A Simple Predator-Prey Population Model with Rich Dynamics

The Dynamics of Pasture-Herbivores-Carnivores with Sigmoidal Density Dependent Harvesting

We investigate biomass-herbivore-carnivore (top predator) interactions in terms of a tritrophic dynamical systems model. The harvesting rates of the herbivores and the top predators are described by means of a sigmoidal function of the herbivores density and the top predator density, respectively. The main focus in this study is on the dynamics as a function of the natural mortality and the maximal harvesting rate of the top predators. We identify parameter regimes for which we have non-existence of equilibrium points as well as necessary conditions for the existence of such states of the modelling framework. The system does not possess any finite equilibrium states in the regime of high herbivore mortality. In the regime of a high consumption rate of the herbivores and low mortality rates of the top predator, an asymptotically stable finite equilibrium state exists. For this positive equilibrium to exist the mortality of the top predator should not exceed a certain threshold level. We also detect regimes producing coexistence of equilibrium states and their respective stability properties. In the regime of negligible harvesting of the top predator level, we observe a finite window of the natural top predator mortality rates for which oscillations in the top predator-, the herbivore- and the biomass level take place. The lower and upper bound of this window correspond to two Hopf bifurcation points. We also identify a bifurcation diagram using the top predator harvesting rate as a control variable. Using this diagram we detect several saddle node- and Hopf bifurcation points as well as regimes for which we have coexistence of interior equilibrium states, bistability and relaxation type of oscillations.

Cyberspace: A Digital Ecosystem

Cyberspace is a new frontier, not just for hackers, but for engineers. It is a digital ecosystem, the next generation of Internet and network applications, promising a whole new world of distributed and open systems that can interact, self-organize, evolve, and adapt. These ecosystems transcend traditional collaborative environments, such as client-server, peer-to-peer, or hybrid models (e.g., web services), to become a self-organized, evolving, interactive environment. Understanding cyberspace as a system is critical if we are to properly design systems to exist within it. Considering it to be a digital ecosystem, where systems can adapt and evolve, will enable systems engineering to become more effective in the future of networks and the Internet. While most systems engineers have only anecdotal experience with large segments of this ecosystem, in today’s world all of them must come to understand it. Engineering any system, or portion of a system, begins with an understanding of the system. This paper presents two interrelated yet distinct foundational models of the ecosystem of cyberspace: a Systemigram to narrate the cyclical nature of cyber warfare, and a modified predator–prey model, as a mathematical model. Systems engineers can utilize these models to design digital “species” that function and adapt within this ecosystem.

Stability Analysis and Numerical Computation of the Fractional Predator–Prey Model with the Harvesting Rate

In this work, a fractional predator-prey model with the harvesting rate is considered. Besides the existence and uniqueness of the solution to the model, local stability and global stability are experienced. A novel discretization depending on the numerical discretization of the Riemann–Liouville integral was introduced and the corresponding numerical discretization of the predator–prey fractional model was obtained. The net reproduction number R 0 was obtained for the prediction and persistence of the disease. The dynamical behavior of the equilibria was examined by using the stability criteria. Furthermore, numerical simulations of the model were performed and their graphical representations are shown to support the numerical discretizations, to visualize the effectiveness of our theoretical results and to monitor the effect of arbitrary order derivative. In our investigations, the fractional operator is understood in the Caputo sense.

An SEI infection model incorporating media impact

To study the impact of media coverage on spread and control of infectious diseases, we use a susceptible-exposed-infective (SEI) model, including individuals' behavior changes in their contacts due to the influences of media coverage, and fully investigate the model dynamics. We define the basic reproductive number ?0 for the model, and show that the modeled disease dies out regardless of initial infections when ?0< 1, and becomes uniformly persistently endemic if ?0>1. When the disease is endemic and the influence of the media coverage is less than or equal to a critical number, there exists a unique endemic equilibrium which is asymptotical stable provided ?0 is greater than and near one. However, if ?0 is larger than a critical number, the model can undergo Hopf bifurcation such that multiple endemic equilibria are bifurcated from the unique endemic equilibrium as the influence of the media coverage is increased to a threshold value. Using numerical simulations we obtain results on the effects of media coverage on the endemic that the media coverage may decrease the peak value of the infectives or the average number of the infectives in different cases. We show, however, that given larger ?0, the influence of the media coverage may as well result in increasing the average number of the infectives, which brings challenges to the control and prevention of infectious diseases.