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 & 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. DawidowiczA. 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. DelgadoM. 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. MohrM. 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. XuT. 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. DawidowiczA. 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. DelgadoM. 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. MohrM. 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. XuT. 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]

Laurent Di Menza, Virginie Joanne-Fabre. An age group model for the study of a population of trees. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020464

[2]

Chao Xing, Jiaojiao Pan, Hong Luo. Stability and dynamic transition of a toxin-producing phytoplankton-zooplankton model with additional food. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020275

[3]

A. M. Elaiw, N. H. AlShamrani, A. Abdel-Aty, H. Dutta. Stability analysis of a general HIV dynamics model with multi-stages of infected cells and two routes of infection. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020441

[4]

Laurence Cherfils, Stefania Gatti, Alain Miranville, Rémy Guillevin. Analysis of a model for tumor growth and lactate exchanges in a glioma. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020457

[5]

Fathalla A. Rihan, Hebatallah J. Alsakaji. Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020468

[6]

Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432

[7]

Weiwei Liu, Jinliang Wang, Yuming Chen. Threshold dynamics of a delayed nonlocal reaction-diffusion cholera model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020316

[8]

Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020317

[9]

Yining Cao, Chuck Jia, Roger Temam, Joseph Tribbia. Mathematical analysis of a cloud resolving model including the ice microphysics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 131-167. doi: 10.3934/dcds.2020219

[10]

Zhouchao Wei, Wei Zhang, Irene Moroz, Nikolay V. Kuznetsov. Codimension one and two bifurcations in Cattaneo-Christov heat flux model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020344

[11]

Shuang Chen, Jinqiao Duan, Ji Li. Effective reduction of a three-dimensional circadian oscillator model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020349

[12]

Barbora Benešová, Miroslav Frost, Lukáš Kadeřávek, Tomáš Roubíček, Petr Sedlák. An experimentally-fitted thermodynamical constitutive model for polycrystalline shape memory alloys. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020459

[13]

Cuicui Li, Lin Zhou, Zhidong Teng, Buyu Wen. The threshold dynamics of a discrete-time echinococcosis transmission model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020339

[14]

Yolanda Guerrero–Sánchez, Muhammad Umar, Zulqurnain Sabir, Juan L. G. Guirao, Muhammad Asif Zahoor Raja. Solving a class of biological HIV infection model of latently infected cells using heuristic approach. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020431

[15]

H. M. Srivastava, H. I. Abdel-Gawad, Khaled Mohammed Saad. Oscillatory states and patterns formation in a two-cell cubic autocatalytic reaction-diffusion model subjected to the Dirichlet conditions. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020433

[16]

Hai-Feng Huo, Shi-Ke Hu, Hong Xiang. Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment. Electronic Research Archive, , () : -. doi: 10.3934/era.2020118

[17]

Youming Guo, Tingting Li. Optimal control strategies for an online game addiction model with low and high risk exposure. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020347

[18]

Omid Nikan, Seyedeh Mahboubeh Molavi-Arabshai, Hossein Jafari. Numerical simulation of the nonlinear fractional regularized long-wave model arising in ion acoustic plasma waves. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020466

[19]

Bernard Bonnard, Jérémy Rouot. Geometric optimal techniques to control the muscular force response to functional electrical stimulation using a non-isometric force-fatigue model. Journal of Geometric Mechanics, 2020  doi: 10.3934/jgm.2020032

[20]

Ebraheem O. Alzahrani, Muhammad Altaf Khan. Androgen driven evolutionary population dynamics in prostate cancer growth. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020426

2019 Impact Factor: 0.953

Metrics

  • PDF downloads (239)
  • HTML views (493)
  • Cited by (1)

Other articles
by authors

[Back to Top]