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A pest management SI model with periodic biological and chemical control concern. (English) Zbl 1104.92054

Summary: We consider an SI model for pest management, with concerns about impulsive releases of infective pests and pesticides sprays. We prove that all solutions of

S ' (t)=rS(t)1-S ( t ) + θ I ( t )/K-βS(t)I 2 (t),I ' (t)=βS(t)I 2 (t)-wI(t),tnτ,ΔS(t)=-μ 1 S(t),ΔI(t)=-μ 2 I(t)+μ,t=nτ,n=1,2,,(1)

are uniformly ultimately bounded and there exist globally asymptotically stable periodic solutions of pest-extinction when

ln1 1-μ 1 >rτ-rμθ1 - exp ( - w t ) Kw1 - (1-μ 2 ) exp (-wτ)-βμ 2 1 - exp ( - 2 w τ ) 2w1 - (1-μ 2 ) exp (-wτ) 2

is satisfied, and a condition for permanence of system (1) is also obtained. It is concluded that the approach of combining impulsive infective releasing with impulsive pesticide spraying provides a reliable tactic basis for practical pest management.

MSC:
92D30Epidemiology
34A37Differential equations with impulses
93C15Control systems governed by ODE
93C95Applications of control theory
34C25Periodic solutions of ODE