具有第三类功能反应特性的Rosenzweig模型的定性分析

<正> 前言一九六三年,Rosenzweig和MacArthur提出了生态学中的捕食者——食饵数学模型 x=f(x)-φ(x·y) y=-ey+kφ(x·y)其中x表示食饵的种群密度,y表示捕食者的种群密度,f(x)表示食饵不受捕食者影响时的增长率,φ(x·y)表示捕食者的捕食率,k叫做食饵转化成捕食者的转化效率。通常取(f(x)=ax-bx~2,φ(x·y)=yφ(x),其中φ(x)叫做捕食者的功能反应函数,则得到模型

THE PREDATOR—PREY SYSTEM WITH HOLLING' 3TH FUNCTIONAL RESPONSE

Regarding φ(x)=(αx~2)/(x~2+β~2) as tolling'3th functional response,by Rosenzweig'model, we discuss the system The results we have obtained are as follows.1) If ke/α+(eβ~2b~2)/(αa~2),and a<3(3~(1/2))bβ,The singular Point (m.f(m)) is asymptotically stable and where m=((eβ~2)/(kα-e))~(1/2) f(x)=-b/αx~2+a/αx-b/αβ~2+(aβ~2)/(αx).3) If k>e/α+(eβ~2b~2)/(αa~2) and a>3(3~(1/2))bβ the singular Point is stable or unstable. if (m.f(m)) is unstable, there exists a~+least one Periodic orbit in the first quadrant of x-y Plane. If there is just one Periodic orbit, it's not Stable. If the Periodic Orbit is not unique, the outer one is semistable from outside, and the inner one is semrslable from inside.