## The dynamics of a prey-dependent consumption model concerning integrated pest management.(English)Zbl 1094.34030

A mathematical model for controlling pest populations is investigated by means of impulsive equations. The model takes into account multi-tactical approach called integrated pest management. The differential system is: \begin{aligned} \frac{dx_1(t)}{dt}&=rx_1(t)\left( 1-\frac{x_1(t)}K\right) -ax_1(t)x_2(t),\\ \frac{dx_2(t)}{dt}&=\frac{\lambda ax_1(t)x_2(t)}{1+ahx_1(t)}-dx_2(t),\end{aligned} for $$t\neq nT,\Delta x_1(t)=-\delta x_1(t),\Delta x_2(t)=\mu$$ for $$t=nT$$, where $$\Delta x(t)=x(t+)-x(t)$$.
The assumptions are: (i) the prey (insect pest) in the absence of any predation grows logistically with carrying capacity $$K>0$$, and with constant intrinsic birth rate $$r>0$$; (ii) the effect of the predation is to reduce the growth rate of the prey by a term proportional ($$a>0$$) with the prey and predator populations; (iii) in the absence of any prey for sustenance the predator’s death rate has exponential decay ($$d>0$$); (iv) the prey contribution to the predator growth rate is $$\frac{\lambda ax_1x_2}{1+ahx_1}$$, where $$a$$ is the searching rate, $$h$$ is the handling time and $$\lambda >0$$ is the rate at which ingested prey in excess of what is needed for maintenance is translated into a predator population increase.
The fundamental idea involved in the construction of this model is that predators may consume an increasingly smaller proportion of killed prey as prey density increases (p 542). It is assumed also that $$\lambda >dh$$ (to have a unique positive equilibrium with global asymptotic stability for the impulseless system of equations). Exterior effects are also included in the model, such as an impulsive reduction of the pest population density after its partial destruction by catching or by poisoning with agricultural chemicals ($$\delta \in [0,1)$$) and an impulsive increase of the predator population density by artificially breeding/releasing the species ($$\mu >0$$).
It is shown that all nonnegative solutions are ultimately bounded (Lemma 4.2) and that the system has a periodic solution $$(0,x_2^{*}(t))$$ (the so-called pest radication solution), where $$x_2^{*}(t)=\mu$$exp$$(-d(t-nT))/(1-$$exp$$(-dT))$$, $$t\in (nT,(n+1)T]$$ and $$n$$ is a positive integer, that has global asymptotic stability (Theorem 4.1) provided that $$T<T_m=\frac 1r$$log$$\frac 1{1-\delta }+\frac{a\mu }{rd}$$. This is established by a linearisation of the system and Floquet theory (the local stability) followed by an iteration technique (formula (4.4)). The system is permanent if $$T>T_m$$ (Theorem 5.1).
A thorough discussion about bifurcation from the periodic solution and the occurence of chaos is done in Sections 6-8 of the paper, revealing the plethora of behaviors produced by the model (attractor crisis, supertransients). The only (minor) remark to this well-written paper is that Lemma 4.2 at page 545 is concerned only with nonnegative solutions of the system as easily derived from its proof.

### MSC:

 34C60 Qualitative investigation and simulation of ordinary differential equation models 34A37 Ordinary differential equations with impulses 92D25 Population dynamics (general) 34D05 Asymptotic properties of solutions to ordinary differential equations 34C25 Periodic solutions to ordinary differential equations 34C23 Bifurcation theory for ordinary differential equations 34C28 Complex behavior and chaotic systems of ordinary differential equations
Full Text:

### References:

 [1] Van Lenteren, J. C.: Integrated pest management in protected crops, in: D. Dent (Ed.), Integrated pest management, Chapman & Hall, London, 311–320, 1995 [2] University of California, Division of Agriculture and Natural Resources. Integrated Pest Management for Alfafa hay, Publication 3312, Publications, Division of Agriculture and Nature Resources, University of Califania, 6701 San Pablo Avenue, Oakland CA 94608–1239, 1981 [3] Parker, F. D.: Management of pest populations by manipulating densities of both host and parasites through periodic releases, In: Huffaker, C. B., Ed. Biological control, New York: Plenum Press, 1971 [4] Barclay, H. J.: Models for pest control using predator release, habitat management and pesticide release in combination. J. Applied Ecology, 19, 337–348 (1982) · doi:10.2307/2403471 [5] Freedman, H. J.: Graphical stability, enrichment, and pest control by a natural enemy. Math. Biosci, 31, 207–225 (1976) · Zbl 0373.92023 · doi:10.1016/0025-5564(76)90080-8 [6] Van Lenteren, J. C.: Measures of success in biological control of anthropoids by augmentation of natural enemies, in: S. Wratten, G. Gurr (Eds.), Measures of success in biological control, Kluwer Academic Publishers, Dordrecht, 77–89, 2000 [7] Grasman, J., Van Herwaarden, O. A., Hemerik, L., Van Lenteren, J. C.: A two–component model of hostparasitoid interactions: determination of the size of inundative releases of parasitoids in biological pest control. Math. Biosci., 169, 207–216 (2001) · Zbl 0966.92026 · doi:10.1016/S0025-5564(00)00051-1 [8] Bainov, D., Simeonov, P.: Impulsive differential equations: periodic solutions and applications, Ptiman Monographs and Surveys in Pure and Applied Mathematics 66, 1993 · Zbl 0815.34001 [9] Lakshmikantham, V., Bainov, D., Simeonov, P.: Theory of impulsive differential equations. World Scientific, Singapore, 1989 · Zbl 0719.34002 [10] Lotka, A. J.: Undated oscillations derived from the law of mass action. J. Amer. Chem. Soc., 42, 1595–1599 (1920) · doi:10.1021/ja01453a010 [11] Volterra, V.: Variations and fluctuations of a number of individuals in animal species living together, Translation In: R. N. Chapman: Animal Ecology, New York: McGraw Hill, 409–448, 1931 [12] Lakmeche, A., Arino, O.: Bifurcation of non trivial periodic solutions of impulsive differential equations arising chemotherapeutic treatment. Dynamics of Continuous, Discrete and Impulsive System, 7, 265–287 (2000) · Zbl 1011.34031 [13] Grebogi, C., Ott, E., Yorke, J. A.: Crises, sudden changes in chaotic attractors and chaotic transient. Physica D, 7, 181–200 (1983) · Zbl 0561.58029 · doi:10.1016/0167-2789(83)90126-4 [14] Riidiger, S.: Practical bifurcation and stability analysis from equilibrium to chaos, Springer–Verlag, New York, 1994 [15] Hasting, A., Figgins, K.: Persistence of transitions in spatially structured ecological models. Science, 263, 1133–1137 (1994) · doi:10.1126/science.263.5150.1133
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.