具有脉冲效应的Holling Ⅲ系统的动力行为

Analysis of a Holling Type Ⅲ Predator System with Impulsive Effect

关键词:  HollingⅢ类功能反应  脉冲效应  全局渐近稳定性  持续生存  分支

Key words:  functional response of Holling type Ⅲ  impulsive effect  globally asymptotical stability  permanence  bifurcation

摘要:

通过周期性释放天敌和化学控制的综合害虫管理(IPM)改进捕食者具有Holling型功能性反应系统:{_{(dy(t)/dt)=-cy(t)+(kαx²(t)y(t)/x²(t)+β²).}^{(dx(t)/dt)=ax(t)-bx2(t)-(αx2(t)y(t)/x2(t)+β2),}
得到一个新的系统:{_{(dy(t)/dt)=-cy(t)+(kαx²(t)y(t)/x²(t)+β²).}^{(dx(t)/dt)=ax(t)-bx2(t)-(αx2(t)y(t)/x2(t)+β2),}}t≠nT,{_{Δy(t)=-p2y(t)+q.}^{Δx(t)=-p₁x(t),}}t=nT.给出当q>0,0≤p₁<1,0≤p₂<1时,新系统的害虫周期全局渐近稳定性与新系统的持续生存条件.研究当q>0,0≤p₁<1,0≤p₂<1时,新系统正周期解的存在性和当q≡0,0<p₁<1,0≤p₂<1时,无捕食者周期解的存在性和稳定性.

Abstract:

By introducing a constant periodic releasing natural enemies and integrated pest management,we devolop the system where the predator has Holling type Ⅲ functional response:{_{(dy(t)/dt)=-cy(t)+(kαx²(t)y(t)/x²(t)+β²)}.^{(dx(t)/dt)=ax(t)-bx2(t)-(αx2(t)y(t)/x2(t)+β2)},
At the sametime,we obtain a new system:{_{(dy(t)/dt)=-cy(t)+(kαx²(t)y(t)/x²(t)+β²).}^{(dx(t)/dt)=ax(t)-bx2(t)-(αx2(t)y(t)/x2(t)+β2),}}t≠nT,{_{Δy(t)=-p2y(t)+q.}^{Δx(t)=-p₁x(t),}}t=nT.When q>0,0 ≤ p₁<1 and 0 ≤ p₂<1,we obtain conditions for global asymptotic stability of pest-eradication periodic solution and permanence of the new system,and also obtain the existence of a postive periodic solution of the new system.Finally,we discuss the existence and stability of the predator-free periodic solution.