## Eco-epidemiology model with age structure and prey-dependent consumption for pest management.(English)Zbl 1179.34049

From the introduction: The prey-dependent consumption predator-prey (natural enemy-pest) model with age structure for the predators and infectious disease in the prey,
\begin{cases} \left.\begin{aligned} & S'(t)=rS(t) \left(1-\frac{S(t)}{K}\right)-\frac{\alpha(t)S(t)}{1+\omega S(t)}-\beta S(t)y_2(t),\\ & I'(t)=\frac{\alpha I(t)S(t)}{1+\omega S(t)}-d_1I(t),\\ & y_1'(t)=\frac{\lambda\beta S(t)y_2(t)}{1+\beta hS(t)}-d_2y_1(t)=my_1(t),\\ &y_2'(t)=my_1(t)-d_2y_2(t),\end{aligned}\right\}\ t\neq nT,\\ \left.\begin{aligned} & \Delta S(t)=0,\quad \Delta I(t)=p,\\ & \Delta y_1(t)=q_1,\quad \Delta y_2(t)=q_2,\end{aligned}\right\}\;t=nT,\quad n=1,2,\dots,\end{cases}
is considered. Infectious pests, immature natural enemies and mature natural enemies are released impulsively. By using Floquet’s theorem, small-amplitude perturbation skills and comparison theorem, we obtain both sufficient conditions for the global asymptotical stability of the susceptible pest-eradication periodic solution and the permanence of the system. The results provide a reliable theoretical tactics for pest management.

### MSC:

 34C60 Qualitative investigation and simulation of ordinary differential equation models 92D30 Epidemiology 92D40 Ecology 34D05 Asymptotic properties of solutions to ordinary differential equations 34A37 Ordinary differential equations with impulses 34C25 Periodic solutions to ordinary differential equations
Full Text:

### References:

 [1] Van Lenteren, J. C., Integrated pest management in protected crops, (Integrated Pest Management (1995), Chapman and Hall: Chapman and Hall London) [2] Zhang, H.; Xu, W. J.; Chen, L. S., A impulsive infective transmission SI model for pest control, Math. Meth. Appl. Sci., 30, 1169-1184 (2007) · Zbl 1155.34328 [3] Barcly, H. J., Models for pest control using predator release, habitat management and pesticide release in combination, J. Appl. Ecol., 19, 337-348 (1982) [4] Tang, S. Y.; Xiao, Y. N.; Chen, L. S.; Cheke, R. A., Integrated pest management models and their dynamical behaviour, Bull. Math. Biol., 67, 115-135 (2005) · Zbl 1334.91058 [5] Debach, P.; Rosen, D., Biological Control by Natural Enemies (1991), Cambridge University Press: Cambridge University Press Cambridge [6] J Cherry, A.; Lomer, C. J.; Djegui, D.; Schulthess, F., Pathogen incidence and their potential as microbial control agents in IPM of maize stemborers in West Africa, Biocontrol, 44, 301-327 (1999) [7] Jiao, J. J.; Chen, L. S., A pest management SI model with periodic biological and chemical control concern, Appl. Math. Comput., 2, 1018-1026 (2006) · Zbl 1104.92054 [8] Yunchang, J. B., Optimal pest management and economic threshold, Agri. Syst., 49, 113-133 (1995) [9] Lakshmikantham, V.; Bainov, D. D.; Simeonov, P. S., Theory of Impulsive Differential Equations (1989), World Science: World Science Singapore · Zbl 0719.34002 [10] Bainov, D. D.; Simeonov, P. S., Impulsive Differential Equations: Periodic Solutions and Applications (1993), Longman Scientific and Technical: Longman Scientific and Technical Burnt Mill · Zbl 0793.34011 [11] Simeonov, P. S.; Bainov, D. D., Stability with respect to part of the variables in system with impulsive effect, J. Math. Anal. Appl., 117, 247-263 (1996) · Zbl 0588.34044 [12] Chen, F. L.; Wen, X. Z., Asymptotic stability for impulsive functional differential equation, J. Math. Anal. Appl., 336, 1149-1160 (2007) · Zbl 1130.34057 [13] Shi, R.; Chen, L., A predator-prey model with disease in the prey and two impulses for integrated pest management, Appl. Math. Modell. (2008)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.