# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2022082
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## Hopf bifurcation for a predator-prey model with age structure and ratio-dependent response function incorporating a prey refuge

 School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China

*Corresponding author: Jixun Chu

Received  May 2021 Revised  March 2022 Early access May 2022

Fund Project: The second author is supported by NSFC grant N0.11280125

In this paper, a predator-prey model with age structure and ratio-dependent response function incorporating a prey refuge is investigated. The model is formulated as an abstract non-densely defined Cauchy problem and a sufficient condition for the existence of the positive age-related equilibrium is given. Then using the integral semigroup theory and the Hopf bifurcation theory for semilinear equations with non-dense domain, it is shown that Hopf bifurcation occurs at the positive age-related equilibrium. Numerical simulations are performed to validate theoretical results and sensitivity analyses are presented. The results show that the prey refuge has a stabilizing effect, that is, the prey refuge is an important factor to maintain the balance between prey and predator population.

Citation: Tongtong Chen, Jixun Chu. Hopf bifurcation for a predator-prey model with age structure and ratio-dependent response function incorporating a prey refuge. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022082
##### References:
 [1] R. Arditi and L. R. Ginzburg, Coupling in predator-prey dynamics: Ratio-dependence, J. Theor. Biol., 139 (1989), 311-326. [2] X. Duan, J. Yin and X. Li, Global Hopf bifurcation of an SIRS epidemic model with age-dependent recovery, Chaos Solitons Fractals, 104 (2017), 613-624.  doi: 10.1016/j.chaos.2017.09.029. [3] A. Ducrot, Z. Liu and P. Magal, Essential growth rate for bounded linear perturbation of non-densely defined Cauchy problems, J. Math. Anal. Appl., 341 (2008), 501-518.  doi: 10.1016/j.jmaa.2007.09.074. [4] H. I. Freedman, Deterministic Mathematical Models in Population Ecology, Marcel Dekker, New York, 1980. [5] X. Fu, Z. Liu and P. Magal, Hopf bifurcation in an age-structured population model with two delays, Commun. Pure Appl. Anal., 14 (2015), 657-676.  doi: 10.3934/cpaa.2015.14.657. [6] L. N. Guin, S. Djilali and S. Chakravarty, Cross-diffusion-driven instability in an interacting species model with prey refuge, Chaos Solitons Fractals, 153 (2021), Paper No. 111501, 16 pp. doi: 10.1016/j.chaos.2021.111501. [7] Z. Guo, H. Huo and H. Xiang, Bifurcation analysis of an age-structured alcoholism model, J. Biol. Dyn., 12 (2018), 1009-1033.  doi: 10.1080/17513758.2018.1535668. [8] Z. Guo, H. Huo and H. Xiang, Hopf bifurcation of an age-structured HIV infection model with logistic target-cell growth, J. Biol. Dyn., 13 (2019), 362-384.  doi: 10.1080/17513758.2019.1602171. [9] R. Han, L. N. Guin and B. Dai, Consequences of refuge and diffusion in a spatiotemporal predator-prey model, Nonlinear Anal. Real World Appl., 60 (2021), 103311, 36 pp. doi: 10.1016/j.nonrwa.2021.103311. [10] M. M. Haque and S. Sarwardi, Dynamics of a harvested prey-predator model with prey refuge dependent on both species, Intl. J. Bif. Chaos, 28 (2018), 1830040, 16 pp. doi: 10.1142/S0218127418300409. [11] G. W. Harrison, Global stability of predator-prey interactions, J. Math. Biol., 8 (1979), 159-171.  