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**A ratio-dependent food chain model and its applications to biological control.**
*(English)*
Zbl 1036.92033

Summary: While biological controls have been successfully and frequently implemented by nature and humans, plausible mathematical models are yet to be found to explain the often observed deterministic extinctions of both pest and control agents in such processes. We study a three trophic level food chain model with ratio-dependent Michaelis-Menten type functional responses. We show that this model is rich in boundary dynamics and is capable of generating such extinction dynamics. A two trophic level Michaelis-Menten type ratio-dependent predator-prey system was globally and systematically analyzed in details recently. A distinct and realistic feature of ratio-dependence is its capability of producing the extinction of prey species, and hence the collapse of the system. Another distinctive feature of this model is that its dynamical outcomes may depend on initial populations levels.

These features, if preserved in a three trophic food chain model, make it appealing for modelling certain biological control processes (where prey is a plant species, middle predator is a pest, and top predator is a biological control agent) where the simultaneous extinction of pest and control agents is the hallmark of their success and which are usually dependent on the amount of control agent.

Our results indicate that this extinction dynamics and sensitivity to initial population levels are not only preserved, but also enriched in the three trophic level food chain model. Specifically, we provide partial answers to questions such as: under what scenarios a potential biological control may be successful, and when it may fail. We also study questions such as what conditions ensure the coexistence of all the three species in the forms of a stable steady state and limit cycle, respectively. A multiple attractor scenario is found.

These features, if preserved in a three trophic food chain model, make it appealing for modelling certain biological control processes (where prey is a plant species, middle predator is a pest, and top predator is a biological control agent) where the simultaneous extinction of pest and control agents is the hallmark of their success and which are usually dependent on the amount of control agent.

Our results indicate that this extinction dynamics and sensitivity to initial population levels are not only preserved, but also enriched in the three trophic level food chain model. Specifically, we provide partial answers to questions such as: under what scenarios a potential biological control may be successful, and when it may fail. We also study questions such as what conditions ensure the coexistence of all the three species in the forms of a stable steady state and limit cycle, respectively. A multiple attractor scenario is found.

### MSC:

92D40 | Ecology |

34D05 | Asymptotic properties of solutions to ordinary differential equations |

34C60 | Qualitative investigation and simulation of ordinary differential equation models |

### Keywords:

Biological control; Predator-prey model; Limit cycle; Chaos; Extinction; Food chain model; Ratio-dependence; Simple food chain
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\textit{S.-B. Hsu} et al., Math. Biosci. 181, No. 1, 55--83 (2003; Zbl 1036.92033)

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