\`x^2+y_1+z_12^34\`
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Spatiotemporal dynamics of a diffusive predator-prey model with generalist predator

  • * Corresponding author: Jianshe Yu

    * Corresponding author: Jianshe Yu 
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  • In this paper, we study the spatiotemporal dynamics of a diffusive predator-prey model with generalist predator subject to homogeneous Neumann boundary condition. Some basic dynamics including the dissipation, persistence and non-persistence(i.e., one species goes extinct), the local and global stability of non-negative constant steady states of the model are investigated. The conditions of Turing instability due to diffusion at positive constant steady states are presented. A critical value $ \rho $ of the ratio $ \frac{d_2}{d_1} $ of diffusions of predator to prey is obtained, such that if $ \frac{d_2}{d_1}>\rho $, then along with other suitable conditions Turing bifurcation will emerge at a positive steady state, in particular so it is with the large diffusion rate of predator or the small diffusion rate of prey; while if $ \frac{d_2}{d_1}<\rho $, both the reaction-diffusion system and its corresponding ODE system are stable at the positive steady state. In addition, we provide some results on the existence and non-existence of positive non-constant steady states. These existence results indicate that the occurrence of Turing bifurcation, along with other suitable conditions, implies the existence of non-constant positive steady states bifurcating from the constant solution. At last, by numerical simulations, we demonstrate Turing pattern formation on the effect of the varied diffusive ratio $ \frac{d_2}{d_1} $. As $ \frac{d_2}{d_1} $ increases, Turing patterns change from spots pattern, stripes pattern into spots-stripes pattern. It indicates that the pattern formation of the model is rich and complex.

    Mathematics Subject Classification: Primary: 35B36, 45M10; Secondary: 92C15.

    Citation:

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  • Figure 1.  Stationary hot spots pattern in model (2). The parameter values are taken as (32) and $ (d_1, d_2) = (0.028, 0.226) $. The zero-flux boundary condition is used and initial condition is small perturbation around the homogeneous steady-state $ E^* = (1.5934, 2.0768) $.

    Figure 2.  Stationary stripes pattern in model (2). The parameter values are taken as (32) and $ (d_1, d_2) = (0.028, 0.27) $. The zero-flux boundary condition is used and initial condition is small perturbation around the homogeneous steady-state $ E^* = (1.5934, 2.0768) $.

    Figure 3.  Stationary spots-stripes pattern in model (2). The parameter values are taken as (32) and $ (d_1, d_2) = (0.028, 0.45) $. The zero-flux boundary condition is used and initial condition is small perturbation around the homogeneous steady-state $ E^* = (1.5934, 2.0768) $

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