter and derive the Hopf bifurcation threshold. It is shown that the delay can induce three types of bistability (node-node bistability, node-cycle bistability and cycle-cycle bistability), periodic oscillations and irregular oscillations triggering spatiotemporal chaos in the diffusive IGP model. Numerical simulations are performed to illustrate our theoretical results and suggest that delay can even trigger the emergence of self-organised spatiotemporal patterns, which evolve from spiral patterns to irregular spatial patterns via spatiotemporal Hopf bifurcation. In addition, the impact of diffusion on the model’s dynamics under certain time delay are also explored. [ABSTRACT FROM AUTHOR]

In this paper, Cournot oligopoly model is examined with an uncertain quadratic cost function. A random quadratic cost function is introduced in this model. The model consists of three competed firms on which each firm in the model wants to maximize its expected profit and also wants to minimize its uncertainty by minimizing the cost. To handle this multi-objective optimization problem, the expectation and worst-case approaches are adopted used to study uncertainty. The existence and uniqueness of the equilibrium are obtained. The asymptotic behavior of the equilibrium point is also investigated. Complete stability and bifurcation analysis are provided. The obtained theoretical results are verified by numerical simulation. [ABSTRACT FROM AUTHOR]

The nonlinear incidence rate can explain the complicated infectious process of disease. And time delay describing the latent period widely exists in the process of disease contagion. In this paper, a spatiotemporal epidemic model with nonlinear incidence rate is investigated. In particular, we considered that the time delay is relatively small. In this case, the characteristic equation are derived, we obtain two mechanisms of instability of the positive constant stationary state, that is, One is the diffusion induced instability, and the other one is delay induced instability. Moreover, the results of numerical simulation validate our theoretical analyses. The obtained results may well catch some major features for epidemic models. [ABSTRACT FROM AUTHOR]

In this paper, we propose a delayed mathematical model for the transmission of Ebola in humans. We consider the transmission of infection between the living humans and from infectious corpses to the living individuals in which the latent period of Ebola is incorporated. We identify the basic reproduction number R 0 for the model, prove that the disease-free equilibrium is always globally asymptotically stable when R 0 < 1, the disease is persistence and a unique endemic equilibrium exists when R 0 > 1. We show that the endemic steady state is locally asymptotically stable under certain condition and globally asymptotically stable in a special case of the model. Numerical simulations are provided to demonstrate and complement the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we derive a nonlocal delayed and diffusive dengue transmission model with a spatial domain being bounded as well as unbounded. We first address the well-posedness to the initial-value problem for the model. In the case of a bounded spatial domain, we establish the threshold dynamics for the spatially heterogeneous system in terms of the basic reproduction number R 0 . Also, a set of sufficient conditions is further obtained for the global attractivity of the endemic steady state where all the parameters are spatially independent. In the case of an unbounded spatial domain, and when the coefficients are all constants, we show that there exist traveling wave solutions of the model. Numerical simulations are performed to illustrate our main analytic results. [ABSTRACT FROM AUTHOR]

HIV-1 infection and treatment may occur in the non-constant environment due to the time-varying drug susceptibility and growth of target cells. In this paper, we propose a within-host virus model with multiple stages for infected cells under time-varying environments, to study how the multiple infected stages affect on the counts of viral load and CD4 + -T cells. We establish the sufficient conditions for both persistent HIV infection and clearance of HIV infection based on two positive constants R * , R * . When the system is under persistent infection, we further obtained detailed estimates of both the lower and upper bounds of the viral load and the counts of CD4 + -T cells. Furthermore, numerical simulations are carried out to verify our analytical results and demonstrate the combined effects of multiple infected stages and non-constant environments, and reflect that both persistence and clearance of infection are possible when R * < 1 < R * holds. In particular, the numerical results exhibit the viral load of system with multiple infected stages may be less than that with single infected stage, and simulate the effect of time-varying environment of the autonomous system with multiple infected stages. We expect that our theoretical and simulation results can provide guidance for clinical therapy for HIV infections. [ABSTRACT FROM AUTHOR]

Evolution problem is always a hot topic in the mathematical biology field. In this paper, we investigate the evolutionary effects of selective disturbance on an evolving trait (e.g. body size and maturation age) of the predator individuals in one-predator two-prey community. By using methods of adaptive dynamics and population dynamics we construct an invasion fitness function and obtain the conditions for evolutionary branching and evolutionary stability under selective disturbance in both monomorphic and dimorphic populations. We further conduct a size-selective disturbance function founded on chi-square distribution to study evolutionary stable coexistence, and considering the evolutionary branching and evolutionary stability by using theoretic analysis and numerical simulations. The evolutionary results from a biological point of view show that (1) two strategies could gradually evolve to form a single ancestral strategy, moreover, higher levels of polymorphism cannot build up during evolution, that is, following first evolutionary branching two species will eventually evolve into two generalist species and reach an evolutionary stable coexistence; (2) smaller disturbance could touch off higher levels of dimorphism during evolution, while large disturbance can go against evolutionary branching and advance evolutionary stability. [ABSTRACT FROM AUTHOR]

In this paper, a new modified Chua oscillator is introduced. The original Chua oscillator is well known for its simple implementation and mathematical modeling. A modification of the oscillator is proposed in order to facilitate the synchronization and the encryption and decryption scheme. The modification consists in changing the nonlinear term of the original oscillator to a smooth and bounded nonlinear function. A bifurcation diagram, a Poincaré map and the Lyapunov exponents are presented as proofs of chaoticity of the newly modified oscillator. An application to secure communications is proposed in which two channels are used. Numerical simulations are performed in order to analyze the communication system. [ABSTRACT FROM AUTHOR]

In this paper, we consider the effect of distributed time delays on dynamics of a mathematical model of gene expression. Both the weak and strong delay kernels are discussed. Sufficient conditions for the local stability of the unique equilibrium are obtained. Taking the average delay as a bifurcation parameter, we investigate the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions by using the method of multiple time scales. Finally, numerical simulation is carried out to illustrate our theoretical results. It shows both subcritical and supercritical Hopf bifurcations can happen. [ABSTRACT FROM AUTHOR]

This paper deals with a three-dimensional plant-herbivore-predator model that incorporates explicitly the plant toxicity in plant-herbivore interactions. The existence and stability conditions of all the feasible equilibria are established. Our results indicate that plant toxicity may play a key role in the dynamical behavior of the system. By adding another plant species with a different toxicity level to this system, we derive threshold conditions on the invasion of the second plant species. The analysis indicates that several parameters may be critical to determine successful invasion. Numerical simulations are also provided to reinforce the theoretical conclusions. [ABSTRACT FROM AUTHOR]