Stochastic modeling of plant-insect interaction dynamics with MEMS-based monitoring and noise effects

The dynamics of plant-insect interactions play a crucial role in the ecosystem, influenced by complex molecular signaling pathways. This study extends existing deterministic models of plant-insect systems by incorporating stochastic elements and molecular interactions, particularly focusing on the roles of Botrytis Induced Kinase-1 (BIK1) and Phyto Alexin Deficient-4 (PAD4) proteins. The model evaluates the effects of constant inhibition, pulsed inhibition, and adaptive feedback control on plant biomass <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m1"><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>y</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math> , insect herbivore density <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m2"><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>y</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math> , PAD4 levels <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m3"><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>y</mml:mi></mml:mrow><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math> , and BIK1 levels <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m4"><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:msub><mml:mrow><mml:mi>y</mml:mi></mml:mrow><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math> . Additionally, we examine the impact of different noise types, including deterministic, Gaussian, and Lévy noise, on system variability and stability. Results indicate that our stochastic model is superior as it shows a significant reduction in BIK1 levels, particularly under higher noise intensities, which enhances PAD4 activity and improves plant defense mechanisms. Moreover, moderate noise intensity <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="m5"><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>σ</mml:mi><mml:mo>=</mml:mo><mml:mn>0.05</mml:mn></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math> provides an optimal balance, sustaining PAD4 levels while effectively controlling insect herbivore populations. We also integrate MEMS-based feedback mechanisms, which dynamically adjust plant biomass and molecular signaling, further stabilizing the system’s response to environmental variability.