DYNAMICAL BEHAVIOR OF AN INNOVATION DIFFUSION MODEL WITH INTRA-SPECIFIC COMPETITION BETWEEN COMPETING ADOPTERS

In this paper,we proposed an innovation diffusion model with three compartments to investigate the diffusion of an innovation(product)in a particular region.The model exhibits two equilibria,namely,the adopter-free and an interior equilibrium.The existence and local stability of the adopter-free and interior equilibria are explored in terms of the effective Basic Influence Number(BIN)R_(A).It is investigated that the adopter free steady-state is stable if R_(A)<1.By consideringτ(the adoption experience of the adopters)as the bifurcation parameter,we have been able to obtain the critical value ofτresponsible for the periodic solutions due to Hopf bifurcation.The direction and stability analysis of bifurcating periodic solutions has been performed by using the arguments of normal form theory and the center manifold theorem.Exhaustive numerical simulations in the support of analytical results have been presented.

STABILITY AND BIFURCATION ANALYSIS OF A DELAYED INNOVATION DIFFUSION MODEL

In this article, a nonlinear mathematical model for innovation diffusion with stage structure which incorporates the evaluation stage （time delay） is proposed. The model is analyzed by considering the effects of external as well as internal influences and other demographic processes such as emigration, intrinsic growth rate, death rate, etc. The asymptotical stability of the various equilibria is investigated. By analyzing the exponential characteristic equation with delay-dependent coefficients obtained through the variational matrix, it is found that Hopf bifurcation occurs when the evaluation period （time delay, T） passes through a critical value. Applying the normal form theory and the center manifold argument, we de- rive the explicit formulas determining the properties of the bifurcating periodic solutions. To illustrate our theoretical analysis, some numerical simulations are also included.

Modeling and Analysis of a Single Species Population with Viral Infection in Polluted Environment

In this paper, a mathematical model is proposed to study the effect of pollutant and virus induced disease on single species animal population and its essential mathematical features are analyzed. It is observed that the susceptible population does not vanish when it is only under the effect of infection but in the polluted environment, it can go to extinction. Also, it has been observed that the replication threshold obtained, increases on account of pollutant concentration consequently decreasing the susceptible population. Further persistence results for the proposed model are obtained and the condition for the existence of the Hopf-bifurcation is derived. Finally, numerical simulation in support of analytical results is carried out.

A Leslie-Gower Holling Type-II Predator-Prey Mathematical Model with Disease in Prey Population Incorporating a Prey Refuge

We formulate and analyze a predator-prey model followed by Leslie-Gower model in which the prey population is infected by disease. We assume that the disease can only spread over prey population. As a result prey population has been classified into two categories, namely susceptible prey, infected prey where as the predator population remains free from infection. To investigate the behaviour of prey population we incorporate prey refuge in this model. Since the prey refuge decreases the predation rate then it has an important effect in our predator-prey interaction model. We have discussed the existence of various equilibrium points and local stability analysis at those equilibrium points. We investigate the Hopf-bifurcation analysis about the interior equilibrium point by taking the rate of infection parameter and the prey refuge parameter as bifurcation parameters. The numerical analysis is carried out to support the analytical results and to discuss some interesting results that our model exhibits.

Effect of random movement and cooperative hunting in the prey-predator system:A dynamical approach

Self-diffusion prerequisite is obtained as the spreading approach of biological populations.Cooperative hunting is a common behavior in predator populations that promotes predation and the coexistence of the prey-predator system.On the other side,the Allee effect among prey may cause the system to become unstable.In this paper,a difusive prey predator system with cooperative hunting and the weak Allee effect in prey populations is discussed.The linear stability and Hopf-bifurcation analysis had been used to examine the system's stability.From the spatial stability of the system,the conditions for Turing instability have been derived.The multiple-scale analysis has been used to derive the amplitude equations of the system.The stability analysis of these amplitude equations leads to the formation of Turing patterns.Finally,numerical simulations are used to analyze spatial patterns forming in 1-D and 2-D.The studies indicate that the model can generate a complex pattern structure and that self-diffusion has a drastic impacton species distribution.

Modeling the effect of time delay in implementation of mitigation policies on the control of atmospheric greenhouse gases

Mitigation of the enhanced greenhouse gas(GHG)concentrations in the Earth's atmosphere is imperative to meet the climate change mitigation objective.Governments of many countries are developing and implementing various mitigation strategies to reduce their GHG emissions.However,a time delay between the formulation and implementation of these mitigation policies can affect their effectiveness in controlling greenhouse gas levels in the atmosphere.This work presents a nonlinear mathematical model to investigate the effect of application of mitigation strategies and the delay involved in their implementation over the reduction of atmospheric greenhouse gases.In model formulation,it is assumed that the mitigation strategies work two-fold;first they reduce the GHG emission rate from the anthropogenic source and second they increase the removal rate of greenhouse gas from the atmosphere.A comprehensive stability analysis of the proposed model system is made to examine its long-term behavior.The model analysis shows that an increase in the implementation rate of mitigation strategies and their efficiencies to cut down the GHG emission rate from point sources and increase the GHG uptake rate lead to reduction in equilibrium GHG concentration.It is found that a long delay in the execution of mitigation policies can destabilize the system dynamics and leads to the generation of periodic oscillations.The expression for the threshold value of the delay parameter at which periodic oscillations arise via Hopf-bifurcation is determined.The stability and direction of bifurcating periodic solutions are discussed.A sensitivity analysis is performed to investigate the effect of changes in key parameters over system dynamics.

