A rumor transmission model with nonmonotonic incidence rate was proposed, which provides excellent explanations of the “psychological” effect with rumor spreading in emergency event. By carrying out a global analysis of the model and studying the stability of the rumor-free equilibrium and the rumor-endemic equilibrium, we showed that either the number of infective individuals tends to zero as time evolves or the rumor persists. Finally, recommendations for policy makers and consulting advice for related commissions are explored in the case study of crazy rumors propagated for the iodized sail shortage panic in China.

This paper analyses the dynamics of a duopoly with quantity-setting firms and different attitudes towards strategic uncertainty. By following the recent literature on decision making under uncertainty, where the Choquet expected utility theory is adopted to allow firms to plan their strategies, we investigate the effects of the interaction between pessimistic and optimistic firms on economic dynamics described by a two-dimensional map. In particular, the study of the local and global behaviour of the map is performed under three assumptions: (1) both firms have complete information on the market demand and adjust production over time depending on past behaviours (static expectations—“best reply” dynamics); (2) both firms have incomplete information and production is adjusted over time by following a mechanism based on marginal profits; and (3) one firm has incomplete information on the market demand and production decisions are based on the behaviour of marginal profits, and the rival has complete information on the market demand and static expectations. In cases 2 and 3 it is shown that complex dynamics and coexistence of attractors may arise. The analysis is carried forward through numerical simulations and the critical lines technique.

The nonlinear dynamic behavior of rain-wind-induced vibration of inclined cable is investigated with the consideration of the equilibrium position of the moving rivulet. The partial differential governing equations of three-degree-of-freedom on the model of rain-wind-induced cable vibration are established, which are proposed for describing the nonlinear interactions among the in-plane, out-of-plane vibration of the cable and the oscillation of the moving rivulet. The Galerkin method is applied to discretize the partial differential governing equations. The approximately analytic solution is obtained by using the method of averaging. The unique correspondence between the wind and the equilibrium position of the rivulet is ascertained. The presence of rivulet at certain positions on the surface of cable is then proved to be one of the trigger for wind-rain-induced cable vibration. The nonlinear dynamic phenomena of the inclined cable subjected to wind and rain turbulence are then studied by varying the parameters including mean wind velocity, Coulomb damping force, damping ratio, the span length, and the initial tension of the inclined cable on the model. The jump phenomenon is also observed which occurs when there are multiple solutions in the system.

Nonautonomous long-short wave equations with quasiperiodic forces are studied. We prove the existence of the uniform attractor for the system by means of energy method, which is widely used to deal with problems who have no continuity (with respect to the initial data) property, as well as to those which Sobolev compact imbedding cannot be applied. Afterwards, we construct an approximate inertial manifold by means of extending phase space method and we estimated the size of the corresponding attracting neighborhood for this manifold.

Matsumoto and Szidarovszky (2011) examined a delayed continuous-time growth model with a special mound-shaped production function and showed a Hopf bifurcation that occurs when time delay passes through a critical value. In this paper, by applying the center manifold theorem and the normal form theory, we obtain formulas for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions. Moreover, Lindstedt’s perturbation method is used to calculate the bifurcated periodic solution, the direction of the bifurcation, and the stability of the periodic motion resulting from the bifurcation.

We introduce a time-to-build technology in a Solow model with nonconstant population. Our analysis shows that the population dynamics may be a source of stability switches and Hopf bifurcations. The analytical results are obtained using the recent technique introduced by Beretta and Kuang (2002) in the studying of delayed differential equations with delay-dependent coefficients in characteristic equation. Numerical simulations are performed in order to illustrate the main dynamic features of the model.

A further generalization of an economic growth model is the main topic of this paper. The paper specifically analyzes the effects on the asymptotic dynamics of the Solow model when two time delays are inserted: the time employed in order that the capital is used for production and the necessary time so that the capital is depreciated. The existence of a unique nontrivial positive steady state of the generalized model is proved and sufficient conditions for the asymptotic stability are established. Moreover, the existence of a Hopf bifurcation is proved and, by using the normal form theory and center manifold argument, the explicit formulas which determine the stability, direction, and period of bifurcating periodic solutions are obtained. Finally, numerical simulations are performed for supporting the analytical results.

