A duopolistic model of dynamic competitive advertising

This paper proposes a differential game model of competitive advertising decisions for non-durable products by extending the Lanchester model and the Deal model of competitive advertising in the literature. The proposed model is compared empirically with the Lanchester model for model fitting and forecast accuracy. It is shown that the model is suitable for an actual market and out-performs the Lanchester model in forecast accuracy. The model provides a sensible modeling alternative to the Lanchester model for the study of dynamic competitive advertising decisions. Necessary and sufficient conditions for open-loop and closed-loop Nash equilibrium solutions to the model are discussed. A numerical algorithm for open-loop and closed-loop Nash strategies to the model is developed.     

Dynamical behavior of an epidemic model with a nonlinear incidence rate

In this paper, we study the global dynamics of an epidemic model with vital dynamics and nonlinear incidence rate of saturated mass action. By carrying out global qualitative and bifurcation analyses, it is shown that either the number of infective individuals tends to zero as time evolves or there is a region such that the disease will be persistent if the initial position lies in the region and the disease will disappear if the initial position lies outside this region. When such a region exists, it is shown that the model undergoes a Bogdanov-Takens bifurcation, i.e., it exhibits a saddle-node bifurcation, Hopf bifurcations, and a homoclinic bifurcation. Existence of none, one or two limit cycles is also discussed.     

Dynamical study and robustness for a nonlinear wastewater treatment model

In this work we deal with a wastewater treatment by using the activated sludge process. The problem is formulated as a nonlinear dynamical system. Firstly, we develop the dynamical study of the model when all parameters are well known. Hence, basic properties of invariance and dissipation are established and, under a suitable condition on parameters, a globally asymptotically stable equilibrium point occurs. Secondly, when the bacterium growth function and the substrate concentration in the feed stream are not well known, the robustness analysis provides the existence of an attractor domain to which all trajectories of the system converge. Finally, we prove that we can reduce the size (volume) of this attractor domain by increasing the recycle rate to a maximum fixed level.     

National and store brand advertising strategies

As the propensity of premium store brands (SBs) increases, retailers must consider different ways to drive sales besides promotional strategies. With this in mind, we consider a national brand (NB) and a (premium) SB co-existing in a market. Each brand has to decide the amount to invest in advertising its product and the prices to charge its customers, which can be determined separately or in unison. When either advertising expenditures or pricing decisions are set, each brand must keep in mind that the advertising efforts and revenue may spillover between the two brands, customers who intend to purchase the NB may end up purchasing the SB and vice versa. We derive an analytical model of the situations described and characterize equilibrium advertising decisions. We find that the characteristics of a premium SB may depend on which marketing/promoting instrument (advertising or pricing) is the primary method for driving demand; and in some situations an NB may be better off to not advertise at all and instead let the premium SB carry out all of the advertising.     

Dynamical analysis of a Lotka-Volterra learning-process model

A Lotka-Volterra learning-process model was proposed by Monteiro and Notargiacomo in [{\it Commum. Nonlinear Sci. Numer. Simulat.} {\bf 47}(2017), 416-420] to approach learning process as an interplay between understanding and doubt. They studied the stability of the boundary equilibria and gave some numerical simulations but no further discussion for bifurcations. In this paper, we study the qualitative properties of the interior equilibria and a singular line segment completely. Moreover, we discuss their bifurcations such as transcritical, pitchfork, Hopf bifurcation on isolated equilibria and transcritical bifurcation without parameters on non-isolated equilibria. Finally, we also demonstrate these analytical theory by numerical simulations.     

A nonlinear Cournot duopoly with advertising

This paper aims at studying local and global dynamics in a nonlinear duopoly with quantity-setting firms and non-cooperative advertising investments that affect the degree of (horizontally) differentiated products. It concentrates on persuasive advertising in a model where each firm has limited information and uses a behavioural rule to set the quantity for the subsequent period. By using some mathematical techniques and numerical simulations, our results show the existence of weak (à la Milnor) attractors, multistability and chaotic dynamics. In the long term, firms may continuously shift from states in which they invest in advertising to states in which advertising investment is absent.     

