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Dynamics of plant disease models with continuous and pulse farming awareness control

  • *Corresponding author: Sanyi Tang

    *Corresponding author: Sanyi Tang
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  • How to develop the mathematical model of plant disease transmission and consider the effect of saturated farming awareness strategies on plant disease control is of great practical significance. To do this, the plant disease transmission models with continuous and pulse farming awareness control strategy have been proposed in the present paper. For the model with continuous control strategies, the threshold values for the existence and stability of multiple equilibria have been given, and the effect of farming awareness on disease persistence is discussed; For the model with nonlinear impulsive control tactics, the existence and stability of the plant disease-free periodic solutions are investigated and the threshold condition is given, further we get the conditions for the permanence of the system and obtain the sufficient conditions under which the positive periodic solution exists by bifurcation theory. Numerical simulations reveal that people's farming awareness, especially global awareness, plays an important role in controlling and eradicating the plant disease. Moreover, the nonlinear impulsive control could result in the complex dynamic behavior, including period doubling, chaos and multiple attractors, which makes it difficult to design a successful plant disease control strategy.

    Mathematics Subject Classification: Primary: 34A37, 34D20, 34H05; Secondary: 92C80, 92D25.

    Citation:

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  • Figure 1.  The effects of aware population recruitment rate $ \alpha $ on the solutions of system (1). The parameter values are fixed as follows: $ r = 0.03, \eta = 0.015, \theta = 0.001, \delta_{1} = 0.05, \delta_{2} = 0.03, \beta = 0.0003, g = 0.003, K = 50, $ and initial values $ (S_{0}, I_{0}, A_{0}) = (48, 2, 0) $

    Figure 2.  The effects of $ T $ on the threshold values $ R_{i},(i = 1,2.) $; Blue represents $ R_{1} $ and black represents $ R_{2} $. The other parameter values are $ r = 0.03, \alpha = 0.5, \eta = 0.015, \theta = 0.03, \delta_{1} = 0.03, \delta_{2} = 0.03, \beta = 0.005, \omega = 0.2, p = 0.5, g = 0.003, K = 50 $

    Figure 3.  Bifurcation diagrams of model (8) with respect to bifurcation parameter $ T $. (A) The effects of $ T $ on the dynamical behavior of infected plants with constant initial values; (B) The effects of $ T $ on the dynamical behavior of infected plants with random initial values. The other parameter values are fixed as follows $ r = 0.06, \alpha = 0.5, \eta = 0.015, \theta = 0.03, \delta_{1} = 0.03, \delta_{2} = 0.03, \beta = 0.005, \omega = 0.2, p = 0.5, g = 0.003, K = 50 $

    Figure 4.  Two coexisting attractors of model (8) with $ T = 54 $. The other parameter values are $ r = 0.06, \alpha = 0.5, \eta = 0.015, \theta = 0.03, \delta_{1} = 0.03, \delta_{2} = 0.03, \beta = 0.005, \omega = 0.2, p = 0.5, g = 0.003, K = 50. $ The initial conditions are (left) $ (S_{0},I_{0},A_{0}) = (10,10,0), $ (right)$ (S_{0},I_{0},A_{0}) = (31,10,0) $

    Figure 5.  The basin of attraction of model (8) corresponding to Fig. 4. The range of initial values is $ 0.01\leq S_{0}\leq 30 $ on horizontal axis, $ 0.01\leq I_{0}\leq 30 $ on vertical axis. The red and green points are attracted to the attractors shown in Fig. 4 from left to right, respectively

    Figure 6.  The effects of the parameter sets on the threshold level $ R_{0}^{A_{0}}. $ The parameter values are as follows: $ \beta = 0.005, $ $ g = 0.003, $ $ \eta = 0.015, $ $ K = 50, $ $ r = 0.03, $ $ \delta_{1} = 0.05, $ $ \delta_{2} = 0.03. $ (A)$ \theta = 0.05; $ (B)$ A_{0} = 10 $

    Figure 7.  The effects of global awareness level $ A_{0} $ and infected plants harvesting rate $ \theta $ on the threshold level $ R_{0}^{A_{0}}. $ The parameter values are as follows: $ \beta = 0.005, $ $ g = 0.003, $ $ \eta = 0.015, $ $ K = 50, $ $ r = 0.03, $ $ \delta_{1} = 0.05, $ $ \delta_{2} = 0.03 $

    Figure 8.  The effects of the parameter sets on the threshold level $ R_{1}^{A_{0}}. $ The parameter values are as follows: $ \beta = 0.005, $ $ g = 0.003, $ $ \eta = 0.015, $ $ K = 50, $ $ r = 0.03, $ $ \delta_{1} = 0.05, $ $ \delta_{2} = 0.03. $ (A)$ T = 7, $ $ \omega = 0.2, $ $ p = 0.5, $ $ \theta = 0.05; $ (B)$ \omega = 0.2, $ $ \theta = 0.05, $ $ p = 0.5, $ $ A_{0} = 10; $ (C)$ T = 7, $ $ \theta = 0.05, $ $ p = 0.5, $ $ A_{0} = 10; $(D)$ \omega = 0.2, $ $ T = 7, $ $ p = 0.5, $ $ A_{0} = 10 $

    Figure 9.  The effects of pulse periods and parameter sets on the threshold level $ R_{1}^{A_{0}}. $ The parameter values are fixed as follows: $ \beta = 0.005, $ $ g = 0.003, $ $ \eta = 0.015, $ $ K = 50, $ $ r = 0.03, $ $ \delta_{1} = 0.05, $ $ \delta_{2} = 0.03. $ (A) $ \omega = 0.2, $ $ \theta = 0.05, $ $ A_{0} = 10; $ (B)$ \omega = 0.2, $ $ \theta = 0.05, $ $ p = 0.5; $ (C) $ p = 0.5, $ $ \theta = 0.05, $ $ A_{0} = 10; $ (D)$ \omega = 0.2, $ $ p = 0.5, $ $ A_{0} = 10 $

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