Hai-Feng Huo

Stability in a predator–prey model with Crowley–Martin function and stage structure for prey

Abstract This paper is concerned with a stage-structured predator–prey system with Crowley–Martin functional response. By analyzing the characteristic equation, the local stability of the equilibria is investigated. The persistence of system and global asymptotic stability of the positive equilibrium are carried out. Finally, several numerical simulations to illustrate the theoretical analysis are also given.

Persistence and Stability for a Generalized Leslie-Gower Model with Stage Structure and Dispersal

A generalized version of the Leslie-Gower predator-prey model that incorporates the prey structure and predator dispersal in two-patch environments is introduced. The focus is on the study of the boundedness of solution, permanence, and extinction of the model. Sufficient conditions for global asymptotic stability of the positive equilibrium are derived by constructing a Lyapunov functional. Numerical simulations are also presented to illustrate our main results.

A Predator-Prey Model with Functional Response and Stage Structure for Prey

A predator-prey system with Holling type II functional response and stage structure for prey is presented. The local and global stability are studied by analyzing the associated characteristic transcendental equation and using comparison theorem. The existence of a Hopf bifurcation at the positive equilibrium is also studied. Some numerical simulations are also given to illustrate our results.

Modelling Alcoholism as a Contagious Disease: A Mathematical Model with Awareness Programs and Time Delay

A dynamic alcohol consumption model with awareness programs and time delay is formulated and analyzed. The aim of this model is to capture the effects of awareness programs and time delay in controlling the alcohol problems. We introduce awareness programs by media in the model as a separate class with growth rate of the cumulative density of them being proportional to the number of mortalities induced by heavy drinking. Susceptible population will isolate themselves and avoid contact with the heavy drinkers or become aware of risk of heavy drinking and decline to drink due to such programs. In particular, we incorporate time delay because the nonconsumer population will take a period of time to become an alcohol consumer. We find that the model has two equilibria: one without alcohol problems and one where alcohol problems are endemic in population. The model analysis shows that though awareness programs cannot eradicate alcohol problems, they are effective measures in controlling the alcohol problems. Further, we conclude that the time delay in alcohol consumption habit which develops in susceptible population may result in Hopf bifurcation by increasing the value of time delay. Some numerical simulation results are also given to support our theoretical predictions.

Hopf bifurcation for a delayed predator–prey diffusion system with Dirichlet boundary condition

A delayed predator–prey diffusion system with Beddington–DeAngelis functional response under Dirichlet boundary condition is investigated. The existence and stability of the positive spatially nonhomogeneous steady-state solution are obtained via the implicit function theorem. Moreover, taking feedback time delay τ as the bifurcation parameter, Hopf bifurcation near the positive steady-state solution is proved to occur at a sequence of critical values, we can show that feedback time delay can induce nonhomogeneous periodic oscillatory patterns. The direction of Hopf bifurcation is forward when parameter m in model (1.2) is sufficiently large. Numerical simulations and numerical solutions are presented to illustrate our theoretical results.