Dynamical analysis and anti-synchronization of a new 6D model with self-excited attractors

A novel 6D dissipative model with an unstable equilibrium point is introduced herein. Some of the dynamic characteristics of the proposed model were explored via analyses and numerical simulations including critical points, stability, Lyapunov exponents, time phase portraits, and circuit implementation. Also, anti-synchronization phenomena were implemented on the new system. Firstly, the error dynamics is found. Then, four different controllers are adopted to stabilize this error relying on the ...     

Dynamical analysis of a plant-herbivore model bifurcation and global stability

The first part of the paper deals with a brief introduction of the plant-herbivore model system along with deterministic analysis of local stability and Hopf-bifurcations. The second part consists of stability analysis of the limit cycle arising from Hopf-bifurcation and uniqueness of limit cycle. The third part deals with the study of global stability of the model system under consideration.     

Dynamical analysis of a delay model of phytoplankton-zooplankton interaction

The interaction of toxic-phytoplankton-zooplankton systems and their dynamical behavior will be considered in this paper based upon nonlinear ordinary differential equation model system. We induced a discrete time delay to the both of the consume response function and distribution of toxic substance term to describe the delay in the conversion of nutrient consumed to species and the fact that the time required for the phytoplankton species to mature before they can produce toxic substances. We generalized the model in [1] and explicit results are derived for globally asymptotically stability of the boundary equilibrium. Using numerical simulation method, we determine there is a parameter range for the delay parameter τ where more complicated dynamics occurs, and this appears to be a new result. Significant outcomes of our numerical findings and their interpretations from ecological point of view are provided in this paper.     

Dynamical analysis of a sex-structured Chlamydia trachomatis transmission model with time delay

The effect of using time delay to model the latency period of Chlamydia trachomatis infection is explored, by designing a deterministic two-sex model for Chlamydia transmission dynamics in a population. The resulting delay differential equation model is shown to undergo the phenomenon of backward bifurcation, where a stable disease-free equilibrium co-exists with one or more stable endemic equilibria when the associated reproduction threshold is less than unity. This phenomenon arises due to the re-infection of individuals who recovered from the disease. Using permanence theory, it is shown that Chlamydia will persist in the population whenever the associated reproduction threshold exceeds unity. It is further shown that long latency period could induce positive (decrease disease burden) or negative (increase disease burden) population-level impact depending on the sign of a certain epidemiological threshold quantity and some other conditions. Furthermore, this study shows that adding a time delay (to model the latency period) does not alter the main equilibrium dynamics (with respect to the effective control or persistence of the disease in the community) of the corresponding non-delayed Chlamydia transmission model considered in our earlier study Sharomi and Gumel (2009) [7].     

Dynamical analysis of a delayed ratio-dependent Holling-Tanner predator-prey model

In this paper, a delayed Holling-Tanner predator-prey model with ratio-dependent functional response is considered. It is proved that the model system is permanent under certain conditions. The local asymptotic stability and the Hopf-bifurcation results are discussed. Qualitative behaviour of the singularity (0,0) is explored by using a blow up transformation. Global asymptotic stability analysis of the positive equilibrium is carried out. Numerical simulations are presented for the support of our analytical findings.     

Dynamical analysis of turbulence in fusion plasmas and nonlinear waves

Turbulence is one of the key problems of classical physics, and it has been the object of intense research in the last decades in a large spectrum of problems involving fluids, plasmas, and waves. In order to review some advances in theoretical and experimental investigations on turbulence a mini-symposium on this subject was organized in the Dynamics Days South America 2010 Conference. The main goal of this mini-symposium was to present recent developments in both fundamental aspects and dynamical analysis of turbulence in nonlinear waves and fusion plasmas. In this paper we present a summary of the works presented at this mini-symposium. Among the questions to be addressed were the onset and control of turbulence and spatio-temporal chaos.     

Dynamical analysis of age-structured pertussis model with covert infection

An age-structured pertussis model with covert infection is proposed to understand the effect of covert infection on the recurrence of pertussis. It is found that vaccination only for young children does not have a decisive effect on whooping cough control. It is shown that although the vaccine coverage rate is relatively high, the model has a backward bifurcation for a larger covert infection rate. In addition, sufficient conditions for the disease-free steady state to be globally asymptotically stable are obtained.     