We examine the impact of three forecasting methods on the bullwhip effect in a two-stage supply chain with one supplier and two retailers. A first order mixed autoregressive-moving average model (ARMA(1, 1)) performs the demand forecast and an order-up-to inventory policy characterizes the inventory decision. The bullwhip effect is measured, respectively, under the minimum mean-squared error (MMSE), moving average (MA), and exponential smoothing (ES) forecasting techniques. The effect of parameters on the bullwhip effect under three forecasting methods is analyzed and the bullwhip effect under three forecasting methods is compared. Conclusions indicate that different forecasting methods lead to different bullwhip effects caused by lead time, underlying parameters of the demand process, market competition, and the consistency of demand volatility between two retailers. Moreover, some suggestions are present to help managers to select the forecasting method that yields the lowest bullwhip effect.

This paper deals with the derivation and the mathematical analysis of an autonomous and nonlinear ordinary differential-based framework. Specifically, the mathematical framework consists of a system of two ordinary differential equations: a logistic equation with a time lag and an equation for the carrying capacity that is assumed here to be time dependent. The qualitative analysis refers to the stability analysis of the coexistence equilibrium and to the derivation of sufficient conditions for the existence of Hopf bifurcations. The results are of great interest in living systems, including biological and economic systems.

Nonlinear dynamics can be used to identify relationships between different firing patterns, which play important roles in the information processing. The present study provides novel biological experimental findings regarding complex bifurcation scenarios from period-1 bursting to period-1 spiking with chaotic firing patterns. These bifurcations were found to be similar to those simulated using the Hindmarsh-Rose model across two separated chaotic regions. One chaotic region lay between period-1 and period-2 burstings. This region has not attracted much attention. The other region is a well-known comb-shaped chaotic region, and it appears after period-2 bursting. After period-2 bursting, the chaotic firings lay in a period-adding bifurcation scenario or in a period-doubling bifurcation cascade. The deterministic dynamics of the chaotic firing patterns were identified using a nonlinear prediction method. These results provided details regarding the processes and dynamics of bifurcation containing the chaotic bursting between period-1 and period-2 burstings and other chaotic firing patterns within the comb-shaped chaotic region. They also provided details regarding the relationships between different firing patterns in parameter space.

Eutrophication removal problems have captured the attention of biologists, mathematicians, and environmental scientists. Within this framework, an impulsive eutrophication controlling system is studied analytically and numerically. A key advantage of the eutrophication system is that it can be quite accurate to describe the interaction effect of some critical factors (fishermen catch and releasing small fry, etc.), which enables a systematic and logical procedure for fitting eutrophication mathematical system to real monitoring data and experiment data. Mathematical theoretical works have been pursuing the investigation of two threshold functions of some critical parameters under the condition of all species persistence, which can in turn provide a theoretical basis for the numerical simulation. Using numerical simulation works, we mainly focus on how to choose the best value of some critical parameters to ensure the sustainability of the eutrophication system so that the eutrophication removal process can be well developed with maximizing economic benefit. These results may be further extended to provide a basis for simulating the algal bloom in the laboratory and understanding the application of some impulsive controlling models about eutrophication removal problems.

Some global existence and uniform asymptotic stability results for fractional functional differential equations are proved. It is worth mentioning that when $\alpha =1$ the initial value problem reduces to a classical dissipative differential equation with delays as in Caraballo et al.'s work (2005).

Uncertainties including the structured and unstructured, especially the nonlinear frictions, always exist in physical servo systems and degrade their tracking accuracy. In this paper, a practical method named adaptive robust controller (ARC) is synthesized with a continuous differentiable friction model for high accuracy motion control of a direct-drive dc motor, which results in a continuous control input and thus is more suitable for application. To further reduce the noise sensitivity and improve the tracking accuracy, a desired compensation version of the proposed adaptive robust controller is also developed and its stability is guaranteed by a proper robust law. The proposed controllers not only account for the structured uncertainties (e.g., parametric uncertainties) but also for the unstructured uncertainties (e.g., unconsidered nonlinear frictions). Furthermore, the controllers theoretically guarantee a prescribed output tracking transient performance and final tracking accuracy in both structured and unstructured uncertainties while achieving asymptotic output tracking in the absence of unstructured uncertainties, which is very important for high accuracy control of motion systems. Extensive comparative experimental results are obtained to verify the high-performance nature of the proposed control strategies.

