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Periodic trajectories for an age-structured prey–predator system with Michaelis–Menten functional response including delays and asymmetric diffusion

Part of: Infinite-dimensional dissipative dynamical systems Parabolic equations and systems

Published online by Cambridge University Press:  12 May 2021

PENG YANG Open the ORCID record for PENG YANG [Opens in a new window]  and
YUANSHI WANG
PENG YANG
Affiliation:
School of Mathematics, Sun Yat-sen University, Guangzhou510275, People’s Republic of China emails: [email protected]; [email protected]
YUANSHI WANG
Affiliation:
School of Mathematics, Sun Yat-sen University, Guangzhou510275, People’s Republic of China emails: [email protected]; [email protected]
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Abstract

This paper studies the periodic trajectories of a novel age-structured prey–predator system with Michaelis–Menten functional response including delays and asymmetric diffusion. To begin with, the system is turned into an abstract non-densely defined Cauchy problem, and a time-lag effect in their interaction is investigated. Next, we acquire that this system appears a periodic orbit near the positive steady state by employing the method of integrated semigroup and the Hopf bifurcation theory for semilinear equations with non-dense domain, which is also the main result of this article. Finally, in order to illustrate our theoretical analysis more vividly, we make some numerical simulations and give some discussions.

Keywords

Periodic trajectoriesage structureMichaelis–Menten functional responsedelaysasymmetric diffusion

MSC classification

Secondary: 35K90: Abstract parabolic equations 37L15: Stability problems 92D25: Population dynamics (general)
Type
Papers
Information
European Journal of Applied Mathematics , Volume 33 , Issue 4 , August 2022 , pp. 587 - 605
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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Footnotes

†

This work was supported by NSF of P.R. China (12071495, 11571382).

References

Anita, S. (2000) Analysis and Control of Age-Dependent Population Dynamics , Mathematical Modelling: Theory and Applications, Vol. 11, Springer, Netherlands.Google Scholar
Auslander, D. M., Oster, G. F. & Huffaker, C. B. (1974) Dynamics of interacting populations. J. Franklin Inst. 297(5), 345–376.CrossRefGoogle Scholar
Chen, L. J., Chen, F. D. & Chen, L. J. (2010) Qualitative analysis of a predator–prey model with Holling type II functional response incorporating a constant prey refuge. Nonlinear Anal. Real World Appl. 11(1), 246–252.CrossRefGoogle Scholar
Chu, J. X., Ducrot, A., Magal, P. & Ruan, S. G. (2009) Hopf bifurcation in a size-structured population dynamic model with random growth. J. Differ. Equations 247(3), 956–1000.CrossRefGoogle Scholar
Cushing, J. M. (1984) Existence and stability of equilibria in age-structured population dynamics. J. Math. Biol. 20(3), 259–276.CrossRefGoogle Scholar
Cushing, J. M. & Saleem, M. (1982) A predator prey model with age structure. J. Math. Biol. 14(2), 231–250.CrossRefGoogle ScholarPubMed
Ducrot, A., Liu, Z. H. & Magal, P. (2008) Essential growth rate for bounded linear perturbation of non-densely defined Cauchy problems. J. Math. Anal. Appl. 341(1), 501–518.CrossRefGoogle Scholar
Ducrot, A., Magal, P. & Ruan, S. G. (2013) Projectors on the generalized eigenspaces for partial differential equations with time delay. Infin. Dimension. Dyn. Syst. 64, 353–390.CrossRefGoogle Scholar
Freedman, H. I. (1980) Deterministic mathematical models in population ecology. Biometrics 22(7), 219–236.Google Scholar
Gurtin, M. E. & Levine, D. S. (1979) On predator–prey interactions with predation dependent on age of prey. Math. Biosci. 47(3), 207–219.CrossRefGoogle Scholar
Hausrath, A. R. (1994) Analysis of a model predator–prey system with refuges. J. Math. Anal. Appl. 181(2) 531–545.CrossRefGoogle Scholar
Iannelli, M. (1995) Mathematical Theory of Age-Structured Population Dynamics , Applied Mathematics Monographs, Vol. 7, Giardini editori e stampatori.Google Scholar
Ko, W. & Ryu, K. (2006) Qualitative analysis of a predator–prey model with Holling type II functional response incorporating a prey refuge. J. Differ. Equations 231(2), 534–550.CrossRefGoogle Scholar
Kumar, K. T. (2005) Stability analysis of a prey–predator model incorporating a prey refuge. Commun. Nonlinear Sci. Numer. Simul. 10(6), 681–691.Google Scholar
Leung, A. (1979) Conditions for global stability concerning a prey–predator model with delay effects. SIAM J. Appl. Math. 36(2), 281–286.CrossRefGoogle Scholar
Liu, Z. H. & Li, N. W. (2015) Stability and bifurcation in a predator–prey model with age structure and delays. J. Nonlinear Sci. 25(4), 937–957.CrossRefGoogle Scholar
Liu, Z. H., Magal, P. & Ruan, S. G. (2011) Hopf bifurcation for non-densely defined cauchy problems. Zeitschrift Fur Angewandte Mathematik Und Physik 62(2), 191–222.CrossRefGoogle Scholar
Macdonald, N. (1977) Time delay in prey–predator models-II. bifurcation theory. Math. Biosci. 33(3), 227–234.CrossRefGoogle Scholar
Magal, P. (2001) Compact attractors for time-periodic age-structured population models. Electr. J. Differ. Equations 2001(65), 229–262.Google Scholar
Magal, P. & Ruan, S. G. (2009) On semilinear cauchy problems with non-dense domain. Adv. Differ. Equations 14(11–12), 1041–1084.Google Scholar
Metz, J. A. J. & Diekmann, O. (1986) The Dynamics of Physiologically Structured Populations , Lecture Notes in Biomathematics, Vol. 68, Springer-Verlag, Berlin, Heidelberg.Google Scholar
Murray, J. D. (1998) Mathematical Biology, Springer-Verlag, Berlin.Google Scholar
Prajneshu, (1980) A stochastic prey–predator model with time delay. Math. Biosci. 52(3), 217–226.CrossRefGoogle Scholar
Thieme, H. R. (1990) “Integrated semigroups” and integrated solutions to abstract cauchy problems. J. Math. Anal. Appl. 152(2), 416–447.CrossRefGoogle Scholar
Thieme, H. R. (1997) Quasi-compact semigroups via bounded perturbation. In: O. Arino, D. Axelrod and M. Kimmel (editors.), Advances in Mathematical Population Dynamics-molecules Cells & Man, World Scientific Publishing, River Edge, NJ, pp. 691–711.Google Scholar
Webb, G. F. (1985) Theory of Nonlinear Age-Dependent Population Dynamics, Chapman & Hall Pure and Applied Mathematics, CRC Press, New York and Basel.Google Scholar