doi: 10.3934/dcdsb.2022025
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Stability and Hopf bifurcation in a prey-predator model with memory-based diffusion

School of Mathematics and Statistics, Central South University, Changsha, Hunan, 410083, China

*Corresponding author: Binxiang Dai

Received  April 2021 Revised  December 2021 Early access February 2022

In this paper, we consider a predator-prey model with memory-based diffusion. We first analyze the stability of all steady states in detail. Then by analyzing the distribution of eigenvalues, we find that the average memory period can cause the stability change of the positive steady state, and Hopf bifurcation occurs at the positive steady state. Moreover, from the central manifold theorem and the normal form theory, we give the direction and stability of Hopf bifurcation. The results show that, under certain conditions, a family of spatially inhomogeneous periodic solutions will bifurcate from the positive steady state when the average memory period appear.

Citation: Shu Li, Zhenzhen Li, Binxiang Dai. Stability and Hopf bifurcation in a prey-predator model with memory-based diffusion. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022025
References:
[1]

F. B. M. Belgacem and C. Cosner, The effect of dispersal along environmental gradients on the dynamics of populations in heterogeneous environment, Canad. Appl. Math. Quart., 3 (1995), 379-397. 

[2]

S. Busenberg and W. Huang, Stability and Hopf bifurcation for a population delay model with diffusion effects, J. Differential Equations, 124 (1996), 80-107.  doi: 10.1006/jdeq.1996.0003.

[3]

R. S. Cantrell and C. Cosner, On the dynamics of predator-prey models with the Beddington-DeAngelis functional response, J. Math. Anal. Appl., 257 (2001), 206-222.  doi: 10.1006/jmaa.2000.7343.

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S. ChenY. Lou and J. Wei, Hopf bifurcation in a delayed reaction-diffusion-advection population model, J. Differential Equations, 264 (2018), 5333-5359.  doi: 10.1016/j.jde.2018.01.008.

[5]

S. S. Chen and J. P. Shi, Stability and Hopf bifurcation in a diffusive logistic population model with nonlocal delay effect, J. Differential Equations, 253 (2012), 3440-3470.  doi: 10.1016/j.jde.2012.08.031.

[6]

F. Dai and B. Liu, Global solution for a general cross-diffusion two-competitive-predator and one-prey system with predator-taxis, Commun. Nonlinear Sci. Numer. Simul., 89 (2020), 105336, 22 pp. doi: 10.1016/j.cnsns.2020.105336.

[7]

D. DuanB. Niu and J. Wei, Hopf-Hopf bifurcation and chaotic attractors in a delayed diffusive predator-prey model with fear effect, Chaos Solitons Fractals, 123 (2019), 206-216.  doi: 10.1016/j.chaos.2019.04.012.

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W. F. FaganM. A. Lewisand M. Auger-Méthé and et al., Spatial memory and animal movement, Ecology Letters, 10 (2013), 1316-1329. 

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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.

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T. Faria, Stability and bifurcation for a delayed predator-prey model and the effect of diffusion, J. Math. Anal. Appl., 254 (2001), 433-463.  doi: 10.1006/jmaa.2000.7182.

[11] B. D. HassardN. D. Kazarinoff and Y. H. Wan, Theory and Applications of Hopf Bifurcation, London Mathematical Society Lecture Note Series, 41. Cambridge University Press, Cambridge-New York, 1981. 
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D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin-New York, 1981.

[13]

C. S. Holling, The functional response of predators to prey density and its role in mimicry and population regulation, Memoirs of the Entomological Society of Canada, 97 (1965), 1-60. 

[14]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.

[15]

F. Li and H. Li, Hopf bifurcation of a predator-prey model with time delay and stage structure for the prey, Math. Comput. Modelling, 55 (2012), 672-679.  doi: 10.1016/j.mcm.2011.08.041.

[16]

Z. Li and B. Dai, Stability and Hopf bifurcation analysis in a Lotka-Volterra competition-diffusion-advection model with time delay effect, Nonlinearity, 34 (2021), 3271-3313.  doi: 10.1088/1361-6544/abe77a.

[17]

X. Liu, et al., Turing-Hopf bifurcations in a predator-prey model with herd behavior, quadratic mortality and prey-taxis, Phys. A, 496 (2018), 446–460. doi: 10.1016/j.physa.2018.01.006.

[18]

A. J. Lotka, The elements of physical biology, HathiTrust, 1925.

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[20]

M. Peng and Z. Zhang, Hopf bifurcation analysis in a predator-prey model with two time delays and stage structure for the prey, Adv. Difference Equ., 2018 (2018), Paper No. 251, 20 pp. doi: 10.1186/s13662-018-1705-9.