This paper is concerned with the mathematical modeling of complex systems characterized by particles refuge. Specifically the paper focuses on the derivation and moments analysis of thermostatted kinetic frameworks with conservative and nonconservative interactions for closed and open complex systems at nonequilibrium. Applications and future research perspectives are discussed in the last section of the paper.

We propose an algae-fish semicontinuous system for the Zeya Reservoir to study the control of algae, including biological and chemical controls. The bifurcation and periodic solutions of the system were studied using a Poincaré map and a geometric method. The existence of order-1 periodic solution of the system is discussed. Based on previous analysis, we investigated the change in the location of the order-1 periodic solution with variable parameters and we described the transcritical bifurcation of the system. Finally, we provided a series of numerical results to illustrate the feasibility of the theoretical results. These results may help to facilitate a better understanding of algal control in the Zeya Reservoir.

The aim of this paper is to show that the Solow model equipped by realistic assumptions on technology and population dynamics is capable of explaining a well-known stylised fact of growth, that is, the presence of persistent oscillations of demoeconomic variables. In particular, our analysis shows that the coexistence of delays between (i) new investment and production and (ii) the birth date and the recruitment in the labour force is a source of cyclical behaviour in capital accumulation.

Nonlinear dynamics of cancer recurrence are known to be governed by several factors as initial tumour size, number of metastatic sites, or quantity of drug resistant cells. The precise extent and location of tumours are very important factors so quantitative and consistent methods of evaluation are needed to assess reponse to patient therapy. Whole-body $I\mathrm{123}$-metaiodobenzylguanidine (mIBG) scintigraphy is used as primary medical image modality to detect neuroblastoma tumours due to its high specificity and sensitivity. However, current oncological guidelines are based on qualitative observer-dependent analysis. This fact makes it difficult to compare results of scintigraphies taken at different moments during therapy or at different institutions. In this paper, we review analytical methods used in neuroblastoma detection and propose an observer-independent method to quantitatively analyse a $I\mathrm{123}$-mIBG scintigraphy.

We study a delayed SIR epidemic model and get the threshold value which determines the global dynamics and outcome of the disease. First of all, for any $\tau$, we show that the disease-free equilibrium is globally asymptotically stable; when , the disease will die out. Directly afterwards, we prove that the endemic equilibrium is locally asymptotically stable for any $\tau =0$; when $R_0>1$, the disease will persist. However, for any $\tau \ne \text{0}$, the existence conditions for Hopf bifurcations at the endemic equilibrium are obtained. Besides, we compare the delayed SIR epidemic model with nonlinear incidence rate to the one with bilinear incidence rate. At last, numerical simulations are performed to illustrate and verify the conclusions.

This paper is concerned with a computer virus model with two delays. Its dynamics are studied in terms of local stability and Hopf bifurcation. Sufficient conditions for local stability of the positive equilibrium and existence of the local Hopf bifurcation are obtained by regarding the possible combinations of the two delays as a bifurcation parameter. Furthermore, explicit formulae for determining direction of the Hopf bifurcation and stability of the bifurcating periodic solutions are obtained by using the normal form method and center manifold theory. Finally, some numerical simulations are presented to support the theoretical results.

The effects of removable devices’ heterouse in different areas on the propagation of malware spreading via removable devices remain unclear. As a result, in this paper, we present a model incorporating the heterogeneous use of removable devices, obtained by dividing the using rate into local area’s rate, neighbour area’s rate and global area’s rate, and then getting the final rate by multiplying the corresponding area ratio. The model’s equilibria and their stability conditions are obtained mathematically and verified by deterministic and stochastic simulations. Simulation results also indicate that the heterogeneity in using rate significantly changes the prospective propagation course of malware. Additionally, the thresholds of removable devices’ using rate in neighbour area are given, which can guide us in designing effective countermalware method.

The present study uses the differential transformation method to solve the governing equations of the coronary artery system and then analyzes the dynamic behavior of the system by means of phase portraits, power spectra, bifurcation diagrams, and Poincaré maps. Also, a master-slave control system is proposed to suppress the nonlinear chaotic behavior of the coronary artery system. The results show that the dynamic behavior of the coronary artery system is significantly dependent on the magnitude of the vibrational amplitude. Specifically, the motion changes from $T$-periodic to 2$T$-periodic, then from 4$T$-periodic to 8$T$-periodic, and finally to chaotic motion with windows of periodic motion as the vibrational amplitude is increased from 0.3 to 0.6 and from 4.5 to 5.9. In addition, it is shown that the proposed control scheme enables the coronary artery system to be synchronized to any state asymptotically such that the risk of cardiopathy is reduced.

