# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2021209
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## On oscillations to a 2D age-dependent predation equations characterizing Beddington-DeAngelis type schemes

 School of Mathematics, Sun Yat-sen University, Guangzhou 510275, China

* Corresponding author: Peng Yang

Received  February 2021 Revised  June 2021 Early access August 2021

Fund Project: The second author is supported by NSF grant of China (12071495, 11571382)

In this study, a 2D age-dependent predation equations characterizing Beddington$-$DeAngelis type schemes are established to investigate the evolutionary dynamics of population, in which the predator is selected to be depicted with an age structure and its fertility function is assumed to be a step function. The dynamic behaviors of the equations are derived from the integrated semigroup method, the Hopf bifurcation theorem, the center manifold reduction and normal form theory of semilinear equations with non-dense domain. It turns out that the equations appear the oscillation phenomenon via Hopf bifurcation (positive equilibrium age distribution lose its stability and give rise to periodic solutions), as the bifurcation parameter moves across certain threshold values. Additionally, the explicit expressions are offered to determine the properties of Hopf bifurcation (the direction the Hopf bifurcation and the stability of the bifurcating periodic solutions). This technique can also be employed to other epidemic and ecological equations. Eventually, some numerical simulations and conclusions are executed to validating the major results of this work.

Citation: Peng Yang, Yuanshi Wang. On oscillations to a 2D age-dependent predation equations characterizing Beddington-DeAngelis type schemes. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021209
##### References:
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show all references

