| Citation: |
Jiarui Zhang, Wei Li, Mi Wang. ANALYSIS OF THE KINETIC PROPERTIES OF THE DISCRETE FOREST DISEASE AND INSECT PEST MODELS[J]. Journal of Applied Analysis & Computation, 2026, 16(1): 136-154. doi: 10.11948/20240542
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ANALYSIS OF THE KINETIC PROPERTIES OF THE DISCRETE FOREST DISEASE AND INSECT PEST MODELS
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Abstract
In this paper, we study the dynamic behavior of discrete forest disease and pest model-spruce aphid model, analyze the properties of the dynamics by using the difference equation theory, including the existence of equilibrium points in the system model, and further analyze the stability and instability conditions of these equilibrium points. In addition, the step h is selected as the bifurcation parameter using the central manifold theorem to analyze the Flip bifurcation and Hopf bifurcation at the equilibrium point, and prove the chaos of the system through the maximum Lyapunov diagram. In order to verify the theoretical proof, the system model is simulated numerically to draw relevant conclusions.
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References
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Jiarui Zhang, Wei Li, Mi Wang. ANALYSIS OF THE KINETIC PROPERTIES OF THE DISCRETE FOREST DISEASE AND INSECT PEST MODELS[J]. Journal of Applied Analysis & Computation, 2026, 16(1): 136-154. doi: 10.11948/20240542
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Figure 1.
The graph of N, S when Flip bifurcation occurs at k = 10000, η = 7.54, β = 14.96, ρ = 0.9, h = 0.262122270347572, Smax = 50, δ = 0.6, r = 10, B(N0, S0) = (1.499951950752, 0.001601641574).
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Figure 2.
The graph of N, S when Hopf bifurcation occurs at k = 50000, η = 1.54, β = 5.77, ρ = 1, h = 0.087593294844698, Smax = 10, δ = 0.9, r = 5, (N0, S0) = (4.319623064185476, 1.360753871629049).
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Figure 3.
The Flip bifurcation is chaotic with the parameters k = 10000, η = 7.54, β = 14.97, ρ = 0.9, h = 0.262122270347572, Smax = 50, δ = 0.6, r = 10, B(N0, S0) = (1.499951950752, 0.001601641574).
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Figure 4.
The parameters in the figure are the maximum Lyapunov exponential map where k = 10000, η = 7.54, β = 14.97, ρ = 0.9, h = 0.262122270347572, Smax = 50, δ = 0.6, r = 10, B(N0, S0) = (1.499951950752, 0.001601641574).
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Figure 5.
The Hopf bifurcation is chaotic with the parameters
$ k = 50000 $ $ \eta = 3.72 $ $ \beta = 8.47 $ $ \rho = 1 $ $ h = 0.149687409244297 $ $ {S_{\max }} = 10 $ $ \delta = 0.9 $ $ r = 5 $ $ B\left( {{N_0},{S_0}} \right) = \left( {4.3149528055,1.3700943889} \right) $ -
Figure 6.
The parameters in the figure are the maximum Lyapunov exponential map where
$ k = 50000 $ $ \eta = 3.72 $ $ \beta = 8.47 $ $ \rho = 1 $ $ h = 0.149687409244297 $ $ {S_{\max }} = 10 $ $ \delta = 0.9 $ $ r = 5 $ $ B\left( {{N_0},{S_0}} \right) = \left( {4.3149528055,1.3700943889} \right) $