We investigate the complex dynamics of cross-diffusion $SI$ epidemic model. We first give the conditions of the local and global stability of the nonnegative constant steady states, which indicates that the basic reproduction number determines whether there is an endemic outbreak or not. Furthermore, we prove the existence and nonexistence of the positive nonconstant steady states, which guarantees the existence of the stationary patterns.

A new integer-order chaotic financial system is extended by introducing a simple investment incentive into a three-dimensional chaotic financial system. A four-dimensional fractional-order chaotic financial system is presented by bringing fractional calculus into the new integer-order financial system. By using weighted integral thought, the fractional order derivative's economics meaning is given. The 0-1 test algorithm and the improved Adams-Bashforth-Moulton predictor-corrector scheme are employed to detect numerically the chaos in the proposed fractional order financial system.

We consider the effect of time delay and cross diffusion on the dynamics of a modified Leslie-Gower predator-prey model incorporating a prey refuge. Based on the stability analysis, we demonstrate that delayed feedback may generate Hopf and Turing instability under some conditions, resulting in spatial patterns. One of the most interesting findings is that the model exhibits complex pattern replication: the model dynamics exhibits a delay and diffusion controlled formation growth not only to spots, stripes, and holes, but also to spiral pattern self-replication. The results indicate that time delay and cross diffusion play important roles in pattern formation.

This paper is concerned with the dynamics of the following abstract retarded evolution equation: $\left(d/dt\right)$ $u\left(t\right)+Au\left(t\right)=F\left(u\left(t-r_1\right),\dots ,u\left(t-r_n\right)\right)+g\left(t\right)$ in a Hilbert space $H$, where $A:D\left(A\right)\subset H\to H$ is a self-adjoint positive-definite operator with compact resolvent and $F:D{\left(A^{\alpha }\right)}^n\to H$ $\left(\alpha \in \left[0,1/2\right]\right)$ is a locally Lipschitz continuous mapping. The dissipativity and pullback attractors are investigated, and the existence of locally almost periodic solutions is established.

Fluctuation dynamics of financial price changes is developed and investigated by oriented percolation system; oriented percolation is percolation with a special direction along which the activity can only propagate one way but not the other. Then, nonlinear behaviors of distribution and leverage effect of return time series are studied for the proposed model and the real stock market by comparison. We also investigate the scaling behaviors of return intervals. And a scaling function of exponential parameter is introduced to analyze fluctuation behaviors of return intervals. The empirical research exhibits that, for proper parameters, the simulation data of the model can fit the real markets to a certain extent.

A Cournot-Bertrand mixed duopoly game model is constructed. The existence and local stable region of the Nash equilibria point are investigated. Complex dynamic properties such as bifurcation and route to chaos are analyzed using parameter basin plots. The strange attractors are also studied when the system is in chaotic states. Furthermore, considering the memory of the market, a delayed Cournot-Bertrand mixed model is considered and the results show that the delayed system has the same Nash equilibrium and has a higher chance of reaching steady states or cycles than the model without delay. So making full use of the historical data can improve the system’s stability.

A delayed predator prey system with refuge and constant rate harvesting is discussed by applying the normal form theory of retarded functional differential equations introduced by Faria and Magalhães. The analysis results show that under some conditions the system has a Bogdanov-Takens singularity. A versal unfolding of the system at this singularity is obtained. Our main results illustrate that the delay has an important effect on the dynamical behaviors of the system.

We investigate the Shilnikov sense homoclinicity in a 3D system and consider the dynamical behaviors in vicinity of the principal homoclinic orbit emerging from a third order simplified system. It depends on the application of the simplest normal form theory and further evolution of the Hopf-zero singularity unfolding. For the Shilnikov sense homoclinic orbit, the complex form analytic expression is accomplished by using the power series of the manifolds surrounding the saddle-focus equilibrium. Then, the second order Poincaré map in a generally analytical style helps to portrait the double pulse dynamics existing in the tubular neighborhood of the principal homoclinic orbit.

A stochastic delay predator-prey model in a polluted environment with impulsive toxicant input is proposed and studied. The thresholds between stability in time average and extinction of each population are obtained. Some recent results are extended and improved greatly. Several simulation figures are introduced to support the conclusions.

