Stability problem for the age-dependent predator-prey model

Previous Article
Inverse observability inequalities for integrodifferential equations in square domains
EECT Home
This Issue
Next Article
Optimal control for a conserved phase field system with a possibly singular potential

March 2018, 7(1): 79-93. doi: 10.3934/eect.2018005

Stability problem for the age-dependent predator-prey model

Antoni Leon Dawidowicz ^1, and Anna Poskrobko ^2,,

1.	Faculty of Mathematics and Computer Science, Jagiellonian University, ul. Łojasiewicza 6, 30-348 Kraków, Poland
2.	Faculty of Computer Science, Bialystok University of Technology, ul. Wiejska 45A, 15-351 Białystok, Poland

* Corresponding author: Anna Poskrobko, a.poskrobko@pb.edu.pl.

Received December 2016 Revised July 2017 Published January 2018

Fund Project: The contribution of Anna Poskrobko was supported by the Bialystok University of Technology grant S/WI/1/2016 and founded by the resources for research by Ministry of Science and Higher Education.

The paper deals with the age-dependent model which is a generalization of the classical Lotka-Volterra model. Age structure of both species, predators and preys is concerned. The model is based on the system of partial differential and integro-differential equations. We study the existence and uniqueness of the solution for the considered population problem. The stability problem for trivial stationary solution of the model is also proved.

Keywords: Population model, Lotka-Volterra model, age-dependent predator-prey model, stability.

Mathematics Subject Classification: Primary: 35F50, 35B35; Secondary: 92D25, 35R09.

Citation: Antoni Leon Dawidowicz, Anna Poskrobko. Stability problem for the age-dependent predator-prey model. Evolution Equations & Control Theory, 2018, 7 (1) : 79-93. doi: 10.3934/eect.2018005

References:

[1]	N. C. Apreutesei, Necessary optimality conditions for a Lotka-Volterra three species system, Math. Model. Nat. Phenom., 1 (2006), 123-135. Google Scholar
[2]	N. C. Apreutesei, Necessary optimality conditions for predator-prey system with a hunter population, Opuscula Math., 30 (2010), 389-397. doi: 10.7494/OpMath.2010.30.4.389. Google Scholar
[3]	N. Bairagi and D. Jana, Age-structured predator-prey model with habitat complexity: Oscillations and control, Dyn. Syst., 27 (2012), 475-499. doi: 10.1080/14689367.2012.723678. Google Scholar
[4]	A. Bielecki, Une remarque sur la méthode de Banach -Caciopoli -Tikhonov dans la théorie des équations différentielles ordinaires, Bull. Acad. Polon. Sci. Cl. Ⅲ., 4 (1956), 261-264. Google Scholar
[5]	S. Busenberg and M. Iannelli, Separable models in age-dependent population-dynamics, J. Math. Biol., 22 (1985), 145-173. doi: 10.1007/BF00275713. Google Scholar
[6]	L. M. Cai, X. Z. Li, X. Y. Song and J. Y. Yu, Permanence and stability of an age-structured prey-predator system with delays, Discrete Dynam. Nat. Soc., 2007 (2007), Art. ID 54861, 15 pp. Google Scholar
[7]	J. M. Cushing and M. Saleem, A predator prey model with age structure, J. Math. Biol., 14 (1982), 231-250. doi: 10.1007/BF01832847. Google Scholar
[8]	A. L. Dawidowicz and A. Poskrobko, Age-dependent single-species population dynamics with delayed argument, Math. Methods Appl. Sci., 33 (2010), 1122-1135. Google Scholar
[9]	A. L. Dawidowicz, A. Poskrobko and J. L. Zalasiński, On the age-dependent predator-prey model, Appl. Math., 38 (2011), 453-467. doi: 10.4064/am38-4-4. Google Scholar
[10]	M. Delgado, M. Molina-Becerra and A. Suárez, Analysis of an age-structured predator-prey model with disease in the prey, Nonlinear Anal. Real World Appl., 7 (2006), 853-871. doi: 10.1016/j.nonrwa.2005.03.031. Google Scholar
[11]	M. Delgado and A. Suárez, Age-dependent diffusive Lotka-Volterra type system, Math. Comput. Modelling, 45 (2007), 668-680. doi: 10.1016/j.mcm.2006.07.013. Google Scholar
[12]	B. Dubey, A prey-predator model with a reserved area, Nonlinear Anal. Model. Control, 12 (2007), 479-494. Google Scholar
[13]	M. Farkas, On the stability of stationary age distributions, Appl. Math. Comput., 131 (2002), 107-123. doi: 10.1016/S0096-3003(01)00131-X. Google Scholar
[14]	U. Foryś, Multi-dimensional Lotka-Volterra systems for carcinogenesis mutations, Math. Methods Appl. Sci., 32 (2009), 2287-2308. doi: 10.1002/mma.1137. Google Scholar
[15]	M. E. Gurtin and D. S. Levine, On predator-prey interactions with predation dependent on age of prey, Math. Biosci., 47 (1979), 207-219. doi: 10.1016/0025-5564(79)90038-5. Google Scholar
[16]	J. Li, Dynamics of age-structured predator-prey population model, J. Math. Anal. Appl., 152 (1990), 399-415. doi: 10.1016/0022-247X(90)90073-O. Google Scholar
[17]	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. Google Scholar
[18]	A. G. McKendrick, Applications of mathematics to medical problems, Proc. Edinburgh Math. Soc., 44 (1925), 98-130. doi: 10.1017/S0013091500034428. Google Scholar
[19]	M. Mohr, M. V. Barbarossa and C. Kuttler, Predator-prey interactions, age structures and delay equations, Math. Model. Nat. Phenom., 9 (2014), 92-107. doi: 10.1051/mmnp/20149107. Google Scholar
[20]	M. Saleem, Predator-prey relationships: Indiscriminate predation, J. Math. Biol., 21 (1984), 25-34. doi: 10.1007/BF00275220. Google Scholar
[21]	M. Saleem and A. K. Tripathi, Asymptotic stability of linear and nonlinear model systems representing age-structured predator-prey interactions, Indian J. Pure Appl. Math., 31 (2000), 1195-1207. Google Scholar
[22]	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. Google Scholar
[23]	E. Venturino, Age-structured predator-prey models, Math. Modelling, 5 (1984), 117-128. doi: 10.1016/0270-0255(84)90020-4. Google Scholar
[24]	V. Volterra, Variations and fluctuations of the number of individuals in animal species living together, ICES Journal of Marine Science,, 3 (1928), 3-51. doi: 10.1093/icesjms/3.1.3. Google Scholar
[25]	J. von Foerster, Some Remarks on Changing Populations In: The Kinetics of Cell Proliferation, Grune & Stratton, New York, 1959.Google Scholar
[26]	W. X. Xu, T. L. Zhangand and Z. B. Xu, Existence of positive periodic solutions of a prey-predator system with several delays, Acta Math. Sci. Ser. A Chinese Ed., 28 (2008), 39-45. Google Scholar

show all references

References:

[1]	N. C. Apreutesei, Necessary optimality conditions for a Lotka-Volterra three species system, Math. Model. Nat. Phenom., 1 (2006), 123-135. Google Scholar
[2]	N. C. Apreutesei, Necessary optimality conditions for predator-prey system with a hunter population, Opuscula Math., 30 (2010), 389-397. doi: 10.7494/OpMath.2010.30.4.389. Google Scholar
[3]	N. Bairagi and D. Jana, Age-structured predator-prey model with habitat complexity: Oscillations and control, Dyn. Syst., 27 (2012), 475-499. doi: 10.1080/14689367.2012.723678. Google Scholar
[4]	A. Bielecki, Une remarque sur la méthode de Banach -Caciopoli -Tikhonov dans la théorie des équations différentielles ordinaires, Bull. Acad. Polon. Sci. Cl. Ⅲ., 4 (1956), 261-264. Google Scholar
[5]	S. Busenberg and M. Iannelli, Separable models in age-dependent population-dynamics, J. Math. Biol., 22 (1985), 145-173. doi: 10.1007/BF00275713. Google Scholar
[6]	L. M. Cai, X. Z. Li, X. Y. Song and J. Y. Yu, Permanence and stability of an age-structured prey-predator system with delays, Discrete Dynam. Nat. Soc., 2007 (2007), Art. ID 54861, 15 pp. Google Scholar
[7]	J. M. Cushing and M. Saleem, A predator prey model with age structure, J. Math. Biol., 14 (1982), 231-250. doi: 10.1007/BF01832847. Google Scholar
[8]	A. L. Dawidowicz and A. Poskrobko, Age-dependent single-species population dynamics with delayed argument, Math. Methods Appl. Sci., 33 (2010), 1122-1135. Google Scholar
[9]	A. L. Dawidowicz, A. Poskrobko and J. L. Zalasiński, On the age-dependent predator-prey model, Appl. Math., 38 (2011), 453-467. doi: 10.4064/am38-4-4. Google Scholar
[10]	M. Delgado, M. Molina-Becerra and A. Suárez, Analysis of an age-structured predator-prey model with disease in the prey, Nonlinear Anal. Real World Appl., 7 (2006), 853-871. doi: 10.1016/j.nonrwa.2005.03.031. Google Scholar
[11]	M. Delgado and A. Suárez, Age-dependent diffusive Lotka-Volterra type system, Math. Comput. Modelling, 45 (2007), 668-680. doi: 10.1016/j.mcm.2006.07.013. Google Scholar
[12]	B. Dubey, A prey-predator model with a reserved area, Nonlinear Anal. Model. Control, 12 (2007), 479-494. Google Scholar
[13]	M. Farkas, On the stability of stationary age distributions, Appl. Math. Comput., 131 (2002), 107-123. doi: 10.1016/S0096-3003(01)00131-X. Google Scholar
[14]	U. Foryś, Multi-dimensional Lotka-Volterra systems for carcinogenesis mutations, Math. Methods Appl. Sci., 32 (2009), 2287-2308. doi: 10.1002/mma.1137. Google Scholar
[15]	M. E. Gurtin and D. S. Levine, On predator-prey interactions with predation dependent on age of prey, Math. Biosci., 47 (1979), 207-219. doi: 10.1016/0025-5564(79)90038-5. Google Scholar
[16]	J. Li, Dynamics of age-structured predator-prey population model, J. Math. Anal. Appl., 152 (1990), 399-415. doi: 10.1016/0022-247X(90)90073-O. Google Scholar
[17]	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. Google Scholar
[18]	A. G. McKendrick, Applications of mathematics to medical problems, Proc. Edinburgh Math. Soc., 44 (1925), 98-130. doi: 10.1017/S0013091500034428. Google Scholar
[19]	M. Mohr, M. V. Barbarossa and C. Kuttler, Predator-prey interactions, age structures and delay equations, Math. Model. Nat. Phenom., 9 (2014), 92-107. doi: 10.1051/mmnp/20149107. Google Scholar
[20]	M. Saleem, Predator-prey relationships: Indiscriminate predation, J. Math. Biol., 21 (1984), 25-34. doi: 10.1007/BF00275220. Google Scholar
[21]	M. Saleem and A. K. Tripathi, Asymptotic stability of linear and nonlinear model systems representing age-structured predator-prey interactions, Indian J. Pure Appl. Math., 31 (2000), 1195-1207. Google Scholar
[22]	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. Google Scholar
[23]	E. Venturino, Age-structured predator-prey models, Math. Modelling, 5 (1984), 117-128. doi: 10.1016/0270-0255(84)90020-4. Google Scholar
[24]	V. Volterra, Variations and fluctuations of the number of individuals in animal species living together, ICES Journal of Marine Science,, 3 (1928), 3-51. doi: 10.1093/icesjms/3.1.3. Google Scholar
[25]	J. von Foerster, Some Remarks on Changing Populations In: The Kinetics of Cell Proliferation, Grune & Stratton, New York, 1959.Google Scholar
[26]	W. X. Xu, T. L. Zhangand and Z. B. Xu, Existence of positive periodic solutions of a prey-predator system with several delays, Acta Math. Sci. Ser. A Chinese Ed., 28 (2008), 39-45. Google Scholar

