# American Institute of Mathematical Sciences

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

## Stability problem for the age-dependent predator-prey model

 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.

Citation: Antoni Leon Dawidowicz, Anna Poskrobko. Stability problem for the age-dependent predator-prey model. Evolution Equations and 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. [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. [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. [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. [5] S. Busenberg and M. Iannelli, Separable models in age-dependent population-dynamics, J. Math. Biol., 22 (1985), 145-173.  doi: 10.1007/BF00275713. [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. [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. [8] A. L. Dawidowicz and A. Poskrobko, Age-dependent single-species population dynamics with delayed argument, Math. Methods Appl. Sci., 33 (2010), 1122-1135. [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. [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. [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. [12] B. Dubey, A prey-predator model with a reserved area, Nonlinear Anal. Model. Control, 12 (2007), 479-494. [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. [14] U. Foryś, Multi-dimensional Lotka-Volterra systems for carcinogenesis mutations, Math. Methods Appl. Sci., 32 (2009), 2287-2308.  doi: 10.1002/mma.1137. [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. [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. [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. [18] A. G. McKendrick, Applications of mathematics to medical problems, Proc. Edinburgh Math. Soc., 44 (1925), 98-130.  doi: 10.1017/S0013091500034428. [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. [20] M. Saleem, Predator-prey relationships: Indiscriminate predation, J. Math. Biol., 21 (1984), 25-34.  doi: 10.1007/BF00275220. [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. [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. [23] E. Venturino, Age-structured predator-prey models, Math. Modelling, 5 (1984), 117-128.  doi: 10.1016/0270-0255(84)90020-4. [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. [25] J. von Foerster, Some Remarks on Changing Populations In: The Kinetics of Cell Proliferation, Grune & Stratton, New York, 1959. [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.

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##### References:
 [1] N. C. Apreutesei, Necessary optimality conditions for a Lotka-Volterra three species system, Math. Model. Nat. Phenom., 1 (2006), 123-135. [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. [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. [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. [5] S. Busenberg and M. Iannelli, Separable models in age-dependent population-dynamics, J. Math. Biol., 22 (1985), 145-173.  doi: 10.1007/BF00275713. [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. [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. [8] A. L. Dawidowicz and A. Poskrobko, Age-dependent single-species population dynamics with delayed argument, Math. Methods Appl. Sci., 33 (2010), 1122-1135. [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. [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. [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. [12] B. Dubey, A prey-predator model with a reserved area, Nonlinear Anal. Model. Control, 12 (2007), 479-494. [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. [14] U. Foryś, Multi-dimensional Lotka-Volterra systems for carcinogenesis mutations, Math. Methods Appl. Sci., 32 (2009), 2287-2308.  doi: 10.1002/mma.1137. [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. [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. [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. [18] A. G. McKendrick, Applications of mathematics to medical problems, Proc. Edinburgh Math. Soc., 44 (1925), 98-130.  doi: 10.1017/S0013091500034428. [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. [20] M. Saleem, Predator-prey relationships: Indiscriminate predation, J. Math. Biol., 21 (1984), 25-34.  doi: 10.1007/BF00275220. [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. [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. [23] E. Venturino, Age-structured predator-prey models, Math. Modelling, 5 (1984), 117-128.  doi: 10.1016/0270-0255(84)90020-4. [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. [25] J. von Foerster, Some Remarks on Changing Populations In: The Kinetics of Cell Proliferation, Grune & Stratton, New York, 1959. [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.
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