Global dynamics of non-smooth Filippov pest-natural enemy system with constant releasing rate

Hao Zhou; Xia Wang; Sanyi Tang; Hao Zhou; Xia Wang; Sanyi Tang

doi:10.3934/mbe.2019366

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 6: 7327-7361. doi: 10.3934/mbe.2019366

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Global dynamics of non-smooth Filippov pest-natural enemy system with constant releasing rate

School of Mathematics and Information Science, Shaanxi Normal University, Xi’an, P.R. China

Received: 12 April 2019 Accepted: 04 August 2019 Published: 09 August 2019

Modelling integrated pest management (IPM) with a threshold control strategy can be achieved with a non-smooth Filippov dynamical system coupled by an untreated subsystem and a treated subsystem which includes chemical and biological tactics. The releasing constant of natural enemies related to biological control generates the complex dynamics. Comprehensive qualitative analyses reveal that the treated subsystem exists with transcritical, saddle-node, Hopf and Bogdanov-Takens bifurcations, for which the threshold conditions and bifurcation curves are provided. Further, by applying techniques of non-smooth dynamical systems including the Filippov convex method and sliding bifurcation techniques, we first obtain the sliding dynamic equation, and then we analyze the existence and stability of regular/virtual equilibria, pseudo-equilibria, boundary equilibria, sliding segments and sliding bifurcations. In particular, if we choose the economic threshold (ET) as the bifurcation parameter, then interesting dynamical behaviors, including boundary equilibrium $\rightarrow$ pseudo-homoclinic $\rightarrow$ touching $\rightarrow$ buckling $\rightarrow$ crossing bifurcations, occur in succession. It is interesting to note that although the number of pests in the untreated subsystem could increase and exceed the economic injury level (EIL), the final size could be less than ET and stabilizes at a relative low level due to side effects of the pesticide on natural enemies. However, the side effects can be effectively avoided by increasing the releasing constant, which can maintain the number of pests below the EIL always and thus achieve the control purpose.
- Filippov system,
- integrated pest management,
- bifurcation,
- sliding dynamics,
- releasing constant
Citation: Hao Zhou, Xia Wang, Sanyi Tang. Global dynamics of non-smooth Filippov pest-natural enemy system with constant releasing rate[J]. Mathematical Biosciences and Engineering, 2019, 16(6): 7327-7361. doi: 10.3934/mbe.2019366

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Abstract

Modelling integrated pest management (IPM) with a threshold control strategy can be achieved with a non-smooth Filippov dynamical system coupled by an untreated subsystem and a treated subsystem which includes chemical and biological tactics. The releasing constant of natural enemies related to biological control generates the complex dynamics. Comprehensive qualitative analyses reveal that the treated subsystem exists with transcritical, saddle-node, Hopf and Bogdanov-Takens bifurcations, for which the threshold conditions and bifurcation curves are provided. Further, by applying techniques of non-smooth dynamical systems including the Filippov convex method and sliding bifurcation techniques, we first obtain the sliding dynamic equation, and then we analyze the existence and stability of regular/virtual equilibria, pseudo-equilibria, boundary equilibria, sliding segments and sliding bifurcations. In particular, if we choose the economic threshold (ET) as the bifurcation parameter, then interesting dynamical behaviors, including boundary equilibrium $\rightarrow$ pseudo-homoclinic $\rightarrow$ touching $\rightarrow$ buckling $\rightarrow$ crossing bifurcations, occur in succession. It is interesting to note that although the number of pests in the untreated subsystem could increase and exceed the economic injury level (EIL), the final size could be less than ET and stabilizes at a relative low level due to side effects of the pesticide on natural enemies. However, the side effects can be effectively avoided by increasing the releasing constant, which can maintain the number of pests below the EIL always and thus achieve the control purpose.

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