[21]

J. Poggiale and P. Auger, Fast oscillating migrations in a predator-prey model, Math. Models Methods Appl. Sci., 6 (1996), 217-226.  doi: 10.1142/S0218202596000559.

[22]

S. Ruan and D. Xiao, Global analysis in a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math., 61 (2000/01), 1445-1472.  doi: 10.1137/S0036139999361896.

[23]

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., 10 (2003), 863-874. 

[24] L. F. ShampineI. Gladwell and S. Thompson, Solving ODEs with Matlab, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511615542.
[25]

J. P. ShiC. C. Wang and H. Wang, Diffusive spatial movement with memory and maturation delays, Nonlinearity, 32 (2019), 3188-3208.  doi: 10.1088/1361-6544/ab1f2f.

[26]

J. ShiC. Wang and H. Wang, Spatial movement with diffusion and memory-based self-diffusion and cross-diffusion, J. Differential Equations, 305 (2021), 242-269.  doi: 10.1016/j.jde.2021.10.021.

[27]

J. P. ShiC. C. WangH. Wang and X. Yan, Diffusive spatial movement with memory, J. Dynam. Differential Equations, 32 (2020), 979-1002.  doi: 10.1007/s10884-019-09757-y.

[28]

Y. SongY. Peng and T. Zhang, The spatially inhomogeneous Hopf bifurcation induced by memory delay in a memory-based diffusion system, J. Differential Equations, 300 (2021), 597-624.  doi: 10.1016/j.jde.2021.08.010.

[29]

Y. L. SongS. H. Wu and H. Wang, Spatiotemporal dynamics in the single population model with memory-based diffusion and nonlocal effect, J. Differential Equations, 11 (2019), 6316-6351.  doi: 10.1016/j.jde.2019.06.025.

[30]

Y. SuJ. Wei and J. Shi, Hopf bifurcations in a reaction-diffusion population model with delay effect, J. Differential Equations, 247 (2009), 1156-1184.  doi: 10.1016/j.jde.2009.04.017.

[31]

J. Tello and D. Wrzosek, Predator-prey model with diffusion and indirect prey-taxis, Math. Models Methods Appl. Sci., 26 (2016), 2129-2162.  doi: 10.1142/S0218202516400108.

[32]

V. Volterra, Variazioni e fluttuazioni del numero d'individui in specie animali conviventi, Mem. Rend. Accad. Naz.Lincei Ser, (1926), 31-113. 

[33]

J. WangJ. Wei and J. Shi, Global bifurcation analysis and pattern formation in homogeneous diffusive predator-prey systems, J. Differential Equations, 260 (2016), 3495-3523.  doi: 10.1016/j.jde.2015.10.036.

[34]

X. Wei and J. Wei, The effect of delayed feedback on the dynamics of an autocatalysis reaction-diffusion system, Nonlinear Anal. Model. Control, 23 (2018), 749-770.  doi: 10.15388/NA.2018.5.7.

[35]

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

[36]

C. Xu and et al., Bifurcation analysis in a delayed Lokta-Volterra predator-prey model with two delays, Nonlinear Dynam., 66 (2011), 169-183.  doi: 10.1007/s11071-010-9919-8.

[37]

R. Yang, Bifurcation analysis of a diffusive predator-prey system with Crowley-Martin functional response and delay, Chaos, Solitons and Fractals, 95 (2017), 131-139.  doi: 10.1016/j.chaos.2016.12.014.

[38]

W. Zuo and J. Wei, Stability and Hopf bifurcation in a diffusive predator-prey system with delay effect, Nonlinear Anal. Real World Appl., 12 (2011), 1998-2011.  doi: 10.1016/j.nonrwa.2010.12.016.

[39]

L. Zhang and S. Fu, Global bifurcation for a Holling-Tanner predator-prey model with prey-taxis, Nonlinear Anal. Real World Appl., 47 (2019), 460-472.  doi: 10.1016/j.nonrwa.2018.12.002.

[40]

J. Zhao and J. Wei, Persistence, Turing instability and Hopf bifurcation in a diffusive plankton system with delay and quadratic closure, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 26 (2016), 1650047, 13 pp. doi: 10.1142/S0218127416500474.

[41]

P. Zhou and D. Xiao, Global dynamics of a classical Lotka-Volterra competition-diffusion-advection system, J. Funct. Anal., 275 (2018), 356-380.  doi: 10.1016/j.jfa.2018.03.006.

