# 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 and Applied Analysis, 2020, 19 (2) : 883-910. doi: 10.3934/cpaa.2020040
##### References:
 [1] W. C. Allee, Animal Aggregations: A Study in General Sociology, University of Chicago Press, 1931. [2] E. Beretta and Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. Anal., 33 (2002), 1144-1165.  doi: 10.1137/S0036141000376086. [3] Y. Du, P. Y. Pang and M. Wang, Qualitative Analysis of a Prey-Predator Model with Stage Structure for the Predator, SIAM J. Appl. Math., 69 (2008), 596-620.  doi: 10.1137/070684173. [4] T. Faria, Normal forms and Hopf bifurcation for partial differential equations with delays, Trans. Amer. Math. Soc., 352 (2000), 2217-2238.  doi: 10.1090/S0002-9947-00-02280-7. [5] H. I. Freedman and J. Wu, Persistence and global asymptotic stability of single species dispersal models with stage structure, Quart. Appl. Math., 49 (1991), 351-371.  doi: 10.1090/qam/1106397. [6] S. A. Gourley and Y. Kuang, A stage structured predator-prey model and its dependence on maturation delay and death rate, J. Math. Biol., 49 (2004), 188-200.  doi: 10.1007/s00285-004-0278-2. [7] J. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977. [8] Y. Jia, Y. Li and J. Wu, Effect of predator cannibalism and prey growth on the dynamic behavior for a predator-stage structured population model with diffusion, J. Math. Anal. Appl., 449 (2017), 1479-1501.  doi: 10.1016/j.jmaa.2016.12.036. [9] J. Jin, J. Shi, J. Wei and F. Yi, Bifurcation of Patterned Solutions in Diffusive Lengyel-Epstein of CIMA Chemical Reaction, Rocky Mountain J. Math., 43 (2013), 1637-1674.  doi: 10.1216/RMJ-2013-43-5-1637. [10] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. [11] C. V. Pao, Convergence of solutions of reaction-diffusion systems with time delays, Nonlinear Anal., 48 (2002), 349-362.  doi: 10.1016/S0362-546X(00)00189-9. [12] Y. Qu and J. Wei, Bifurcation analysis in a time-delay model for prey-predator growth with stage-structure, Nonlinear Dynam., 49 (2007), 285-294.  doi: 10.1007/s11071-006-9133-x. [13] S. Ruan and J. Wei, On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 251 (2003), 863-874. [14] Y. Saito and Y. Takeuchi, A time-delay model for Prey-predator growth with stage structure, Can. Appl. Math. Q., 11 (2003), 293-302. [15] C. M. Taylor and A. Hastings, Allee effects in biological invasions, Ecol. Lett., 8 (2005), 895-908. [16] J. Wang, J. Shi and J. Wei, Dynamics and pattern formation in a diffusive predator-prey system with strong Allee effect in prey, J. Differential Equations, 251 (2011), 1276-1304.  doi: 10.1016/j.jde.2011.03.004. [17] J. Wang, J. Shi and J. Wei, Predator-prey system with strong Allee effect in prey, J. Math. Biol., 3 (2011), 291-331.  doi: 10.1007/s00285-010-0332-1. [18] L. Wang, Y. Pei and G. Feng, Mathematical analysis of an eco-epidemiological predator-prey model with stage-structure and latency, J. Appl. Math. Comput., 57 (2017), 1-18.  doi: 10.1007/s12190-017-1102-7. [19] W. Wang and L. Chen, A predator-prey system with stage-structure for predator, Comput. Math. Appl., 33 (1997), 83-91.  doi: 10.1016/S0898-1221(97)00056-4. [20] X. Wang and Z. Li, Dynamics for a type of general reaction-diffusion model, Nonlinear Anal., 67 (2007), 2699-2711.  doi: 10.1016/j.na.2006.09.034. [21] J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4050-1. [22] Y. Xiao and L. Chen, Global Stability of a predator-prey System with Stage Structure for the Predator, Acta Math. Sin. (Engl. Ser.), 20 (2004), 63-70.  doi: 10.1007/s10114-002-0234-2. [23] X. Xu and J. Wei, Bifurcation analysis of a spruce budworm model with diffusion and physiological structures, J. Differential Equations, 262 (2017), 5206-5230.  doi: 10.1016/j.jde.2017.01.023. [24] S. Yan and S. Guo, Dynamics of a Lotka-Volterra competition-diffusion model with stage structure and spatial heterogeneity, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1559-1579.  doi: 10.3934/dcdsb.2018059. [25] R. Yang, M. Liu and C. Zhang., A diffusive toxin producing phytoplankton model with maturation delay and three-dimensional patch, Comput. Math. Appl., 73 (2017), 824-837.  doi: 10.1016/j.camwa.2017.01.006. [26] F. Yi, J. Wei and J. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system, J. Differential Equations, 246 (2009), 1944-1977.  doi: 10.1016/j.jde.2008.10.024. [27] G. Zhang, W. Li and G. Lin, Traveling waves in delayed predator-prey systems with nonlocal diffusion and stage structure, Math. Comput. Modelling, 49 (2009), 1021-1029.  doi: 10.1016/j.mcm.2008.09.007.

