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doi: 10.3934/eect.2022034
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## Asymptotic analysis of an age-structured predator-prey model with ratio-dependent Holling Ⅲ functional response and delays

 1 School of Science, Nanjing University of Posts and Telecommunications, Nanjing 210023, China 2 School of Mathematical Sciences, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200241, China

*Corresponding author: Yuan Yuan

Received  March 2022 Revised  June 2022 Early access July 2022

Fund Project: This work is supported by NNSF of China (grant No. 12101323), Science and Technology Commission of Shanghai Municipality (STCSM) (grant No. 18dz2271000), Natural Science Foundation of Jiangsu Province of China (grant No. BK20200749), Nanjing University of Posts and Telecommunications Science Foundation (grant No. NY220093)

This paper studies the dynamical behavior of a radio-dependent predator-prey model with age structure and two delays. The model is first formulated as an abstract non-densely defined Cauchy problem and the conditions for existence of the positive equilibrium point are derived. Then, through determining the distribution of eigenvalues, the globally asymptotic stability of the boundary equilibrium and the locally asymptotic stability for the positive equilibrium are obtained, respectively. In addition, it is also shown that a non-trivial periodic oscillation phenomenon through Hopf bifurcation appears under some conditions. Finally, some numerical examples are provided to illustrate the obtained results.

Citation: Dongxue Yan, Yuan Yuan, Xianlong Fu. Asymptotic analysis of an age-structured predator-prey model with ratio-dependent Holling Ⅲ functional response and delays. Evolution Equations and Control Theory, doi: 10.3934/eect.2022034
##### References:
 [1] H. R. Akcakaya, Population cycles of mammals: Evidence for a ratio-dependent pre-dation hypothesis, Ecol. Monogr., 62 (1992), 119-142. [2] H. R. Akcakaya, R. Arditi and L. R. Ginzburg, Ratio-dependent prediction: Anabstraction that works, Ecol., 76 (1995), 995-1004. [3] R. Arditi and L. R. Ginzburg, Coupling in predator-prey dynamics: Ratio-dependence, J. Theoret. Biol., 139 (1989), 311-326. [4] R. Arditi, L. R. Ginzburg and H. R. Akcakaya, Variation in plankton densities among lakes: A case for ratio-dependent models, Amer. Nat., 138 (1991), 1287-1296. [5] R. Arditi and H. Saiah, Empirical evidence of the role of heterogeneity in ratio-dependent consumption, Econogy, 73 (1992), 1544-1551. [6] E. Beretta and Y. Kuang, Global analyses in some delayed ratio-dependent predator-prey systems, Nonl. Anal., 32 (1998), 381-408.  doi: 10.1016/S0362-546X(97)00491-4. [7] J. Chen and H. Zhang, The qualitative analysis of two species predator-prey model with Holling's type Ⅲ functional response, Appl. Math. Mech., 7 (1986), 77-86.  doi: 10.1007/BF01896254. [8] S. Chen, J. Shi and J. Wei, The effect of delay on a diffusive predator-prey systemwith Holling type-Ⅱ predator functional response, Comm. Pure Appl. Anal., 12 (2013), 481-501.  doi: 10.3934/cpaa.2013.12.481. [9] J. Chu, A. Ducrot, P. Magal and S. Ruan, Hopf bifurcation in a size structured population dynamic model with random growth, J. Diff. Equ., 247 (2009), 956-1000.  doi: 10.1016/j.jde.2009.04.003. [10] J. M. Cushing and M. Saleem, A predator prey model with age structure, J. Math. Biol., 14 (1982), 231-250.  