[42]

L. K. A. L, Global $L^{\infty}$ estimates for a class of reaction-diffusion systems, J. Math. Anal. Appl., 217 (1998), 72-94.  doi: 10.1006/jmaa.1997.5702.

show all references

References:
[1]

F. B. M. Belgacem and C. Cosner, The effect of dispersal along environmental gradients on the dynamics of populations in heterogeneous environment, Canad. Appl. Math. Quart., 3 (1995), 379-397. 

[2]

S. Busenberg and W. Huang, Stability and Hopf bifurcation for a population delay model with diffusion effects, J. Differential Equations, 124 (1996), 80-107.  doi: 10.1006/jdeq.1996.0003.

[3]

R. S. Cantrell and C. Cosner, On the dynamics of predator-prey models with the Beddington-DeAngelis functional response, J. Math. Anal. Appl., 257 (2001), 206-222.  doi: 10.1006/jmaa.2000.7343.

[4]

S. ChenY. Lou and J. Wei, Hopf bifurcation in a delayed reaction-diffusion-advection population model, J. Differential Equations, 264 (2018), 5333-5359.  doi: 10.1016/j.jde.2018.01.008.

[5]

S. S. Chen and J. P. Shi, Stability and Hopf bifurcation in a diffusive logistic population model with nonlocal delay effect, J. Differential Equations, 253 (2012), 3440-3470.  doi: 10.1016/j.jde.2012.08.031.

[6]

F. Dai and B. Liu, Global solution for a general cross-diffusion two-competitive-predator and one-prey system with predator-taxis, Commun. Nonlinear Sci. Numer. Simul., 89 (2020), 105336, 22 pp. doi: 10.1016/j.cnsns.2020.105336.

[7]

D. DuanB. Niu and J. Wei, Hopf-Hopf bifurcation and chaotic attractors in a delayed diffusive predator-prey model with fear effect, Chaos Solitons Fractals, 123 (2019), 206-216.  doi: 10.1016/j.chaos.2019.04.012.

[8]

W. F. FaganM. A. Lewisand M. Auger-Méthé and et al., Spatial memory and animal movement, Ecology Letters, 10 (2013), 1316-1329. 

[9]

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.

[10]

T. Faria, Stability and bifurcation for a delayed predator-prey model and the effect of diffusion, J. Math. Anal. Appl., 254 (2001), 433-463.  doi: 10.1006/jmaa.2000.7182.

[11] B. D. HassardN. D. Kazarinoff and Y. H. Wan, Theory and Applications of Hopf Bifurcation, London Mathematical Society Lecture Note Series, 41. Cambridge University Press, Cambridge-New York, 1981. 
[12]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, Berlin-New York, 1981.

[13]

C. S. Holling, The functional response of predators to prey density and its role in mimicry and population regulation, Memoirs of the Entomological Society of Canada, 97 (1965), 1-60. 

[14]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.

[15]

F. Li and H. Li, Hopf bifurcation of a predator-prey model with time delay and stage structure for the prey, Math. Comput. Modelling, 55 (2012), 672-679.  doi: 10.1016/j.mcm.2011.08.041.

[16]

Z. Li and B. Dai, Stability and Hopf bifurcation analysis in a Lotka-Volterra competition-diffusion-advection model with time delay effect, Nonlinearity, 34 (2021), 3271-3313.  doi: 10.1088/1361-6544/abe77a.

[17]

X. Liu, et al., Turing-Hopf bifurcations in a predator-prey model with herd behavior, quadratic mortality and prey-taxis, Phys. A, 496 (2018), 446–460. doi: 10.1016/j.physa.2018.01.006.

[18]

A. J. Lotka, The elements of physical biology, HathiTrust, 1925.

[19]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Progress in Nonlinear Differential Equations and their Applications, 16. Birkhäuser Verlag, Basel, 1995.

[20]

M. Peng and Z. Zhang, Hopf bifurcation analysis in a predator-prey model with two time delays and stage structure for the prey, Adv. Difference Equ., 2018 (2018), Paper No. 251, 20 pp. doi: 10.1186/s13662-018-1705-9.

[21]

J. Poggiale and P. Auger, Fast oscillating migrations in a predator-prey model, Math. Models Methods Appl. Sci., 6 (1996), 217-226.  doi: 10.1142/S0218202596000559.

[22]

S. Ruan and D. Xiao, Global analysis in a predator-prey system with nonmonotonic functional response, SIAM J. Appl. Math., 61 (2000/01), 1445-1472.  doi: 10.1137/S0036139999361896.

[23]

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., 10 (2003), 863-874. 

[24] L. F. ShampineI. Gladwell and S. Thompson, Solving ODEs with Matlab, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511615542.
[25]

J. P. ShiC. C. Wang and H. Wang, Diffusive spatial movement with memory and maturation delays, Nonlinearity, 32 (2019), 3188-3208.  doi: 10.1088/1361-6544/ab1f2f.

