一类具阶段结构的捕食系统正周期解的存在性

Existence of Positive Periodic Solution for a Predator-prey System with Stage Structure

关键词:  捕食系统  阶段结构  自食  正周期解  重合度

Key words:  predator-prey system  stage structure  cannibalism  positive periodic solution  coincidence degree

摘要:

建立具有阶段结构和自食现象而且食饵和捕食者均受密度制约的周期捕食系统:
⎛x'(t)=x(t)[β₁(t)-a(t)x(t)]-b(t)x(t)z₂(t),
⎜z'₁(t)=β₂(t)z₂(t)-s(t)z₁(t)-c(t)z₁(t)z₂(t),
⎨
⎜z'₂(t)=r(t)z₁(t)-d(t)z₂²(t)+e(t)z₁(t)z₂(t)+h(t)x(t)z₂(t),
⎝x(0)>0,z₁(0)>0,z₂(0)>0,并且利用重合度理论得到正周期解存在的充分条件为d^ls^l > e^Mβ₂^M+(b^M/β₁^l)(r^Mβ₂^M+(h^Mβ₁^Ms^l/a^l)).

Abstract:

In this paper,a class of predator-prey system with stage structure and cannibalism is established,which is restricted by prey's and predators' density:
⎛x'(t)=x(t)[β₁(t)-a(t)x(t)]-b(t)x(t)z₂(t),
⎜z'₁(t)=β₂(t)z₂(t)-s(t)z₁(t)-c(t)z₁(t)z₂(t),
⎨
⎜z'₂(t)=r(t)z₁(t)-d(t)z₂²(t)+e(t)z₁(t)z₂(t)+h(t)x(t)z₂(t),
⎝x(0)>0,z₁(0)>0,z₂(0)>0.
By means of the theory of coincidence degree,a sufficien condition is obtained for the existence of positive periodic solution:d^ls^l > e^Mβ₂^M+(b^M/β₁^l)(r^Mβ₂^M+(h^Mβ₁^Ms^l/a^l)).