We study the duopoly model proposed by Matsumoto and Nonaka (2006), in which two firms produce two complementary goods, and there are externalities of different signs. We analyze the topological complexity of the model by computing its topological entropy with prescribed accuracy, and, in addition, we show when such topological complexity is physically observable by characterizing the attractors. Finally, we exploit the fact that reaction maps have negative Schwarzian derivative to show the existence of absolutely continuous (with respect to Lebesgue measure) ergodic measures, and, as an economic application, we compute the average profit for almost all initial conditions.

This paper investigates the adaptive $Q$-$S$ synchronization of the fractional-order chaotic systems with nonidentical structures. Based on the stability of fractional-order systems and adaptive control technique, a general formula for designing the controller and parameters update law is proposed to achieve adaptive $Q$-$S$ synchronization between two different chaotic systems with different structures. The effective scheme parameters identification and $Q$-$S$ synchronization of chaotic systems can be realized simultaneously. Furthermore, two typical illustrative numerical simulations are given to demonstrate the effectiveness of the proposed scheme, for each case, we design the controller and parameter update laws in detail. The numerical simulations are performed to verify the effectiveness of the theoretical results.

This paper is concerned with a semiratio-dependent predator-prey system with nonmonotonic functional response and two delays. It is shown that the positive equilibrium of the system is locally asymptotically stable when the time delay is small enough. Change of stability of the positive equilibrium will cause bifurcating periodic solutions as the time delay passes through a sequence of critical values. The properties of Hopf bifurcation such as direction and stability are determined by using the normal form method and center manifold theorem. Numerical simulations confirm our theoretical findings.

The destruction of the natural environment has been drawing more and more attention. Developing low-carbon industry chains is an effective solution to the conflict between rapid economic growth and high CO₂ emissions. Summarizing various traditional and new industry chain sustainable development, live pig industry was chosen as a typical industry chain to study low-carbon development using a system dynamics and random chance-constrained model (SD-RCCM). Leshan, a world natural and cultural heritage area in China, was selected as a typical city to analyze the low-carbon pig industry. Three different programs based on distribution ratios were selected to study this industry. The results showed that program 1, which considers both environmental and economic benefits, realizes sustainable development. In order to extend the pig industry chain and fully utilize pig ordure and other waste, introducing a Clean Development Mechanism (CDM) and household biogas exploitation program is recommended.

We consider the stochastic optimal control problems under G-expectation. Based on the theory of backward stochastic differential equations driven by G-Brownian motion, which was introduced in Hu et al. (2012), we can investigate the more general stochastic optimal control problems under G-expectation than that were constructed in Zhang (2011). Then we obtain a generalized dynamic programming principle, and the value function is proved to be a viscosity solution of a fully nonlinear second-order partial differential equation.

This paper considers a Kaleckian type model of business cycle based on a nonlinear delay differential equation, whose associated characteristic equation is a transcendental equation with delay dependent coefficients. Using the conventional analysis introduced by Beretta and Kuang (2002), we show that the unique equilibrium can be destabilized through a Hopf bifurcation and stability switches may occur. Then some properties of Hopf bifurcation such as direction, stability, and period are determined by the normal form theory and the center manifold theorem.

Using bifurcation methods and the Abelian integral, we investigate the number of the limit cycles that bifurcate from the period annulus of the singular point when we perturb the planar ordinary differential equations of the form $\dot{x}=-yC(x,y)$, $\dot{y}=xC(x,y)$ with an arbitrary polynomial vector field, where $C(x,y)=1-x^3$ or $C(x,y)=1-x^4$.

We propose an ecoepidemiological prey predator model, where selective harvest effort on predator population is considered. Vaccination and taxation are introduced as control instruments, which are utilized to control number of susceptible prey population and protect predator population from overexploitation, respectively. Conditions which influence nonnegativity and boundedness of solutions are studied. Global stability analysis around disease-free equilibrium is discussed based on robust Bendixson criterion, which is theoretically beneficial to studying coexistence and interaction mechanism of population within harvested ecoepidemiological system. By using Pontryagin’s maximum principle, an optimal control strategy is derived to maximize the total discounted net economic revenue to society as well as protect prey population from infectious disease. Numerical simulations are carried out to show the consistency with theoretical analysis.