doi: 10.1007/BF00279719. [12] X. Jiang, Z. She and S. Ruan, Global dynamics of a predator-prey system with density-dependent mortality and ratio-dependent functional response, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 1967-1990.  doi: 10.3934/dcdsb.2020041. [13] Y. Kuang and E. Beretta, Global qualitative analysis of a ratio-dependent predator-prey system, J. Math. Biol., 36 (1998), 389-406.  doi: 10.1007/s002850050105. [14] B. Li and Y. Kuang, Heteroclinic bifurcation in the Michaelis-Menten-type ratio-dependent predator-prey system, SIAM J. Appl. Math., 67 (2007), 1453-1464.  doi: 10.1137/060662460. [15] X. Li, G. Hu, X. Li and Z. Feng, Positive steady states of a ratio-dependent predator-prey system with cross-diffusion, Math. Biosci. Eng., 16 (2019), 6753-6768.  doi: 10.3934/mbe.2019337. [16] Z. Liu and N. Li, Stability and bifurcation in a predator-prey model with age structure and delays, J. Nonlinear Sci., 25 (2015), 937-957.  doi: 10.1007/s00332-015-9245-x. [17] Z. Liu and P. Magal, Bogdanov-Takens bifurcation in a predator-prey model with age structure, Z. Angew. Math. Phys., 72 (2021), 24 pp. doi: 10.1007/s00033-020-01434-1. [18] Z. Liu, P. Magal and S. Ruan, Hopf bifurcation for non-densely defined Cauchy problems, Z. Angew. Math. Phys., 62 (2011), 191-222.  doi: 10.1007/s00033-010-0088-x. [19] Z. Liu, P. Magal and S. Ruan, Normal forms for semilinear equations with non-dense domain with applications to age structured models, J. Differential Equations, 257 (2014), 921-1011.  doi: 10.1016/j.jde.2014.04.018. [20] Z. Liu, P. Magal and S. Ruan, Oscillations in age-structured models of consumer-resource mutualisms, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 537-555.  doi: 10.3934/dcdsb.2016.21.537. [21] Z. Liu, P. Magal and D. Xiao, Bogdanov-Takens bifurcation in a predator-prey model, Z. Angew. Math. Phys., 67 (2016), 29 pp. doi: 10.1007/s00033-016-0724-1. [22] Z. Liu, H. Tang and P. Magal, Hopf bifurcation for a spatially and age structured population dynamics model, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1735-1757.  doi: 10.3934/dcdsb.2015.20.1735. [23] Z. Liu and R. Yuan, Zero-Hopf bifurcation for an infection-age structured epidemic model with a nonlinear incidence rate, Sci. China Math., 60 (2017), 1371-1398.  doi: 10.1007/s11425-016-0371-8. [24] Z. Ma, W. Li, Y. Zhao, W. Wang, H. Zhang and Z. Li, Effects of prey refuges on a predator-prey model with a class of functional responses: The role of refuges, Math. Biosci., 218 (2009), 73-79.  doi: 10.1016/j.mbs.2008.12.008. [25] P. Magal, Compact attractors for time-periodic age-structured population models, Electron. J. Differential Equations, 2001 (2001), 1-35. [26] P. Magal and S. Ruan, Center Manifolds for Semilinear Equations with Non-Dense Domain and Applications on Hopf Bifurcation in Age Structured Models, Mem. Amer. Math. Soc. 2009. doi: 10.1090/S0065-9266-09-00568-7. [27] P. Magal and S. Ruan, On semilinear Cauchy problems with non-dense domain, Adv. Differential Equations, 14 (2009), 1041-1084. [28] J. Maynard-Smith, Models in Ecology, CUP Archive, 1978. [29] H. Molla, M. S. Rahman and S. Sarwardi, Dynamics of a predator-prey model with Holling type Ⅱ functional response incorporating a prey refuge depending on both the species, Int. J. Nonlinear Sci. Numer. Simul., 20 (2019), 1-16.  doi: 10.1515/ijnsns-2017-0224. [30] D. Mukherjee and C. Maji, Bifurcation analysis of a Holling type Ⅱ predator-prey model with refuge, Chinese J. Phys., 65 (2020), 153-162.  doi: 10.1016/j.cjph.2020.02.012. [31] R. Peng, Qualitative analysis on a diffusive and ratio-dependent predator-prey model, IMA J. Appl. Math., 78 (2013), 566-586.  