[26]

J. ShiC. Wang and H. Wang, Spatial movement with diffusion and memory-based self-diffusion and cross-diffusion, J. Differential Equations, 305 (2021), 242-269.  doi: 10.1016/j.jde.2021.10.021.

[27]

J. P. ShiC. C. WangH. Wang and X. Yan, Diffusive spatial movement with memory, J. Dynam. Differential Equations, 32 (2020), 979-1002.  doi: 10.1007/s10884-019-09757-y.

[28]

Y. SongY. Peng and T. Zhang, The spatially inhomogeneous Hopf bifurcation induced by memory delay in a memory-based diffusion system, J. Differential Equations, 300 (2021), 597-624.  doi: 10.1016/j.jde.2021.08.010.

[29]

Y. L. SongS. H. Wu and H. Wang, Spatiotemporal dynamics in the single population model with memory-based diffusion and nonlocal effect, J. Differential Equations, 11 (2019), 6316-6351.  doi: 10.1016/j.jde.2019.06.025.

[30]

Y. SuJ. Wei and J. Shi, Hopf bifurcations in a reaction-diffusion population model with delay effect, J. Differential Equations, 247 (2009), 1156-1184.  doi: 10.1016/j.jde.2009.04.017.

[31]

J. Tello and D. Wrzosek, Predator-prey model with diffusion and indirect prey-taxis, Math. Models Methods Appl. Sci., 26 (2016), 2129-2162.  doi: 10.1142/S0218202516400108.

[32]

V. Volterra, Variazioni e fluttuazioni del numero d'individui in specie animali conviventi, Mem. Rend. Accad. Naz.Lincei Ser, (1926), 31-113. 

[33]

J. WangJ. Wei and J. Shi, Global bifurcation analysis and pattern formation in homogeneous diffusive predator-prey systems, J. Differential Equations, 260 (2016), 3495-3523.  doi: 10.1016/j.jde.2015.10.036.

[34]

X. Wei and J. Wei, The effect of delayed feedback on the dynamics of an autocatalysis reaction-diffusion system, Nonlinear Anal. Model. Control, 23 (2018), 749-770.  doi: 10.15388/NA.2018.5.7.

[35]

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

[36]

C. Xu and et al., Bifurcation analysis in a delayed Lokta-Volterra predator-prey model with two delays, Nonlinear Dynam., 66 (2011), 169-183.  doi: 10.1007/s11071-010-9919-8.

[37]

R. Yang, Bifurcation analysis of a diffusive predator-prey system with Crowley-Martin functional response and delay, Chaos, Solitons and Fractals, 95 (2017), 131-139.  doi: 10.1016/j.chaos.2016.12.014.

[38]

W. Zuo and J. Wei, Stability and Hopf bifurcation in a diffusive predator-prey system with delay effect, Nonlinear Anal. Real World Appl., 12 (2011), 1998-2011.  doi: 10.1016/j.nonrwa.2010.12.016.

[39]

L. Zhang and S. Fu, Global bifurcation for a Holling-Tanner predator-prey model with prey-taxis, Nonlinear Anal. Real World Appl., 47 (2019), 460-472.  doi: 10.1016/j.nonrwa.2018.12.002.

[40]

J. Zhao and J. Wei, Persistence, Turing instability and Hopf bifurcation in a diffusive plankton system with delay and quadratic closure, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 26 (2016), 1650047, 13 pp. doi: 10.1142/S0218127416500474.

[41]

P. Zhou and D. Xiao, Global dynamics of a classical Lotka-Volterra competition-diffusion-advection system, J. Funct. Anal., 275 (2018), 356-380.  doi: 10.1016/j.jfa.2018.03.006.

[42]

L. K. A. L, Global $L^{\infty}$ estimates for a class of reaction-diffusion systems, J. Math. Anal. Appl., 217 (1998), 72-94.  doi: 10.1006/jmaa.1997.5702.

Figure 1.  Numerical simulation of (5) for $ \tau = 0 $ with parameter condition $ {\bf(P1)} $, the unique positive equilibrium $ E_* = (0.8, 0.2) $ of (5) is locally asymptotically stable
Figure 2.  Numerical simulation of (5) for $ \tau = 3 $ with parameter condition $ {\bf(P1)} $, the unique positive equilibrium $ E_* = (0.8, 0.2) $ of (5) is locally asymptotically stable
Figure 3.  Numerical simulation of (5) for $\tau=0$ with parameter condition ${\bf(P2)}$, the unique positive equilibrium $E_*=(0.25,0.75)$ of (5) is locally asymptotically stable
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