Vol. 11, No. 4 (2022): Regular Issue

In the present paper we consider a family of non-Volterra quadratic stochastic operators depending on the parameters $\alpha, \beta$ and study their trajectory behaviors. We find all fixed and periodic points for an $(\alpha,\beta)$- quadratic stochastic operator on the two-dimensional simplex. A complete description of the set of limit points is given, and we show that such operators have the ergodic property.

In this work, it is proved the existence of at least three weak solutions can be obtained for a new class of nonlinear fractional boundary value systems by using variational methods combined with a critical point theory due to Bonano and Marano while two examples in $% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{3}$ and $% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{4}$ are given to illustrate our main results applications.

In the present paper we consider a differentiated-good Stackelberg model, when the follower firm engages in an R$\&$D process that gives an endogenous cost-reducing innovation. The aim is two-fold: the first is to study the case when there is a technology transfer between the innovator and the non-innovator firm based on a fixed-fee licensing contract, and the second is to study the case when there is a technology transfer between the innovator and the non-innovator firm based on a two-part tariff licensing contract. The main result of the paper is that the degree of the differentiation of the goods is the key factor in the decisions of the innovator firm, influencing its licensing strategy. In particular, we find that for the innovator firm is better a fixed-fee or a two-part tariff licensing contract than no-licensing, even if the innovation is drastic. In the case of a fixed-fee licensing, the main variables of this duopoly model increase with the differentiation of the goods all the time. It turns out that in the case of a two-part tariff licensing, this conclusion does not fit all the time. The findings of this paper extend the literature on contract auctions when the innovating firm has different options for licensing its innovation.

In this paper, we consider two competing fish population which are subject to be harvested. The growth of them follows the logistic growth function. From different sources release toxic substances which are harmful to each fish species. The boundedness equilibria, stability, bionomic equilibrium, optimal harvesting policy and optimal control have been studied. We have shown that the dynamical outcomes of the interacting fish species will much sensitive to the system parameters and their initial population volumes. Counter-intuitive results on role played by toxic coefficients are highly gated. Finally, some numerical example are cited to illustrate the effect of toxicity upon the species and optimal control theory is applied to minimize the harmful toxic effect

In this paper, the effect of delay on a network of indirectly coupled Euler's beams is analyzed. The system consists of a number of hinged-hinged beams indirectly interconnected to piezoelectric patches in a parallel conformation. A stability analysis is provided in order to give the threshold values of the parameter space for the onset of unstable motion. It is found that the stable region is reduced as the time delay increases. Disturbance-induced by time-delay on the synchronization state and the strong amplitude reduction (SAR) state is also presented. It is conventionally known that delay induces instability in coupled systems, but we find here that this delay can also contribute to stabilize these systems by synchronizing them.

A mathematical model addressing the phase transition scenario from stability to chaos in a three-species food chain model with alternative food source is presented in this research manuscript. The biological phenomenon of diverse solution trajectory formation has been uncovered here in terms of coupled nonlinear ordinary differential equations through positive initial conditions. The present paper is mainly focused on the existence and stability of coexistence equilibrium point as obtained by the system parameters and the interpretation of the result in terms of their possible ecological implications. The dynamics of the model reveal chaotic long-term behaviour over a broad range of parameters. The existence of a strange attractor and computation of the largest Lyapunov exponent also exhibit the presence of chaotic dynamics of our model. The final numerical simulations verify the correctness of our theoretical analysis.

In this manuscript, we establish the existence and exponential stability of neutral stochastic integrodifferential equation driven by fractional Brownian motion and Poisson jumps in a Hilbert space. We prove an existence results and establishes some conditions ensuring the exponential decay to zero in mean square for the mild solution by means of the Banach fixed point principle and the theory of resolvent operator developed by Grimmer (1982). An example illustrates the potential benefits of these results.

In this paper, dynamic behaviors and chaos control of a Fractional Order Glucose-Insulin Regulatory System (FOGIRS) recently introduced in the literature are studied. Some interesting behaviors such as periodic motions, antimonotonicity, and period-doubling route to chaos are observed. Besides, it is found that the minimum effective dimension for the system to generate chaos is 2.904 in the conmensurate order case and 2.882 in the incommensurate order case. To control chaos in this system, two controllers are designed. First, a feedback controller to stabilize the chaotic system to one of its equilibrium points, and a robust adaptive fractional order sliding mode controller is used to globally stabilize the chaotic FOGIRS with uncertain dynamics. Numerical simulations are used to show the effectiveness of those controllers.

In this paper, the Brusselator reactor model with time delay and network structure is introduced and investigated. In view of understanding spatiotemporal dynamics of the model, dynamical analysis about the reactor is first carried out, such as stability and bifurcation analysis near the homogeneous steady state via choosing time delay as the critical parameter. It is found that time delay can induce stable and unstable states of the homogeneous steady state, and the model experiences the Hopf bifurcation at such critical value. Then the center manifold reduction and the normal form theory are applied to explore the detailed properties of the Hopf bifurcation. It is noted that such Brusselator model possesses the supercritical and subcritical Hopf bifurcation. Numerical simulations are performed to verify the validity of theoretical analysis.

A wide variety of Gronwall inequalities of fractional variable order is presented. These are of left and right sides, applications follow. This new research strongly supports the new trend of Fractional Calculus of variable order, see [1].

We study the effect of harvesting on the stability of a predator-prey system. Alternative resource is provided to predator population to reduce extinction probability of predator. The model is extended incorporating gestation time delay of predator as the transformation of energy from prey to predator is not instantaneous. The essential mathematical analysis with the help of equilibria, stability analysis and Hopf bifurcation theory are studied. By applying the normal form theory and the center manifold theorem, the direction of Hopf bifurcation is determined. To verify analytical findings, numerical experiments are done using biological meaningful parameter values. Results illustrate that the proposed system experiences Hopf bifurcation around positive equilibrium point when time delay crosses a sequence of critical values. These critical values can be controlled by supplying alternative resource as well as harvesting effort. The model can be applied in plankton and fishery management systems.

In this paper, we apply a nonlinear time transformation to find accurate analytical solutions of a generalized Duffing-harmonic oscillator having a rational form for the restoring force. The transformation considered here has the interesting and useful feature that it leads to fully explicit solutions, in contrast to most such transformations. Its combination with the single term harmonic balance is found to be equivalent to the cubication method. The accuracy of the results can be improved by using the Newton linearization technique when looking for an-harmonic approximate solutions. It is observed that this process not only increases the accuracy of the solutions, but it can also enlarge the range of validity of the approximations in the space of parameters as compared to the method of cubication.