doi: 10.1093/imamat/hxr066. [32] S. Ruan, Y. Tang and W. Zhang, Versal unfoldings of predator-prey systems with ratio-dependent functional response, J. Differential Equations, 249 (2010), 1410-1435.  doi: 10.1016/j.jde.2010.06.015. [33] G. D. Ruxton, Short term refuge use and stability of predator-prey models, Theor. Popul. Biol., 47 (1995), 1-17. [34] N. Sk, P. K. Tiwari and S. Pal, A delay nonautonomous model for the impacts of fear and refuge in a three species food chain model with hunting cooperation, Math. Comput. Simulation, 192 (2022), 136-166.  doi: 10.1016/j.matcom.2021.08.018. [35] N. Sk, P. K. Tiwari, S. Pal and M. Martcheva, A delay non-autonomous model for the combined effects of fear, prey refuge and additional food for predator, J. Biol. Dyn., 15 (2021), 580-622.  doi: 10.1080/17513758.2021.2001583. [36] H. Tang and Z. Liu, Hopf bifurcation for a predator-prey model with age structure, Appl. Math. Model., 40 (2016), 726-737.  doi: 10.1016/j.apm.2015.09.015. [37] H. R. Thieme, Semiflows generated by Lipschitz perturbations of non-densely defined operators, Differential Integral Equations, 3 (1990), 1035-1066. [38] Z. Wang and Z. Liu, Hopf bifurcation of an age-structured compartmental pest-pathogen model, J. Math. Anal. Appl., 385 (2012), 1134-1150.  doi: 10.1016/j.jmaa.2011.07.038. [39] D. Yan, H. Cao, X. Xu and X. Wang, Hopf bifurcation for a predator-prey model with age structure, Phys. A, 526 (2019), 120953, 15 pp. doi: 10.1016/j.physa.2019.04.189. [40] P. Yang, Hopf bifurcation of an age-structured prey-predator model with Holling type Ⅱ functional response incorporating a prey refuge, Nonlinear Anal. Real World Appl., 49 (2019), 368-385.  doi: 10.1016/j.nonrwa.2019.03.014. [41] P. Yang and Y. Wang, Existence and properties of Hopf bifurcation in an age-dependent predation system with prey harvesting, Commun. Nonlinear Sci. Numer. Simul., 91 (2020), 105395, 40 pp. doi: 10.1016/j.cnsns.2020.105395. [42] P. Yang and Y. Wang, Hopf bifurcation of an infection-age structured eco-epidemiological model with saturation incidence, J. Math. Anal. Appl., 477 (2019), 398-419.  doi: 10.1016/j.jmaa.2019.04.038. [43] P. Yang and Y. Wang, Hopf-zero bifurcation in an age-dependent predator-prey system with Monod-Haldane functional response comprising strong Allee effect, J. Differential Equations, 269 (2020), 9583-9618.  doi: 10.1016/j.jde.2020.06.048. [44] P. Yang and Y. Wang, Periodic solutions of a delayed eco-epidemiological model with infection-age structure and Holling type Ⅱ functional response, Intl. J. Bif. Chaos, 30 (2020), 2050011, 20 pp. doi: 10.1142/S021812742050011X. [45] Y. Yang and T. Zhang, Dynamic analysis of a modified stochastic predator-prey system with general ratio-dependent functional response, Bull. Korean Math. Soc., 53 (2016), 103-117.  doi: 10.4134/BKMS.2016.53.1.103. [46] X. Zhang and Z. Liu, Bifurcation analysis of an age structured HIV infection model with both virus-to-cell and cell-to-cell transmissions, Intl. J. Bif. Chaos, 28 (2018), 1850109, 20 pp. doi: 10.1142/S0218127418501092. [47] X. Zhang and Z. Liu, Hopf bifurcation for a susceptible-infective model with infection-age structure, J. Nonlinear Sci., 30 (2020), 317-367.  doi: 10.1007/s00332-019-09575-y. [48] X. Zhang and Z. Liu, Periodic oscillations in age-structured ratio-dependent predator-prey model with Michaelis-Menten type functional response, Phys. D, 389 (2019), 51-63.  doi: 10.1016/j.physd.2018.10.002.

