Research On Insect Interspecific Dynamical Systems With Age-structure And Delay

Insect interspecific dynamical model, like single species Richards model andpredator-prey model, plays a key role in modern biomathematics and pest ecologicalregulation. Therefore, many scholars made effective discussions in this field. However, thereare few papers for insect interspecific dynamical models with age-structure and delay untilnow. As we all know, insects are heterothermic animals, so their number (or density) and statedepend on past period’s number and state, which are called delay effect. Furthermore,previous research neglected insect age-structure so that corresponding models did not depictreal world since some insects’ larva consumes prey and adult insects do not eat the prey.Meanwhile, in the previous research there are fewer multiple species mathematical modelswith multiple delays because the single population models with delay are hard to set up fortheir complication and people’s aware limitation. The population dynamical systems withdelay are not only the content to study in mathematics, but also the main content for pestpopulation ecological regulation in agriculture. So this doctorial dissertation’s object is tostudy insect interspecific dynamical models with age-structure and delay effect, give analysisfor systems’ stability, find critical point to see critical property and the rate between naturalenemy and the harmful insect, and provide theoretical good explanations for pest speciesregulation. The main results of this doctoral dissertation include several aspects as follows:1. Firstly built multiple Richards model with delay by functional differentialequations.The research on three competitive species simplified from Richards model showsthat the globally asymptotical stability for three competitive species is unrelated withasymmetrical parameter and shape parameter of Richards model.2. Firstly built a mathematical model with age-structure and delay on Cotton Aphid andSeven-Spot Ladybird Beetle, and analyzed this system boundary and stability with functionaldifferential equations theory. The system’s equilibrium point and the rate between theladybird and cotton aphid y1y2:Nare sought out to give theoretical basis to pest insectecological regulation. With computer simulation, one can find the delay r has no effect oncoexistence between cotton aphid and ladybird, but has great effect on stability of equilibriumpoint. When the dealy r is greater than00, then the system has positive solution. On the other hand, the parameter b, which is maximum per capita daily egg production, is also animportant factor in biological control. The greater the parameter b is, the easier the cottonaphid is controled by its natural enemy. These results show that the development period hasgreat effect on the coexistence, and if the development period is long enough, then the cottonaphid and ladybird are coexistent with periodic form.3. Firstly gave Scaeva pyrastri-aphid dynamical system with age-structure and delay.The dissertation discussed the system’s equilibrium point stability and Hopf bifurcation withdelay differential equations and computer numerical simulation. These results mean that theequilibrium point E10,0,0is always unstable; under conditions (H3)-(H5), a sequence ofHopf bifurcations occur at the equilibrium point E2K,0,0. For positive equilibriumpointE the rate of ladybird and cotton aphid is, and it is locallyasymptotically stable and is hard to prove theoretically its global asymptotically stable. Thedissertation also found the delay has great effect on two populations by numerical simulation.When the delay is greater than some fixed positive constant, the periodic oscillation betweenScaeva pyrastri and aphid occurred, meaning that the development period of Scaeva pyrastrihas great effect on the population dynamical system. So, one knows that the dynamicalbehavior between Scaeva pyrastri and aphid must be periodic according with the real world.But this result needs to be proved in the future with mathematical theory and real populationdata.Above result, in a way, is general in real world.The innovation points of this dissertation are as follows:(1) Firstly built multiple Richards model with delay, and the result shows that the globallyasymptotically stable of three competitive species is unrelated with the asymmetricalparameter and shape parameter;(2) Firstly built a mathematical model with age-structure and delay for Cotton Aphid andSeven-Spot Ladybird Beetle, and the result shows that the development period has effect oncoexistence, and if the period time is long enough, then the cotton aphid and ladybird arecoexistent with periodic form.(3) Firstly gave Scaeva pyrastri-aphid dynamical system with age-structure and delay. Theresult shows that the development period of Scaeva pyrastri has great effect on the populationdynamical system. So, one knows that the dynamical behavior between Scaeva pyrastri andaphid must be periodic stability which is coordinated with the real world.