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Optimal control for a conserved phase field system with a possibly singular potential
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 |
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.
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.
|
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.
|
[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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