Dynamics of tritrophic interaction with volatile compounds in plants and nonlocal diffusion

In this paper, we study a tritrophic model with nonlocal diffusion. This model describes the interaction between crop, pest, and the pest natural enemy, in which the release of volatile organic compounds (VOCs) by the crop is explicitly taken into account. Furthermore, we consider that the movement of the pest and its natural enemy is described by nonlocal diffusion. We prove that a solution exists, is unique, bounded, positive, and there exists an aphid-free point. We calculate a threshold quantity $ \mathcal{R} $, and it is shown that the principal eigenvalue problem of the nonlocal diffusion operator has a principal eigenvalue associated with a positive eigenfunction. For $ \mathcal{R}<1 $, we prove that the aphid-free point is locally asymptotically stable, and when $ \mathcal{R}>1 $, the aphid-free point is unstable. We also prove the uniform persistence of the system. Three equilibrium points were obtained, and we prove that the model has one ecosystem collapse equilibrium point $ E_0 $ and is unstable. We use the Lyapunov approach to prove the global stability of the aphid-free point under certain assumptions. Illustrative numerical examples are used to verify the theoretical outcomes.