Study On Reaction-diffusion Models For Insects Control And Their Dynamics

Sterile insect release method (SIRM) is one of the biological control methods forinsects. The idea of this method is to release the insects of the same type that are s-terile and let them compete with fertile individuals for mating, with the aim of dilutingthe productive capacity of the pests, consequently, controlling the pests population, eveneradicating the pests. SIRM has been well respected in the area of pests control due to itsnon-polluting for nature and its efectiveness in certain situations. SIRM has been proveda very useful technique for pest control and applied widely in areas incloding the sup-pression of infectious disease and crop protection. To guide the implementation of SIRMand evaluate the efectiveness of the SIRM, biologists and mathematicians built variousmodels describing SIRM to research.In this paper, we utilize monotone method for parabolic equations and elliptic equa-tions, theory of monotone dynamical systems, Hopf bifurcation theorem, abstract bifurca-tion theorem and Lyapunov function to study three SIRM models and one Holling-Tannerpredator-prey model. We discuss the possibility of success and failure for controlling thepests via SIRM, derive the sufcient conditions for success of SIRM, and estimate thecost of SIRM when it succeeds. After considering the efect of predation on SIRM, westudy a Holling-Tanner predator-prey model, researching the Hopf bifurcation and Turinginstability of which. The details are as follows:(1) For the SIRM with general assumptions, we study a reaction-difusion modelof which on a bounded space domain, derive the sufcient and necessary conditions forsuccess or failure of SIRM. Using the releasing rate of sterile insects as the discussion pa-rameter, we obtain the conditions for existence and stability of coexistence steady states,and prove that the system will undergo saddle-node bifurcation applying abstract bifur-cation theorem. By comparison theorem for parabolic equations and Poincare′-Bendixsontheorem, we prove that when the releasing rate is larger than a certain critical value, thefertile-free steady state becomes the unique steady state which is globally asymptoticallystable. Thus we provide not only the sufcient and necessary condition for success ofSIRM, but also the cost estimate.(2) We consider a model for SIRM with release only on the boundary of the habitat,for a kind of reaction-difusion equations with positive Neumann boundary condition, prove the existence of nonconstant positive steady states, and show that when nonconstantpositive steady states do not exist, the boundary steady state is the unique steady state forthe system and it is globally asymptotically stable. The form of the model we consider isa reaction-difusion equations with positive Neumann boundary condition, whose steadystates are all nonconstant. Constructing a pair of upper and lower solutions, we obtainthe existence of nonconstant positive steady states. Applying the theory of monotonedynamical systems, we prove that when positive steady states do not exist, the uniquefertile-free steady state is globally asymptotically stable. Thus, in theory, we prove thatSIRM with release only on the boundary are also able to control the pests successfullyand give the cost estimate.(3) For the SIRM model under predation and the Holling-Tanner model in form ofordinary diferential equations, we obtain the attractability of a equilibrium for all the pos-itive solutions. The systems of the two models mentioned above possess no monotonici-ty, therefore the monotone method can not be applied. Hence by constructing Lyapunovfunction, we prove that the equilibrium of the relative systems attracts all the positivesolutions, solving the dynamical problems of the relative models.(4) For the Holling-Tanner model and the SIRM model under predation, we performa detailed analysis of stability, Hopf bifurcation and Turing instability, derive conditionsdetermining the existence of Hopf bifurcation, the direction of the bifurcation, the sta-bility of the bifurcating periodic orbit and the occurring of Truing instability. For theordinary diferential equation form of Holling-Tanner model, we show the coexistence ofa stable equilibrium, an unstable periodic orbit and a stable periodic orbit. For the partialdiferential equation form, we obtain the sufcient and necessary conditions for occurringof Truing instability of constant steady states and spatial homogenous periodic orbits, andshow that the solutions starting nearby a Truing unstable periodic orbit are attracted by anonconstant steady state.