具有Beddington-DeAngelis功能性反应的三维离散顺环捕食系统的持久性

Permanence of a Three-Species Clockwise Chain Predator-Prey System with Beddington-DeAngelis Functional Response

研究一类具有Bedd ington-DeAngelis功能性反应的三维顺环捕食系统的持久性问题。首先,建立具有B-D功能性反应的三维顺环捕食系统的半离散化数学模型,具体为x1(n+1)=x1(n)exp[r1(n)-a1(n)x1(n)-c1(n)+db11((nn))xx22((nn))+x1(n)+c3(n)k+3(dn3)(bn3)(xn1)(xn3)(n+)x3(n)]x2(n+1)=x2(n)exp[r2(n)-a2(n)x2(n)-c2(n)+db22((nn))xx33((nn))+x2(n)+c1(n)k+1(dn1)(bn1)(xn2)(xn1)(n+)x1(n)]x3(n+1)=x3(n)exp[r3(n)-a3(n)x3(n)-c3(n)+db33((nn))xx11((nn))+x3(n)+c2(n)k+2(dn2)(bn2)(xn3)(xn2)(n+)x2(n)]。然后,利用不等式技巧,得到系统永久持续生存性的一个充分条件,即:假设条件r1Lc1L>b1UM2,r2Lc2L>b2UM3,r3Lc3L>b3UM1成立,则此半离散化三维顺环捕食系统是永久持续生存的,其中M1=maxr1U+a1Lk3Ub3U,exp(r1U-a11L+k3UbU3),M2=maxrU2+a2Lk1UbU1,exp(r2U-a1L2+k1UbU1),M3=maxr3U+a3kL2Ub2U,exp(r3U-a13L+k2Ub2U)均为正常数。所获得结论将连续情形推广到了半离散化模型。

This paper studies the permanence of a predator-prey system with three-species clockwise chain and Beddington-DeAngelis functional response. Firstly, A mathematical model of a predator-prey system with three-species clockwise chain and B-D functional response is formulated in the following form of semi-discretisations {x1（n＋1）=x1（n）exp{[r1（n）-a1（n）x1（n）-b1（n）x2（n）/c1（n）＋d1（n）x2（n）＋x1（n）＋k3（n）＋b3（n）x3（n）/c3（n）d3（n）x1（n）＋x3（n）]} x2（n＋1）=x2（n）exp{[r2（n）-a2（n）x2（n）-b2（n）x3（n）/c2（n）＋d2（n）x3（n）＋x2（n）＋k1（n）＋b1（n）x1（n）/c1（n）d1（n）x2（n）＋x1（n）]}.x3（n＋1）=x3（n）exp{[r3（n）-a3（n）x3（n）-b3（n）x1（n）/c3（n）＋d3（n）x1（n）＋x3（n）＋k2（n）＋b2（n）x2（n）/c2（n）d2（n）x3（n）＋x2（n）]}Then, a sufficientcondition ensuring the uniform permanence of the system is obtained by using some skills of inequalities ,that is, the system with three-species clockwise chain in form of semi-discretisations is permanent provided that r1^Lc1^L〉b1^UM2,r2^Lc2^L〉b2^UM3,r3^Lc3^L〉b3^UM1,where M1=max{r1^U＋k3^Ub3^U/a1^L,exp（r1^U-1＋k3^Ub3^U）/a1^L},M2=max{r2^U＋k1^Ub1^U/a2^L,exp（r2^U-1＋k1^Ub1^U）/a2^L},M3=max{r3^U＋k2^Ub2^U/a3^L,exp（r3^U-1＋k2^Ub2^U）/a3^L}is positively invariant. The results of the corresponding continuous systems in some relevant references are extended to ones of the systems in form of semi-discretisations.