三阶段时滞种群生长模型的渐近稳定性

Stability for Single Population Growth Model with Three-stage Structures and Time Delay

根据种群生长的阶段性,引入时滞建立了一类三阶段结构的时滞种群生长模型:{_1(t)=αx_3(t)-γx_1(t)-αe^(-γτ)x_3(t-τ)_2(t)=αe^(-γτ)x_3(t-τ)-bx_2(t)-αx_2(t),_3(t)=ax_2(t)-cx_3(t)-dx_3~2(t),初始条件:{x_1(t)=φ1(t)≥0,x2(t)=φ_2(t)≥0x_3(t)=φ_3(t)≥0,t∈[-τ,0]。利用微分方程稳定性理论分析了系统的零平衡点和正平衡点的局部稳定性。利用有效的Liapunov函数得到零平衡点和正平衡点的全局稳定性:1)当aαe^(-γτ)<(b+a)c时,系统有唯一平衡点E_0,且它是局部稳定的;当aαe^(-γτ)>(b+a)c时,E_0是不稳定的,此时系统除了E_0外,还存在唯一正平衡点E_*,且它是局部稳定的。2)当αe(-γτ)≤c,则系统的平衡点E_0是全局渐进稳定的,当αe^(-γτ)≥(a+b/a-b)c,a>b,则系统的正平衡点E_*是全局渐进稳定的。所得结论对人工控制种群的发展具有一定的指导意义。

According to characteristics of populations, a class of three-stage structure of populations system with time delay is estab-lished ：{x·1（t）=αx3（t）-γx1（t）-αe^-πτx3（t-τ） x·2（t）=αe^-πx3（t-τ）-bx2（t）αx2（t） ·x3（t）=αx2（t）-cx3（t）-dx3^2（t）, Initial conditions ：{x1（t）=φ1（t）≥0,x2（t）=φ2（t）≥0 x3（t）=φ3（t）≥0,t∈[-τ,0].By using the stability theory of differential equations, analysising the local asymptotic stability of the zero equilibrium and the positive equilibrium. By using the effective Liapunov function, the global asymptotic stability of the zero equilibrium and the positive equilibrium are proved. 1）If aae^-π〈 （b＋ a） c, the system has only one equilibrium E0, local asymptotic stable. If aae^-π〉 （b＋ a） c, E0 is not stable. Except E0, there is only one the positive equilibrium E. , local asymptotic stable. 2）If ae^-π≤c, the equilibrium E0 is global asymptotic stable. If ae｜^-π≥a＋b/a-bc,where a〉b , the positive equilibrium E. is global asymptotic stable. The conclusion is directive significance for the development of artificial control population.