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##### References:
 [1] R. Arditi and L. R. Ginzburg, Coupling in predator-prey dynamics: Ratio-dependence, J. Theor. Biol., 139 (1989), 311-326. [2] X. Duan, J. Yin and X. Li, Global Hopf bifurcation of an SIRS epidemic model with age-dependent recovery, Chaos Solitons Fractals, 104 (2017), 613-624.  doi: 10.1016/j.chaos.2017.09.029. [3] A. Ducrot, Z. Liu and P. Magal, Essential growth rate for bounded linear perturbation of non-densely defined Cauchy problems, J. Math. Anal. Appl., 341 (2008), 501-518.  doi: 10.1016/j.jmaa.2007.09.074. [4] H. I. Freedman, Deterministic Mathematical Models in Population Ecology, Marcel Dekker, New York, 1980. [5] X. Fu, Z. Liu and P. Magal, Hopf bifurcation in an age-structured population model with two delays, Commun. Pure Appl. Anal., 14 (2015), 657-676.  doi: 10.3934/cpaa.2015.14.657. [6] L. N. Guin, S. Djilali and S. Chakravarty, Cross-diffusion-driven instability in an interacting species model with prey refuge, Chaos Solitons Fractals, 153 (2021), Paper No. 111501, 16 pp. doi: 10.1016/j.chaos.2021.111501. [7] Z. Guo, H. Huo and H. Xiang, Bifurcation analysis of an age-structured alcoholism model, J. Biol. Dyn., 12 (2018), 1009-1033.  doi: 10.1080/17513758.2018.1535668. [8] Z. Guo, H. Huo and H. Xiang, Hopf bifurcation of an age-structured HIV infection model with logistic target-cell growth, J. Biol. Dyn., 13 (2019), 362-384.  doi: 10.1080/17513758.2019.1602171. [9] R. Han, L. N. Guin and B. Dai, Consequences of refuge and diffusion in a spatiotemporal predator-prey model, Nonlinear Anal. Real World Appl., 60 (2021), 103311, 36 pp. doi: 10.1016/j.nonrwa.2021.103311. [10] M. M. Haque and S. Sarwardi, Dynamics of a harvested prey-predator model with prey refuge dependent on both species, Intl. J. Bif. Chaos, 28 (2018), 1830040, 16 pp. doi: 10.1142/S0218127418300409. [11] G. W. Harrison, Global stability of predator-prey interactions, J. Math. Biol., 8 (1979), 159-171.  doi: 10.1007/BF00279719. [12] X. Jiang, Z. She and S. Ruan, Global dynamics of a predator-prey system with density-dependent mortality and ratio-dependent functional response, Discrete Contin. Dyn. Syst. Ser. B, 26 (2021), 1967-1990.  doi: 10.3934/dcdsb.2020041. [13] Y. Kuang and E. Beretta, Global qualitative analysis of a ratio-dependent predator-prey system, J. Math. Biol., 36 (1998), 389-406.  doi: 10.1007/s002850050105. [14] B. Li and Y. Kuang, Heteroclinic bifurcation in the Michaelis-Menten-type ratio-dependent predator-prey system, SIAM J. Appl. Math., 67 (2007), 1453-1464.  doi: 10.1137/060662460. [15] X. Li, G. Hu, X. Li and Z. Feng, Positive steady states of a ratio-dependent predator-prey system with cross-diffusion, Math. Biosci. Eng., 16 (2019), 6753-6768.  doi: 10.3934/mbe.2019337. [16] Z. Liu and N. Li, Stability and bifurcation in a predator-prey model with age structure and delays, J. Nonlinear Sci., 25 (2015), 937-957.  doi: 10.1007/s00332-015-9245-x. [17] Z. Liu and P. Magal, Bogdanov-Takens bifurcation in a predator-prey model with age structure, Z. Angew. Math. Phys., 72 (2021), 24 pp. doi: 10.1007/s00033-020-01434-1. [18] Z. Liu, P. Magal and S. Ruan, Hopf bifurcation for non-densely defined Cauchy problems, Z. Angew. Math. Phys., 62 (2011), 191-222.  