##### References:
 [1] S. Aniţa, Analysis and Control of Age-Dependent Population Dynamics, Kluwer Academic Publishers, Dordrecht, 2000. doi: 10.1007/978-94-015-9436-3.  Google Scholar [2] J. R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, J. Anim. Ecol., 44 (1975), 331-340.  doi: 10.2307/3866.  Google Scholar [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.  Google Scholar [4] S. N. Chow and J. K. Hale, Methods of Bifurcation Theory, Springer-Verlag, New York, 1982.  Google Scholar [5] J. Chu, Z. Liu, P. Magal and S. Ruan, Normal forms for an age structured model, J. Dyn. Differ. Equ., 28 (2016), 733-761.  doi: 10.1007/s10884-015-9500-8.  Google Scholar [6] P. H. Crowley and E. K. Martin, Functional responses and interference within and between year classes of a dragonfly population, J. N. Am. Benthol. Soc., 8 (1989), 211-221.  doi: 10.2307/1467324.  Google Scholar [7] D. L. Deangelis, R. A. Goldstein and R. V. O'neill, A model for tropic interaction, Ecology, 56 (1975), 881-892.  doi: 10.2307/1936298.  Google Scholar [8] 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.  Google Scholar [9] C. S. Holling, The functional response of predators to prey density and its role in mimicry and population regulation, Mem. Entomol. Sci. Can., 97 (1965), 5-60.  doi: 10.4039/entm9745fv.  Google Scholar [10] H.-F. Huo, P. Yang and H. Xiang, Dynamics for an SIRS epidemic model with infection age and relapse on a scale-free network, J. Franklin Inst., 356 (2019), 7411-7443.  doi: 10.1016/j.jfranklin.2019.03.034.  Google Scholar [11] T.-W. Hwang, Global analysis of the predator$-$prey system with Beddington-Deangelis functional response, J. Math. Anal. Appl., 281 (2003), 395-401.  doi: 10.1016/S0022-247X(02)00395-5.  Google Scholar [12] M. Iannelli, Mathematical Theory of Age-Structured Population Dynamics, Giardini Editori E Stampatori, Pisa, 1995. Google Scholar [13] 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.  Google Scholar [14] Z. Liu, P. Magal and S. Ruan, Normal forms for semilinear equations with non-dense domain with applications to age structured models, J. Differ. Equ., 257 (2014), 921-1011.  doi: 10.1016/j.jde.2014.04.018.  Google Scholar [15] A. J. Lotka, Elements of Physical Biology, Williams and Wilkins, Baltimore, 1925. Google Scholar [16] P. Magal, Compact attractors for time-periodic age-structured population models, Electron. J. Differ. Equ., 2001 (2001), No. 65, 35 pp.  Google Scholar [17] P. Magal and S. Ruan, Center Manifolds for Semilinear Equations with Non-dense Domain and Applications to Hopf bifurcation in Age Structured Models, Mem. Am. Math. Soc., 202 2009. doi: 10.1090/S0065-9266-09-00568-7.  Google Scholar [18] P. Magal and S. Ruan, On semilinear cauchy problems with non-dense domain, Adv. Differ. Equat., 14 (2009), 1041-1084.   Google Scholar [19] J. A. J. Metz and O. Diekmann, The Dynamics of Physiologically Structured Populations, Springer-Verlag, Berlin Heidelberg, 1986. doi: 10.1007/978-3-662-13159-6.  Google Scholar [20] J. D. Murray, Mathematical Biology, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-08539-4.  Google Scholar [21] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar [22] G. T. Skalski and J. F. Gilliam, Functional response with predator interference: Viable alternatives to the Holling type II model, Ecology, 82 (2001), 3083-3092.  doi: 10.1890/0012-9658(2001)082[3083:FRWPIV]2.0.CO;2.  Google Scholar [23] H. Tang and Z. H. 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.  Google Scholar [24] H. R. Thieme, Quasi-compact semigroups via bounded perturbation, in: O. Arino, D. Axelrod, M. Kimmel (Eds.), Advances in Mathematical Population Dynamics-molecules Cells and Man, World Scientific Publishing, River Edge, NJ., (1997) 691–711.  Google Scholar [25] H. R. Thiemea, "Integrated semigroups" and integrated solutions to abstract Cauchy problems, J. Math. Anal. Appl., 152 (1990), 416-447.  doi: 10.1016/0022-247X(90)90074-P.  Google Scholar [26] H. R. Thiemea, Semiflows generated by Lipschitz perturbations of non-densely defined operators, Differ. Integral Equ., 3 (1990), 1035-1066.   Google Scholar [27] V. Volterra, Variazioni e fluttuazioni delnumero d'individui in specie animali conviventi, 2 (1926), 31–113. Google Scholar [28] 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.  Google Scholar [29] G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, Marcel Dekker, Inc., New York, 1985.  Google Scholar [30] 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.  Google Scholar [31] 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.  Google Scholar [32] 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. doi: 10.1016/j.cnsns.2020.105395.  Google Scholar [33] 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. Differ. Equ., 269 (2020), 9583-9618.  doi: 10.1016/j.jde.2020.06.048.  Google Scholar [34] P. Yang and Y. Wang, Periodic solutions of a delayed eco-epidemiological model with infection-age structure and Holling type II functional response, Int. J. Bifurcation Chaos, 30 (2020), 2050011, 20 pp. doi: 10.1142/S021812742050011X.  Google Scholar [35] X. Zhang and Z. Liu, Bifurcation analysis of an age structured HIV infection model with both virus-to-cell and cell-to-cell transmissions, Int. J. Bifurcation Chaos, 28 (2018), 1850109, 20 pp. doi: 10.1142/S0218127418501092.  Google Scholar [36] X. Zhang and Z. Liu, Periodic oscillations in age-structured ratio-dependent predator-prey model with Michaelis$-$Menten type functional response, Physica D, 389 (2018), 51-63.  doi: 10.1016/j.physd.2018.10.002.  Google Scholar [37] 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.  Google Scholar
Numerical solutions of equations (BD) as $\tau = 5 < \tau_0 = 5.2509$: (a) image of the function $q(x)$; (b) solution structure of the predator; (c) solution structure of the prey; (d) phase portrait of equations (BD); (e) distribution function of the predator; (f) trajectory diagram of equations (BD)
Numerical solutions of equations (BD) as $\tau = 7 > \tau_0 = 5.2509$: (a) image of the function $q(x)$; (b) periodic structure of the predator; (c) periodic structure of the prey; (d) phase portrait of equations (BD); (e) distribution function of the predator; (f) trajectory diagram of equations (BD)
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