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Impact of non-smooth threshold control on a reaction–diffusion predator–prey model with time delay

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Abstract

To explore the efficacy of integrated pest management, we modify the predator–prey (pest–natural enemy) model by incorporating Holling III functional response and transform it into a non-smooth Filippov control system. Unlike conventional Filippov systems, the model takes into consideration time delay and spatial heterogeneity. Consequently, we establish and examine a delayed reaction–diffusion Filippov prey–predator model. Firstly, the dynamics of the two subsystems are analyzed, which includes the existence and stability of the equilibrium points, along with determining the adequate conditions for local Hopf bifurcation. Subsequently, we implement a detailed investigation of the sliding mode dynamics and stability of the pseudoequilibrium. Theoretical and numerical simulations indicate that on the one hand, the threshold level should be prescribed adequately to reduce the pest population equal to or below the threshold level. On the other hand, reading from the boundary node and boundary focus bifurcations, slightly varying the economic threshold may save a failure control strategy by dragging the number of the pests from a regular equilibrium above the threshold to a boundary equilibrium or a pseudoequilibrium equal to the threshold. Furthermore, the sequent appearance of global sliding bifurcations including touching, sliding switching and crossing bifurcations expound that the incorporation of time delay not only complicates the dynamics of the system, but also brings more challenge for pest control.

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Funding

This work is supported by Shandong Provincial Natural Science Foundation of China (No. ZR2023YQ002).

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  1. School of Mathematics and Statistics, Shandong Normal University, **an, 250014, People’s Republic of China

    Yuzhi Liu & You** Yang

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  1. Yuzhi Liu
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  2. You** Yang
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Correspondence to You** Yang.

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Liu, Y., Yang, Y. Impact of non-smooth threshold control on a reaction–diffusion predator–prey model with time delay. Nonlinear Dyn 112, 14637–14656 (2024). https://doi.org/10.1007/s11071-024-09796-1

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