Dynamics in a diffusive predator-prey system with stage structure and strong allee effect

Yuying Liu; Yuxiao Guo; Junjie Wei

doi:10.3934/cpaa.2020040

Article Contents

2020, Volume 19, Issue 2: 883-910. Doi: 10.3934/cpaa.2020040

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Dynamics in a diffusive predator-prey system with stage structure and strong allee effect

1.
School of Mathematics, Harbin Institute of Technology, Harbin, Heilongjiang, 150001, China
2.
Department of Mathematics, Harbin Institute of Technology, Weihai, Shandong, 264209, China
3.
School of Science, Jimei University, Xiamen, Fujian, 361021, China

^* Corresponding author
^* Corresponding author

Received: November 30, 2018

Revised: March 31, 2019

Published: October 2019

The authors are supported by the National Natural Science Foundation of China (No. 11771109)

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Abstract

In this paper, we consider the dynamics of a diffusive predator-prey system with stage structure and strong Allee effect. The upper-lower solution method and the comparison principle are used in proving the nonnegativity of the solutions. Then the stability and the attractivity basin of the boundary equilibria are obtained, by which we investigated the bistable phenomena. The existence and local stability of the positive constant steady-state are investigated, and the existence of Hopf bifurcation is studied by analyzing the distribution of eigenvalues. On the center manifold, we studied the criticality of the Hopf bifurcation by the normal form theory. Some numerical simulations are carried out for illustrating the theoretical results.

Keywords:

Mathematics Subject Classification: Primary: 34K18, 37L10; Secondary: 35B35.

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Figure 1. The graphs of $ s_1(u)+s_2(u) $ and $ H_n(\tau) $ on $ (\underline{\tau}, \bar{\tau}) $

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Figure 2. The graphs of $ S_n^m $ on $ I_n $

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Figure 3. For system (4), the positive steady state $ E_3 $ is locally asymptotically stable, where $ \tau = 20 \in(\tau_{k}, \tau_{max}) $, $ u_0(x, t) = 7.7+0.5\cos2x $, $ v_0(x, t) = 12-0.5\cos2x $

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Figure 4. For system (4), the bifurcating periodic solutions are asymptotically stable, where $ \tau = 10.8426<\tau_{1}\approx10.8427 $ and is close to $ \tau_1 $. The initial values are $ u_0(x, t) = 6.92+0.5\cos(2x) $, $ v_0(x, t) = 13.63-0.5\cos(2x) $

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Figure 5. For system (4), the transient spatially homogeneous periodic solutions occur, where $ \tau = 8.9734>\tau_{0}\approx8.9733 $ and is close to $ \tau_{0} $. The initial values are $ u_0(x, t) = 6.77+0.8\cos2x $, $ v_0(x, t) = 13.73-0.8\cos2x $

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Figure 6. $ E_2(K, 0) $ is locally asymptotically stable, where $ \tau = 43>\tau_{max}\approx42.0034 $, $ u_0(x, t) = 9.5+0.3\cos2x $, $ v_0(x, t) = 8+\cos2x $

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Figure 7. $ E_0(0, 0) $ attracts the solutions which have big initial value $ v_0(x, t) $, where $ \tau = 43>\tau_{max}\approx42.0034 $, $ u_0(x, t) = 9.5+0.3\cos2x $, $ v_0(x, t) = 16+\cos2x $

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