Stability analysis of mutualistic-parasitic coupled system

Abstract

The co鄄evolution between parasite and its host is one of the most important research field in both population ecology and biological forecasting, such as crop cultivation, livestock breeding, excessive copies of pathogenic cells, and so on. The main models for studying host鄄parasite interactions include: (1) The classical Lotka鄄Volterra model and Leslie model, which showed that the host鄄parasite system could have diversified dynamical behaviors, including local asymptotic stability, global asymptotic stability, limit cycle, bifurcation phenomenon, chaotic phenomena, and so on; (2) Epidemic models, which are developed to explain whether the spread of virus will depend on the threshold value. If the amount of virus is higher than the threshold value, infectious will be maintained, whist the infectious will tend to disappear, if the amount of virus is lower than the threshold value; (3) The Nicholson鄄Bailey model with discrete time variable. The model demonstrates that host and parasite population system might form a coupling vibration. The oscillation in this model is not stable, and any disturbance might lead to non鄄equilibrium of the system. The improved models will display more diversified dynamical behaviors such as Hopf bifurcation, period鄄doubling bifurcation, chaotic phenomenon, etc. All above鄄mentioned models for co鄄evolution between hosts and parasites are based on an assumption that the increase