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
References:
[1]

W. C. Allee, Animal Aggregations: A Study in General Sociology, University of Chicago Press, 1931. Google Scholar

[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.  Google Scholar

[3]

Y. DuP. 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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[7]

J. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977.  Google Scholar

[8]

Y. JiaY. 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.  Google Scholar

[9]

J. JinJ. ShiJ. 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.  Google Scholar

[10] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[15]

C. M. Taylor and A. Hastings, Allee effects in biological invasions, Ecol. Lett., 8 (2005), 895-908.   Google Scholar

[16]

J. WangJ. 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.  Google Scholar

[17]

J. WangJ. 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.  Google Scholar

[18]

L. WangY. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4050-1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

R. YangM. 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.  Google Scholar

[26]

F. YiJ. 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.  Google Scholar

[27]

G. ZhangW. 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.  Google Scholar

show all references

References:
[1]

W. C. Allee, Animal Aggregations: A Study in General Sociology, University of Chicago Press, 1931. Google Scholar

[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.  Google Scholar

[3]

Y. DuP. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

J. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977.  Google Scholar

[8]

Y. JiaY. 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.  Google Scholar

[9]

J. JinJ. ShiJ. 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.  Google Scholar

[10] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[15]

C. M. Taylor and A. Hastings, Allee effects in biological invasions, Ecol. Lett., 8 (2005), 895-908.   Google Scholar

[16]

J. WangJ. 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.  Google Scholar

[17]

J. WangJ. 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.  Google Scholar

[18]

L. WangY. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4612-4050-1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

R. YangM. 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.  Google Scholar

[26]

F. YiJ. 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.  Google Scholar

[27]

G. ZhangW. 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.  Google Scholar

Figure 1.  The graphs of $ s_1(u)+s_2(u) $ and $ H_n(\tau) $ on $ (\underline{\tau}, \bar{\tau}) $
Figure 2.  The graphs of $ S_n^m $ on $ I_n $
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 $
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) $
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 $
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 $
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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