Impact of time delay and cooperation strategy on the stability of a predator-prey model with Holling type III functional response

In this paper,we propose and analyze a delayed predator-prey model with Holling type III functional response taking into account cooperation behavior in predators.The time delay is introduced in the attack rate to represent the time necessary to trigger the attack.Each analytical result is followed by an ecological interpretation.We investigate the effect of hunting cooperation on both the number and the level of positive steady states.We observe that the level of the positive equilibrium decreases when increasing the hunting cooperation parameter.Then,we study the impact of the delay as well as the cooperation in hunting on the dynamics of the system.We prove that the presence of delay in the attack rate induces stability switches around the coexisting equilibrium when predators cooperate.In addition,we consider the discrete delay as a bifurcation parameter and prove that the model undergoes a Hopf-bifurcation at the coexisting equilibrium when the delay crosses some critical values.Numerical simulations are presented to confirm our analytical findings.

Bifurcation analysis and optimal control of an epidemic model with limited number of hospital beds

This paper deals with a three-dimensional nonlinear mathematical model to analyze an epidemic's future course when the public healthcare facilities,specifically the number of hospital beds,are limited.The feasibility and stability of the obtained equilibria are analyzed,and the basic reproduction number(Ro)is obtained.We show that the system exhibits transcritical bifurcation.To show the existence of Bogdanov-Takens bifurcation,we have derived the normal form.We have also discussed a generalized Hopf(or Bautin)bifurcation at which the first Lyapunov coefficient evanescences.To show the existence of saddle-node bifurcation,we used Sotomayor's theorem.Furthermore,we have identified an optimal layout of hospital beds in order to control the disease with minimum possible expenditure.An optimal control setting is studied analytically using optimal control theory,and numerical simulations of the optimal regimen are presented as well.

EFFECT OF KAIROMONE ON PREDATOR-PREY DYNAMICS -- A DELAY MODEL

In most of the predator-prey systems, prey individuals make transitions between vulnerable and invulnerable states or locations. This transition is regulated by various inducible defense mechanisms. Diel vertical migration （DVM） in zooplankton is the most effective and instantaneous defense observed in zooplankton population. Zooplankton shows downward vertical migration in the daytime in the presence of predators （or predator kairomones） to avoid predation （i.e. refuge use）, and it enters into the surface water again at night to graze phytoplankton. The dynamics of the planktonic ecosystem under DVM of zooplankton along with fish kairomone and the multiple delays due to migration for vulnerable and invulnerable prey and reproduction in the predator population is of considerable interest both in theoretical and experimental ecologists. By developing mathematical model, we analyze such a system. The conditions for which the system enters into Hopf-bifurcation are obtained. Moreover, the conditions for which the bifurcating branches are supercritical are also derived. Our results indicate that DVM along with the effect of kairomone and multiple delays with a certain range are responsible to enhance the stability of the system around the positive interior equilibrium point.

Comparison between two tritrophic food chain models with multiple delays and anti-predation effect

Exploring the predator prey linkage in food chain system is the most familiar research work in population biology.Recently,some research experiments show that predator-prey interaction not only governed by direct hunting but also influenced by some indirect effect such as fear effect(felt by prey)that may change the physiological behavior of prey.Based upon this fact,we consider a tritrophic food chain model incorporating with anti-predation response(fear effect)and multiple time delays for biomass conversion from prey to middle predator and middle to top predator.We analyze the resulting delay differential equations and explore how the anti-predation response level affects the population dynamics.We also investigate the effect of delay parameters,for which the model system switches its stability through Hopf-bifurcation.We compare all of our results between two different food chain models consisting of two different functional responses.Some numerical simulations are performed to validate the effectiveness of the derived theoretical results.

Research on Rumor Spreading Model with Time Delay and Control Effect

Information flow retains a critical role in decision making among investors. In this paper,we employ a diffusion model based on epidemiology theory to study the rumor spreading process within investors. The paper introduce the feedback mechanism of classical control theory into the model, which helps to reflect the interaction between rumor spreaders and information supervision.Further we apply a time delay factor to give investors access to transparent information and change their behavior. Subsequently, the stability of the rumor disappearance equilibrium and the rumor existence equilibrium are analyzed and the condition for the system undergoes a Hopf-bifurcation is given. The mathematical arguments are subjected to numerical simulations to present the ideal case scenarios. The results suggest that, increase the general strength of information supervision and the proportion coefficient associated with the infected population in the short-term delay are conducive to better control.