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**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.

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.

Reviewer: Octavian Mustafa (Craiova)

### 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 |

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\textit{B. Liu} et al., Acta Math. Sin., Engl. Ser. 21, No. 3, 541--554 (2005; Zbl 1094.34030)

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