# American Institute of Mathematical Sciences

February  2020, 19(2): 883-910. doi: 10.3934/cpaa.2020040

## 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

Received  December 2018 Revised  April 2019 Published  October 2019

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

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.

Citation: Yuying Liu, Yuxiao Guo, Junjie Wei. Dynamics in a diffusive predator-prey system with stage structure and strong allee effect. Communications on Pure & Applied Analysis, 2020, 19 (2) : 883-910. doi: 10.3934/cpaa.2020040
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##### References:
The graphs of $s_1(u)+s_2(u)$ and $H_n(\tau)$ on $(\underline{\tau}, \bar{\tau})$
The graphs of $S_n^m$ on $I_n$
For system (4), the positive steady state $E_3$ is locally asymptotically stable, where $\tau = 20 \in(\tau_{k}, \tau_{max})$, $u_0(x, t) = 7.7+0.5\cos2x$, $v_0(x, t) = 12-0.5\cos2x$
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 $u_0(x, t) = 6.92+0.5\cos(2x)$, $v_0(x, t) = 13.63-0.5\cos(2x)$
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 $u_0(x, t) = 6.77+0.8\cos2x$, $v_0(x, t) = 13.73-0.8\cos2x$
$E_2(K, 0)$ is locally asymptotically stable, where $\tau = 43>\tau_{max}\approx42.0034$, $u_0(x, t) = 9.5+0.3\cos2x$, $v_0(x, t) = 8+\cos2x$
$E_0(0, 0)$ 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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