doi: 10.1007/s00033-010-0088-x. [19] Z. Liu, P. Magal and S. Ruan, Normal forms for semilinear equations with non-dense domain with applications to age structured models, J. Differential Equations, 257 (2014), 921-1011.  doi: 10.1016/j.jde.2014.04.018. [20] Z. Liu, P. Magal and S. Ruan, Oscillations in age-structured models of consumer-resource mutualisms, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 537-555.  doi: 10.3934/dcdsb.2016.21.537. [21] Z. Liu, P. Magal and D. Xiao, Bogdanov-Takens bifurcation in a predator-prey model, Z. Angew. Math. Phys., 67 (2016), 29 pp. doi: 10.1007/s00033-016-0724-1. [22] Z. Liu, H. Tang and P. Magal, Hopf bifurcation for a spatially and age structured population dynamics model, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 1735-1757.  doi: 10.3934/dcdsb.2015.20.1735. [23] Z. Liu and R. Yuan, Zero-Hopf bifurcation for an infection-age structured epidemic model with a nonlinear incidence rate, Sci. China Math., 60 (2017), 1371-1398.  doi: 10.1007/s11425-016-0371-8. [24] Z. Ma, W. Li, Y. Zhao, W. Wang, H. Zhang and Z. Li, Effects of prey refuges on a predator-prey model with a class of functional responses: The role of refuges, Math. Biosci., 218 (2009), 73-79.  doi: 10.1016/j.mbs.2008.12.008. [25] P. Magal, Compact attractors for time-periodic age-structured population models, Electron. J. Differential Equations, 2001 (2001), 1-35. [26] P. Magal and S. Ruan, Center Manifolds for Semilinear Equations with Non-Dense Domain and Applications on Hopf Bifurcation in Age Structured Models, Mem. Amer. Math. Soc. 2009. doi: 10.1090/S0065-9266-09-00568-7. [27] P. Magal and S. Ruan, On semilinear Cauchy problems with non-dense domain, Adv. Differential Equations, 14 (2009), 1041-1084. [28] J. Maynard-Smith, Models in Ecology, CUP Archive, 1978. [29] H. Molla, M. S. Rahman and S. Sarwardi, Dynamics of a predator-prey model with Holling type Ⅱ functional response incorporating a prey refuge depending on both the species, Int. J. Nonlinear Sci. Numer. Simul., 20 (2019), 1-16.  doi: 10.1515/ijnsns-2017-0224. [30] D. Mukherjee and C. Maji, Bifurcation analysis of a Holling type Ⅱ predator-prey model with refuge, Chinese J. Phys., 65 (2020), 153-162.  doi: 10.1016/j.cjph.2020.02.012. [31] R. Peng, Qualitative analysis on a diffusive and ratio-dependent predator-prey model, IMA J. Appl. Math., 78 (2013), 566-586.  doi: 10.1093/imamat/hxr066. [32] S. Ruan, Y. Tang and W. Zhang, Versal unfoldings of predator-prey systems with ratio-dependent functional response, J. Differential Equations, 249 (2010), 1410-1435.  doi: 10.1016/j.jde.2010.06.015. [33] G. D. Ruxton, Short term refuge use and stability of predator-prey models, Theor. Popul. Biol., 47 (1995), 1-17. [34] N. Sk, P. K. Tiwari and S. Pal, A delay nonautonomous model for the impacts of fear and refuge in a three species food chain model with hunting cooperation, Math. Comput. Simulation, 192 (2022), 136-166.  doi: 10.1016/j.matcom.2021.08.018. [35] N. Sk, P. K. Tiwari, S. Pal and M. Martcheva, A delay non-autonomous model for the combined effects of fear, prey refuge and additional food for predator, J. Biol. Dyn., 15 (2021), 580-622.  