互惠-寄生耦合系统的稳定性

Stability analysis of mutualistic-parasitic coupled system

寄生物与宿主之间协同进化的研究早已在生命科学领域引起广泛关注。现有研究寄生物与其宿主协同进化的模型几乎都是基于寄生物将会导致宿主种群减少的前提建立的。然而,寄生物在很多情况下也会促进宿主种群的增长,比如低密度的寄生物能提高宿主的免疫力从而提高宿主的存活率。基于这一前提假设,在经典的Lotka-Volterra模型和Leslie型捕食者-食饵模型基础上,引入寄生物对宿主的固有促进水平参数K,建立了一类互惠-寄生耦合模型。利用微分方程稳定性理论对模型进行分析,发现系统平衡点的稳定性与固有促进水平K密切相关。分析显示,在不同的固有促进水平K下,寄生物与宿主将会以稳定均衡或周期振荡的形式持续共存。数值模拟实验表明在一定条件下系统会出现Hopf分岔现象,并且随着固有促进水平K的增大,系统还会出现稳定的极限环,即随着固有促进水平K的增大,寄生物与宿主由稳定共存转变为变振幅、变周期的振荡共存。当固有促进水平参数K为零时,我们的模型就转化为经典的Leslie型捕食者-食饵模型。

The co-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-parasite interactions include： （i） The classical Lotka-.Volterra model and Leslie model, which showed that the host-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-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-equilibrium of the system. The improved models will display more diversified dynamical behaviors such as Hopf bifurcation, period-doubling bifurcation, chaotic phenomenon, etc. All above-mentioned models for co-evolution between hosts and parasites are based on an assumption that the increase of parasite population will decrease the host population. However, in many host-parasite systems, parasites might favorite the host population increase in some situations. For example, in the system between Escherichia coli or Lactobacillus and their hosts, the parasite with low-density will enhance immunity of hosts therefore improving host＇s survival rate, while the parasites with high-density will produce more carcinogens and toxins, causing detrimental effects on hosts. Based on the assumption that parasites might facilitate the host population increase, here we introduce the inherent promotion level K （ maximum promotion effect of parasite on host） into an integration model of Lotka-Voherra and Leslie type, then establish a mutualistic-parasitic coupled system. Using differential equation stability theory, we find that the behavior dynamics of the system closely associate with the inherent promotion level parameter K. Analysis shows that hosts and parasites will coexist with stable equilibrium, if the inherent promotion level K is a relative low value, or coexist with quasi-periodic oscillation, if the inherent promotion level K is a relatively high value. Numerical simulation shows that the system will display Hopf bifurcation at K =αl（1 ＋r）/（13to） ＋ to（1 -r） , and will exhibit a stable limit cycle at K 〉 α1（1 ＋r）/（βω） ＋ ω（1 - r） . Namely, with the increasing of the inherent promotion level K, stable coexistence between hosts and parasites might transform into the coexistence of variable-amplitude oscillations and variable-period oscillations. If the inherent promotion level K was zero, the model developed here will be transformed to be the classical Leslie prey-predator model.