The Hopf bifurcation for a predator-prey system with $\theta$-logistic growth and prey refuge is studied. It is shown that the ODEs undergo a Hopf bifurcation at the positive equilibrium when the prey refuge rate or the index-$\theta$ passed through some critical values. Time delay could be considered as a bifurcation parameter for DDEs, and using the normal form theory and the center manifold reduction, explicit formulae are derived to determine the direction of bifurcations and the stability and other properties of bifurcating periodic solutions. Numerical simulations are carried out to illustrate the main results.

A novel chaotic image encryption scheme based on the time-delay Lorenz system is presented in this paper with the description of Circulant matrix. Making use of the chaotic sequence generated by the time-delay Lorenz system, the pixel permutation is carried out in diagonal and antidiagonal directions according to the first and second components. Then, a pseudorandom chaotic sequence is generated again from time-delay Lorenz system using all components. Modular operation is further employed for diffusion by blocks, in which the control parameter is generated depending on the plain-image. Numerical experiments show that the proposed scheme possesses the properties of a large key space to resist brute-force attack, sensitive dependence on secret keys, uniform distribution of gray values in the cipher-image, and zero correlation between two adjacent cipher-image pixels. Therefore, it can be adopted as an effective and fast image encryption algorithm.

A novel family of numerical integration of closed Newton-Cotes quadrature rules is presented which uses the derivative value at the midpoint. It is proved that these kinds of quadrature rules obtain an increase of two orders of precision over the classical closed Newton-Cotes formula, and the error terms are given. The computational cost for these methods is analyzed from the numerical point of view, and it has shown that the proposed formulas are superior computationally to the same order closed Newton-Cotes formula when they reduce the error below the same level. Finally, some numerical examples show the numerical superiority of the proposed approach with respect to closed Newton-Cotes formulas.

The problem discussed is the stability of two input-output feedforward and feedback relations, under an integral-type constraint defining an admissible class of feedback controllers. Sufficiency-type conditions are given for the positive, bounded and of closed range feed-forward operator to be strictly positive and then boundedly invertible, with its existing inverse being also a strictly positive operator. The general formalism is first established and the linked to properties of some typical contractive and pseudocontractive mappings while some real-world applications and links of the above formalism to asymptotic hyperstability of dynamic systems are discussed later on.

A worm spread model concerning impulsive control strategy is proposed and analyzed. We prove that there exists a globally attractive virus-free periodic solution when the vaccination rate is larger than ${\theta }_1$. Moreover, we show that the system is uniformly persistent if the vaccination rate is less than ${\theta }_1$. Some numerical simulations are also given to illustrate our main results.

The instantaneous availability of a periodically inspected system with several minimal repairs before a replacement or a perfect repair is studied. First, we investigate two concrete minimal repair models with periodic inspection and constant repair time and give their instantaneous availability formulas. Then the instantaneous availability of the proposed model with constant repair times is presented by a set of recursive formulas, and its piecewise monotonicity is also shown. An example is presented to illustrate its application.

We study the existence of periodic orbit for a differential system describing the effects of indirect predation over two preys. Besides discussing a generalized version of the model, we present some remarks and numerical experiments for the nonautonomous version of the two models.

Hopf bifurcation of a delayed predator-prey system with prey infection and the modified Leslie-Gower scheme is investigated. The conditions for the stability and existence of Hopf bifurcation of the system are obtained. The state feedback and parameter perturbation are used for controlling Hopf bifurcation in the system. In addition, direction of Hopf bifurcation and stability of the bifurcated periodic solutions of the controlled system are obtained by using normal form and center manifold theory. Finally, numerical simulation results are presented to show that the hybrid controller is efficient in controlling Hopf bifurcation.

This paper is concerned with blow-up phenomena and persistence properties for an integrable two-component Dullin-Gottwald-Holm shallow water system. We give sufficient conditions on the initial data which guarantee blow-up phenomena of solutions in finite time for both periodic and nonperiodic cases, respectively. Furthermore, the persistence properties of solutions to the system are investigated.

This paper is concerned with a Gause-type predator-prey system with two delays. Firstly, we study the stability and the existence of Hopf bifurcation at the coexistence equilibrium by analyzing the distribution of the roots of the associated characteristic equation. A group of sufficient conditions for the existence of Hopf bifurcation is obtained. Secondly, an explicit formula for determining the stability and the direction of periodic solutions that bifurcate from Hopf bifurcation is derived by using the normal form theory and center manifold argument. Finally, some numerical simulations are carried out to illustrate the main theoretical results.