[1]	Rui Xu, M.A.J. Chaplain, F.A. Davidson. Periodic solutions of a discrete nonautonomous Lotka-Volterra predator-prey model with time delays. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 823-831. doi: 10.3934/dcdsb.2004.4.823
[2]	Michael Y. Li, Xihui Lin, Hao Wang. Global Hopf branches and multiple limit cycles in a delayed Lotka-Volterra predator-prey model. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 747-760. doi: 10.3934/dcdsb.2014.19.747
[3]	S. Nakaoka, Y. Saito, Y. Takeuchi. Stability, delay, and chaotic behavior in a Lotka-Volterra predator-prey system. Mathematical Biosciences & Engineering, 2006, 3 (1) : 173-187. doi: 10.3934/mbe.2006.3.173
[4]	Xiaoli Liu, Dongmei Xiao. Bifurcations in a discrete time Lotka-Volterra predator-prey system. Discrete & Continuous Dynamical Systems - B, 2006, 6 (3) : 559-572. doi: 10.3934/dcdsb.2006.6.559
[5]	Miljana JovanoviĆ, Marija KrstiĆ. Extinction in stochastic predator-prey population model with Allee effect on prey. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2651-2667. doi: 10.3934/dcdsb.2017129
[6]	Yuzo Hosono. Traveling waves for the Lotka-Volterra predator-prey system without diffusion of the predator. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 161-171. doi: 10.3934/dcdsb.2015.20.161
[7]	Hanwu Liu, Lin Wang, Fengqin Zhang, Qiuying Li, Huakun Zhou. Dynamics of a predator-prey model with state-dependent carrying capacity. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4739-4753. doi: 10.3934/dcdsb.2019028
[8]	Xiao He, Sining Zheng. Protection zone in a modified Lotka-Volterra model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2027-2038. doi: 10.3934/dcdsb.2015.20.2027
[9]	P. Auger, N. H. Du, N. T. Hieu. Evolution of Lotka-Volterra predator-prey systems under telegraph noise. Mathematical Biosciences & Engineering, 2009, 6 (4) : 683-700. doi: 10.3934/mbe.2009.6.683
[10]	Sílvia Cuadrado. Stability of equilibria of a predator-prey model of phenotype evolution. Mathematical Biosciences & Engineering, 2009, 6 (4) : 701-718. doi: 10.3934/mbe.2009.6.701
[11]	Shanjing Ren. Global stability in a tuberculosis model of imperfect treatment with age-dependent latency and relapse. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1337-1360. doi: 10.3934/mbe.2017069
[12]	Ovide Arino, Manuel Delgado, Mónica Molina-Becerra. Asymptotic behavior of disease-free equilibriums of an age-structured predator-prey model with disease in the prey. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 501-515. doi: 10.3934/dcdsb.2004.4.501
[13]	Youssef Amal, Martin Campos Pinto. Global solutions for an age-dependent model of nucleation, growth and ageing with hysteresis. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 517-535. doi: 10.3934/dcdsb.2010.13.517
[14]	Peng Feng. On a diffusive predator-prey model with nonlinear harvesting. Mathematical Biosciences & Engineering, 2014, 11 (4) : 807-821. doi: 10.3934/mbe.2014.11.807
[15]	Ronald E. Mickens. Analysis of a new class of predator-prey model. Conference Publications, 2001, 2001 (Special) : 265-269. doi: 10.3934/proc.2001.2001.265
[16]	Yukio Kan-On. Global bifurcation structure of stationary solutions for a Lotka-Volterra competition model. Discrete & Continuous Dynamical Systems - A, 2002, 8 (1) : 147-162. doi: 10.3934/dcds.2002.8.147
[17]	Yueding Yuan, Yang Wang, Xingfu Zou. Spatial dynamics of a Lotka-Volterra model with a shifting habitat. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-39. doi: 10.3934/dcdsb.2019076
[18]	Kaigang Huang, Yongli Cai, Feng Rao, Shengmao Fu, Weiming Wang. Positive steady states of a density-dependent predator-prey model with diffusion. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3087-3107. doi: 10.3934/dcdsb.2017209
[19]	Xiaoling Li, Guangping Hu, Zhaosheng Feng, Dongliang Li. A periodic and diffusive predator-prey model with disease in the prey. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 445-461. doi: 10.3934/dcdss.2017021
[20]	Yang Lu, Xia Wang, Shengqiang Liu. A non-autonomous predator-prey model with infected prey. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3817-3836. doi: 10.3934/dcdsb.2018082

American Institute of Mathematical Sciences