doi: 10.1080/17513758.2021.2001583. [36] H. Tang and Z. Liu, Hopf bifurcation for a predator-prey model with age structure, Appl. Math. Model., 40 (2016), 726-737.  doi: 10.1016/j.apm.2015.09.015. [37] H. R. Thieme, Semiflows generated by Lipschitz perturbations of non-densely defined operators, Differential Integral Equations, 3 (1990), 1035-1066. [38] Z. Wang and Z. Liu, Hopf bifurcation of an age-structured compartmental pest-pathogen model, J. Math. Anal. Appl., 385 (2012), 1134-1150.  doi: 10.1016/j.jmaa.2011.07.038. [39] D. Yan, H. Cao, X. Xu and X. Wang, Hopf bifurcation for a predator-prey model with age structure, Phys. A, 526 (2019), 120953, 15 pp. doi: 10.1016/j.physa.2019.04.189. [40] P. Yang, Hopf bifurcation of an age-structured prey-predator model with Holling type Ⅱ functional response incorporating a prey refuge, Nonlinear Anal. Real World Appl., 49 (2019), 368-385.  doi: 10.1016/j.nonrwa.2019.03.014. [41] P. Yang and Y. Wang, Existence and properties of Hopf bifurcation in an age-dependent predation system with prey harvesting, Commun. Nonlinear Sci. Numer. Simul., 91 (2020), 105395, 40 pp. doi: 10.1016/j.cnsns.2020.105395. [42] P. Yang and Y. Wang, Hopf bifurcation of an infection-age structured eco-epidemiological model with saturation incidence, J. Math. Anal. Appl., 477 (2019), 398-419.  doi: 10.1016/j.jmaa.2019.04.038. [43] P. Yang and Y. Wang, Hopf-zero bifurcation in an age-dependent predator-prey system with Monod-Haldane functional response comprising strong Allee effect, J. Differential Equations, 269 (2020), 9583-9618.  doi: 10.1016/j.jde.2020.06.048. [44] P. Yang and Y. Wang, Periodic solutions of a delayed eco-epidemiological model with infection-age structure and Holling type Ⅱ functional response, Intl. J. Bif. Chaos, 30 (2020), 2050011, 20 pp. doi: 10.1142/S021812742050011X. [45] Y. Yang and T. Zhang, Dynamic analysis of a modified stochastic predator-prey system with general ratio-dependent functional response, Bull. Korean Math. Soc., 53 (2016), 103-117.  doi: 10.4134/BKMS.2016.53.1.103. [46] X. Zhang and Z. Liu, Bifurcation analysis of an age structured HIV infection model with both virus-to-cell and cell-to-cell transmissions, Intl. J. Bif. Chaos, 28 (2018), 1850109, 20 pp. doi: 10.1142/S0218127418501092. [47] X. Zhang and Z. Liu, Hopf bifurcation for a susceptible-infective model with infection-age structure, J. Nonlinear Sci., 30 (2020), 317-367.  doi: 10.1007/s00332-019-09575-y. [48] X. Zhang and Z. Liu, Periodic oscillations in age-structured ratio-dependent predator-prey model with Michaelis-Menten type functional response, Phys. D, 389 (2019), 51-63.  doi: 10.1016/j.physd.2018.10.002.
Numerical simulations of system (1) with $\tau = 1 < {\tau _0}$ : (a) The solution of the prey; (b) The solution of the predator; (c) The phase portrait of system (1); (d) The distribution function of the predators $u\left( {t, a} \right)$
Numerical simulations of system (1) with $\tau = 2.2 > {\tau _0}$ : (a) The solution of the prey; (b) The solution of the predator; (c) The phase portrait of system (1); (d) The distribution function of the predators $u\left( {t, a} \right)$
The effect of the refuge term $m$ on the dynamics of the predator and prey populations
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