An empirical study on the Lanchester model of combat for competitive advertising decisions

This paper presents an empirical study on the Lanchester model of combat for competitive advertising decisions. Three issues are evaluated: (i) the specification of the market share response model; (ii) the effect of inflation on the estimation of the response model; and (iii) the performance of competitive strategies. It is shown that (a) the square root function that is used in previous studies is often inappropriate to characterize the market share response model; (b) market share variations are more responsive to current advertising expenditures; (c) closed-loop Nash equilibrium strategies are better competitive advertising strategies for firms to maximize profits than open-loop Nash equilibrium strategies; and (d), finally, general perfect equilibria Nash equilibrium strategies developed by Case are usually not good competitive advertising strategies for firms to maximize profits.     

Global analysis of a vector-host epidemic model with nonlinear incidences

In this paper, an epidemic model with nonlinear incidences is proposed to describe the dynamics of diseases spread by vectors (mosquitoes), such as malaria, yellow fever, dengue and so on. The constant human recruitment rate and exponential natural death, as well as vector population with asymptotically constant population, are incorporated into the model. The stability of the system is analyzed for the disease-free and endemic equilibria. The stability of the system can be controlled by the threshold number R₀. It is shown that if R₀ is less than one, the disease free equilibrium is globally asymptotically stable and in such a case the endemic equilibrium does not exist; if R₀ is greater than one, then the disease persists and the unique endemic equilibrium is globally asymptotically stable. Our results imply that the threshold condition of the system provides important guidelines for accessing control of the vector diseases, and the spread of vector epidemic in an efficient way can be prevented. The contribution of the nonlinear saturating incidence to the basic reproduction number and the level of the endemic equilibrium are also analyzed, respectively.     

Dynamical analysis of a new microbial pesticide model with the Monod growth rate

In this paper, a new mathematical model for the entomopathogenic nematode with the Monod growth rate is formulated. Firstly, continuous release of the entomopathogenic nematode is considered. The existence of limit cycles, the Hopf bifurcation and the stability of the periodic solution created by the bifurcation are proved. The sufficient conditions for the globally asymptotical stability of system are obtained. Secondly, impulsive release of the entomopathogenic nematode is also considered. By using the Floquet’s theorem and the small amplitude perturbations, we show that the pest-free periodic solution is locally stable if some conditions are satisfied. In a certain limiting case, it is shown that a nontrivial periodic solution emerges via a supercritical bifurcation. Finally, our findings are confirmed by means of numerical simulations. Thus, we provide mathematical evidence on how to release the entomopathogenic nematode in order to control pests at acceptably low levels.     

The polynomial hierarchy and a simple model for competitive analysis

The multi-level linear programs of Candler, Norton and Townsley are a simple class of sequenced-move games, in which players are restricted in their moves only by common linear constraints, and each seeks to optimize a fixed linear criterion function in his/her own continuous variables and those of other players. All data of the game and earlier moves are known to a player when he/she is to move. The one-player case is just linear programming.We show that questions concerning only the value of these games exhibit complexity which goes up all levels of the polynomial hierarchy and appears to increase with the number of players.For three players, the games allow reduction of the ₂ and ₂ levels of the hierarchy. These levels essentially include computations done with branch-and-bound, in which one is given an oracle which can instantaneously solve NP-complete problems (e.g., integer linear programs). More generally, games with (p + 1) players allow reductions of _p and _p in the hierarchy.An easy corollary of these results is that value questions for two-player (bi-level) games of this type is NP-hard.The author's work has been supported by the Alexander von Humboldt Foundation and the Institut fur Okonometrie und Operations Research of the University of Bonn, Federal Republic of Germany; grant ECS8001763 of the National Science Foundation, USA; and a grant from the Georgia Tech Foundation.     

Dynamical behavior analysis of a two-dimensional discrete predator-prey model with prey refuge and fear factor

This paper investigates the dynamics of an improved discrete Leslie-Gower predator-prey model with prey refuge and fear factor. First, a discrete Leslie-Gower predator-prey model with prey refuge and fear factor has been introduced. Then, the existence and stability of fixed points of the model are analyzed. Next, the bifurcation behaviors are discussed, both flip bifurcation and Neimark-Sacker bifurcation have been studied. Finally, some simulations are given to show the effectiveness of the theoretical results.