doi: 10.1007/BF01832847. [11] D. L. DeAngelis, Dynamics of Nutrient Cycling and Food Webs, Springer, Berlin, 2012. doi: 10.1007/978-94-011-2342-6. [12] A. Ducrot, P. Magal and S. Ruan, Projectors on the generalized eigenspaces for partial differential equations with time delay, Inf. Dim. Dyn. Syst., 64 (2013), 353-390.  doi: 10.1007/978-1-4614-4523-4_14. [13] A. P. Gutierrez, The physiological basis of ratio-dependent predator-prey theory: A metabolic pool model of Nicholson's blowflies as an example, Econogy, 73 (1992), 1552-1563. [14] J. Li, Dynamics of age-structured predator-prey population models, J. Math. Anal. Appl., 152 (1990), 399-415.  doi: 10.1016/0022-247X(90)90073-O. [15] Z. Liu and N. Li, Stability and bifurcation in a predator-prey model with age structure and delays, J. Nonl. Science, 25 (2015), 937-957.  doi: 10.1007/s00332-015-9245-x. [16] 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. [17] 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., 202 (2009). doi: 10.1090/S0065-9266-09-00568-7. [18] P. Magal and S. Ruan, Theory and Applications of Abstract Semilinear Cauchy Problems, Springer-Verlag, New York, 2018. doi: 10.1007/978-3-030-01506-0. [19] M. Martcheva and H. R. Thieme, Progression age enhanced backward bifurcation in an epidemic model with super-infection, J. Math. Biol., 46 (2003), 385-424.  doi: 10.1007/s00285-002-0181-7. [20] A. F. Nindjin, M. A. Aziz-Alaoui and M. Cadivel, Analysis of a predator-prey model with modified Leslie-Grower and Holling-type Ⅱ schemes with time delay, Nonl. Anal. (RWA), 7 (2006), 1104-1118.  doi: 10.1016/j.nonrwa.2005.10.003. [21] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [22] H. Qolizadeh Amirabad, O. Rabieimotlagh and H. M. Mohammadinejad, Permanency in predator-prey models of Leslie type with ratio-dependent simplified Holling type-Ⅳ functional response, Math. Comput. Simulation, 157 (2019), 63-76.  doi: 10.1016/j.matcom.2018.09.023. [23] Y. Song and S. Yuan, Bifurcation analysis in a predator-prey system with time delay, Nonl. Anal.: (RWA), 7 (2006), 265-284.  doi: 10.1016/j.nonrwa.2005.03.002. [24] 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. [25] H. R. Thieme, Convergence results and a Poincar$\acute{e}$-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755-763.  doi: 10.1007/BF00173267. [26] L. Wang, C. Dai and M. Zhao, Hopf bifurcation in an age-structured prey-predator model with Holling Ⅲ response function, Math. Biosc. Eng., 18 (2021), 3144-3159.  doi: 10.3934/mbe.2021156. [27] 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. [28] D. Xiao and W. Li, Stability and bifurcation in a delayed ratio-dependent predator-prey syste, Proc. Edinburgh Math. Soc., 46 (2003), 205-220.  doi: 10.1017/S0013091500001140. [29] 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. [30] X. Zhang and Z. Liu, Hopf bifurcation analysis in a predator-prey model with predator-age structure and predator-prey reaction time delay, Appl. Math. Model., 91 (2021), 530-548.  doi: 10.1016/j.apm.2020.08.054. [31] J. Zhou, Bifurcation analysis of a diffusive predator-prey model with ratio-dependent Holling type Ⅲ functional response, Nonl. Dynam., 81 (2015), 1535-1552.  doi: 10.1007/s11071-015-2088-z. [32] G. Zhu and J. Wei, Global stability and bifurcation analysis of a delayed predator-prey system with prey immigration, Electron. J. Qual. Theory Differ. Equ., 13 (2016), Paper No. 13, 20 pp. doi: 10.14232/ejqtde.2016.1.13.