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##### References:
 [1] W. C. Allee, Animal Aggregations: A Study in General Sociology, University of Chicago Press, 1931. [2] E. Beretta and Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. Anal., 33 (2002), 1144-1165.  doi: 10.1137/S0036141000376086. [3] Y. Du, P. Y. Pang and M. Wang, Qualitative Analysis of a Prey-Predator Model with Stage Structure for the Predator, SIAM J. Appl. Math., 69 (2008), 596-620.  doi: 10.1137/070684173. [4] T. Faria, Normal forms and Hopf bifurcation for partial differential equations with delays, Trans. Amer. Math. Soc., 352 (2000), 2217-2238.  doi: 10.1090/S0002-9947-00-02280-7. [5] H. I. Freedman and J. Wu, Persistence and global asymptotic stability of single species dispersal models with stage structure, Quart. Appl. Math., 49 (1991), 351-371.  doi: 10.1090/qam/1106397. [6] S. A. Gourley and Y. Kuang, A stage structured predator-prey model and its dependence on maturation delay and death rate, J. Math. Biol., 49 (2004), 188-200.  doi: 10.1007/s00285-004-0278-2. [7] J. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977. [8] Y. Jia, Y. Li and J. Wu, Effect of predator cannibalism and prey growth on the dynamic behavior for a predator-stage structured population model with diffusion, J. Math. Anal. Appl., 449 (2017), 1479-1501.  doi: 10.1016/j.jmaa.2016.12.036. [9] J. Jin, J. Shi, J. Wei and F. Yi, Bifurcation of Patterned Solutions in Diffusive Lengyel-Epstein of CIMA Chemical Reaction, Rocky Mountain J. Math., 43 (2013), 1637-1674.  doi: 10.1216/RMJ-2013-43-5-1637. [10] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. [11] C. V. Pao, Convergence of solutions of reaction-diffusion systems with time delays, Nonlinear Anal., 48 (2002), 349-362.  doi: 10.1016/S0362-546X(00)00189-9. [12] Y. Qu and J. Wei, Bifurcation analysis in a time-delay model for prey-predator growth with stage-structure, Nonlinear Dynam., 49 (2007), 285-294.  doi: 10.1007/s11071-006-9133-x. [13] S. Ruan and J. Wei, On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 251 (2003), 863-874. [14] Y. Saito and Y. Takeuchi, A time-delay model for Prey-predator growth with stage structure, Can. Appl. Math. Q., 11 (2003), 293-302. [15] C. M. Taylor and A. Hastings, Allee effects in biological invasions, Ecol. Lett., 8 (2005), 895-908. [16] J. Wang, J. Shi and J. Wei, Dynamics and pattern formation in a diffusive predator-prey system with strong Allee effect in prey, J. Differential Equations, 251 (2011), 1276-1304.  doi: 10.1016/j.jde.2011.03.004. [17] J. Wang, J. Shi and J. Wei, Predator-prey system with strong Allee effect in prey, J. Math. Biol., 3 (2011), 291-331.  doi: 10.1007/s00285-010-0332-1. [18] L. Wang, Y. Pei and G. Feng, Mathematical analysis of an eco-epidemiological predator-prey model with stage-structure and latency, J. Appl. Math. Comput., 57 (2017), 1-18.  doi: 10.1007/s12190-017-1102-7. [19] W. Wang and L. Chen, A predator-prey system with stage-structure for predator, Comput. Math. Appl., 33 (1997), 83-91.  doi: 10.1016/S0898-1221(97)00056-4. [20] X. Wang and Z. Li, Dynamics for a type of general reaction-diffusion model, Nonlinear Anal., 67 (2007), 2699-2711.  doi: 10.1016/j.na.2006.09.034. [21] J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4050-1. [22] Y. Xiao and L. Chen, Global Stability of a predator-prey System with Stage Structure for the Predator, Acta Math. Sin. (Engl. Ser.), 20 (2004), 63-70.  doi: 10.1007/s10114-002-0234-2. [23] X. Xu and J. Wei, Bifurcation analysis of a spruce budworm model with diffusion and physiological structures, J. Differential Equations, 262 (2017), 5206-5230.  doi: 10.1016/j.jde.2017.01.023. [24] S. Yan and S. Guo, Dynamics of a Lotka-Volterra competition-diffusion model with stage structure and spatial heterogeneity, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 1559-1579.  doi: 10.3934/dcdsb.2018059. [25] R. Yang, M. Liu and C. Zhang., A diffusive toxin producing phytoplankton model with maturation delay and three-dimensional patch, Comput. Math. Appl., 73 (2017), 824-837.  doi: 10.1016/j.camwa.2017.01.006. [26] F. Yi, J. Wei and J. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system, J. Differential Equations, 246 (2009), 1944-1977.  doi: 10.1016/j.jde.2008.10.024. [27] G. Zhang, W. Li and G. Lin, Traveling waves in delayed predator-prey systems with nonlocal diffusion and stage structure, Math. Comput. Modelling, 49 (2009), 1021-1029.  doi: 10.1016/j.mcm.2008.09.007.
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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