show all references

##### References:
 [1] H. R. Akcakaya, Population cycles of mammals: Evidence for a ratio-dependent pre-dation hypothesis, Ecol. Monogr., 62 (1992), 119-142. [2] H. R. Akcakaya, R. Arditi and L. R. Ginzburg, Ratio-dependent prediction: Anabstraction that works, Ecol., 76 (1995), 995-1004. [3] R. Arditi and L. R. Ginzburg, Coupling in predator-prey dynamics: Ratio-dependence, J. Theoret. Biol., 139 (1989), 311-326. [4] R. Arditi, L. R. Ginzburg and H. R. Akcakaya, Variation in plankton densities among lakes: A case for ratio-dependent models, Amer. Nat., 138 (1991), 1287-1296. [5] R. Arditi and H. Saiah, Empirical evidence of the role of heterogeneity in ratio-dependent consumption, Econogy, 73 (1992), 1544-1551. [6] E. Beretta and Y. Kuang, Global analyses in some delayed ratio-dependent predator-prey systems, Nonl. Anal., 32 (1998), 381-408.  doi: 10.1016/S0362-546X(97)00491-4. [7] J. Chen and H. Zhang, The qualitative analysis of two species predator-prey model with Holling's type Ⅲ functional response, Appl. Math. Mech., 7 (1986), 77-86.  doi: 10.1007/BF01896254. [8] S. Chen, J. Shi and J. Wei, The effect of delay on a diffusive predator-prey systemwith Holling type-Ⅱ predator functional response, Comm. Pure Appl. Anal., 12 (2013), 481-501.  doi: 10.3934/cpaa.2013.12.481. [9] J. Chu, A. Ducrot, P. Magal and S. Ruan, Hopf bifurcation in a size structured population dynamic model with random growth, J. Diff. Equ., 247 (2009), 956-1000.  doi: 10.1016/j.jde.2009.04.003. [10] J. M. Cushing and M. Saleem, A predator prey model with age structure, J. Math. Biol., 14 (1982), 231-250.  doi: 10.1007/BF01832847. [11] D. L. DeAngelis, Dynamics of Nutrient Cycling and Food Webs, Springer, Berlin, 2012. doi: 10.1007/978-94-011-2342-6. [12] A. Ducrot, P. Magal and S. Ruan, Projectors on the generalized eigenspaces for partial differential equations with time delay, Inf. Dim. Dyn. Syst., 64 (2013), 353-390.  doi: 10.1007/978-1-4614-4523-4_14. [13] A. P. Gutierrez, The physiological basis of ratio-dependent predator-prey theory: A metabolic pool model of Nicholson's blowflies as an example, Econogy, 73 (1992), 1552-1563. [14] J. Li, Dynamics of age-structured predator-prey population models, J. Math. Anal. Appl., 152 (1990), 399-415.  doi: 10.1016/0022-247X(90)90073-O. [15] Z. Liu and N. Li, Stability and bifurcation in a predator-prey model with age structure and delays, J. Nonl. Science, 25 (2015), 937-957.  doi: 10.1007/s00332-015-9245-x. [16] 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. [17] 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., 202 (2009). doi: 10.1090/S0065-9266-09-00568-7. [18] P. Magal and S. Ruan, Theory and Applications of Abstract Semilinear Cauchy Problems, Springer-Verlag, New York, 2018. doi: 10.1007/978-3-030-01506-0. [19] M. Martcheva and H. R. Thieme, Progression age enhanced backward bifurcation in an epidemic model with super-infection, J. Math. Biol., 46 (2003), 385-424.  doi: 10.1007/s00285-002-0181-7. [20] A. F. Nindjin, M. A. Aziz-Alaoui and M. Cadivel, Analysis of a predator-prey model with modified Leslie-Grower and Holling-type Ⅱ schemes with time delay, Nonl. Anal. (RWA), 7 (2006), 1104-1118.  doi: 10.1016/j.nonrwa.2005.10.003. [21] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [22] H. Qolizadeh Amirabad, O. Rabieimotlagh and H. M. Mohammadinejad, Permanency in predator-prey models of Leslie type with ratio-dependent simplified Holling type-Ⅳ functional response, Math. Comput. Simulation, 157 (2019), 63-76.  doi: 10.1016/j.matcom.2018.09.023. [23] Y. Song and S. Yuan, Bifurcation analysis in a predator-prey system with time delay, Nonl. Anal.: (RWA), 7 (2006), 265-284.  doi: 10.1016/j.nonrwa.2005.03.002. [24] 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. [25] H. R. Thieme, Convergence results and a Poincar$\acute{e}$-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755-763.  doi: 10.1007/BF00173267. [26] L. Wang, C. Dai and M. Zhao, Hopf bifurcation in an age-structured prey-predator model with Holling Ⅲ response function, Math. Biosc. Eng., 18 (2021), 3144-3159.  doi: 10.3934/mbe.2021156. [27] 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. [28] D. Xiao and W. Li, Stability and bifurcation in a delayed ratio-dependent predator-prey syste, Proc. Edinburgh Math. Soc., 46 (2003), 205-220.  doi: 10.1017/S0013091500001140. [29] 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. [30] X. Zhang and Z. Liu, Hopf bifurcation analysis in a predator-prey model with predator-age structure and predator-prey reaction time delay, Appl. Math. Model., 91 (2021), 530-548.  doi: 10.1016/j.apm.2020.08.054. [31] J. Zhou, Bifurcation analysis of a diffusive predator-prey model with ratio-dependent Holling type Ⅲ functional response, Nonl. Dynam., 81 (2015), 1535-1552.  doi: 10.1007/s11071-015-2088-z. [32] G. Zhu and J. Wei, Global stability and bifurcation analysis of a delayed predator-prey system with prey immigration, Electron. J. Qual. Theory Differ. Equ., 13 (2016), Paper No. 13, 20 pp. doi: 10.14232/ejqtde.2016.1.13.
(a) Time series of $V(t)$. (b) Time series of $P(t)$. (c) Solutions of the system (1.1) go to the boundary equilibrium. (d) Time series of $p(t,a)$. In the four pictures, the lines $a,\; b,\; c$ correspond to three different sets of initial values
(a) Time series of $V(t)$. (b) Time series of $P(t)$. (c) Solutions of the system (1.1) go to the positive steady state when $\tau_1 = \tau_2 = 0.3$. (d) Time series of $p(t,a)$
(a) Time series of $V(t)$. (b) Time series of $P(t)$. (c) Solutions of the system (1.1) oscillates periodically when $\tau_1 = \tau_2 = 1.2$. (d) Time series of $p(t,a)$
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