Geodesic
Stability analysis of a tritrophic model with stage structure in the prey population
Blé, Gamaliel1 ; Dela-Rosa, Miguel Angel 2 ; Loreto-Hernández, Iván 2
1 División Académica de Ciencias Básicas, UJAT, Km 1, Carretera Cunduacán-Jalpa de Méndez, Cunduacán, Tabasco, c.p. 86690, México
2 División Académica de Ciencias Básicas, CONACyT-UJAT, Km 1, Carretera Cunduacán-Jalpa de Méndez, Cunduacán, Tabasco, c.p. 86690, México
Journal of nonlinear sciences and its applications, Tome 12 (2019) no. 12, p. 765-790.

Voir la notice de l'article provenant de la source International Scientific Research Publications

We analyze the role of the age structure of a prey in the dynamics of a tritrophic model. We study the effect of predation on a non-reproductive prey class, when the reproductive class of the prey has a defense mechanism. We consider two cases accordingly to the interaction between predator and reproductive class of the prey. In the first case, the functional response is Holling type II and it is possible to show up to two positive equilibria. When we consider a defense mechanism the functional response is Holling type IV. In both cases, we show sufficient parameter conditions to have a stable limit cycle obtained by a supercritical Hopf bifurcation. Some numerical simulations are carried out.
DOI : 10.22436/jnsa.012.12.01
Classification : 37G15, 34C23, 92D40, 92D25
Keywords: Hopf's Bifurcation, tritrophic model, coexistence of species, prey age structure

Blé, Gamaliel 1 ; Dela-Rosa, Miguel Angel  2 ; Loreto-Hernández, Iván  2

1 División Académica de Ciencias Básicas, UJAT, Km 1, Carretera Cunduacán-Jalpa de Méndez, Cunduacán, Tabasco, c.p. 86690, México
2 División Académica de Ciencias Básicas, CONACyT-UJAT, Km 1, Carretera Cunduacán-Jalpa de Méndez, Cunduacán, Tabasco, c.p. 86690, México 
@article{JNSA_2019_12_12_a0,
     author = {Bl\'e, Gamaliel and Dela-Rosa, Miguel Angel  and Loreto-Hern\'andez, Iv\'an },
     title = {Stability analysis of a tritrophic model with stage structure in the prey 	population},
     journal = {Journal of nonlinear sciences and its applications},
     pages = {765-790},
     publisher = {mathdoc},
     volume = {12},
     number = {12},
     year = {2019},
     doi = {10.22436/jnsa.012.12.01},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.22436/jnsa.012.12.01/}
}
TY  - JOUR
AU  - Blé, Gamaliel
AU  - Dela-Rosa, Miguel Angel 
AU  - Loreto-Hernández, Iván 
TI  - Stability analysis of a tritrophic model with stage structure in the prey 	population
JO  - Journal of nonlinear sciences and its applications
PY  - 2019
SP  - 765
EP  - 790
VL  - 12
IS  - 12
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.22436/jnsa.012.12.01/
DO  - 10.22436/jnsa.012.12.01
LA  - en
ID  - JNSA_2019_12_12_a0
ER  - 
%0 Journal Article
%A Blé, Gamaliel
%A Dela-Rosa, Miguel Angel 
%A Loreto-Hernández, Iván 
%T Stability analysis of a tritrophic model with stage structure in the prey 	population
%J Journal of nonlinear sciences and its applications
%D 2019
%P 765-790
%V 12
%N 12
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.22436/jnsa.012.12.01/
%R 10.22436/jnsa.012.12.01
%G en
%F JNSA_2019_12_12_a0
Blé, Gamaliel; Dela-Rosa, Miguel Angel ; Loreto-Hernández, Iván . Stability analysis of a tritrophic model with stage structure in the prey 	population. Journal of nonlinear sciences and its applications, Tome 12 (2019) no. 12, p. 765-790. doi : 10.22436/jnsa.012.12.01. http://geodesic.mathdoc.fr/articles/10.22436/jnsa.012.12.01/

[1] Beddington, J. R.; Free, C. A. Age structure effects in predator-prey interactions, Theoret. Population Biology, Volume 9 (1976), pp. 15-24 | DOI | Zbl

[2] Castellanos, V.; Castillo-Santos, F. E.; Dela-Rosa, M. A.; Loreto-Hernández, I. Hopf and Bautin Bifurcation in a Tritrophic Food Chain Model with Holling Functional Response Types III and IV, Internat. J. Bifur. Chaos Appl. Sci. Engrg., Volume 28 (2018), pp. 1-24 | DOI | Zbl

[3] Cushing, J. M.; Saleem, M. A predator prey model with age structure, J. Math. Biol., Volume 14 (1982), pp. 231-250 | DOI

[4] Dawes, J. H. P.; Souza, M. O. A derivation of Holling's type I, II and III functional responses in predator prey systems, J. Theoret. Biol., Volume 327 (2013), pp. 11-22 | DOI | Zbl

[5] Falconi, M. The effect of the prey age structure on a predator-prey system, Sci. Math. Jpn., Volume 64 (2006), pp. 267-275 | Zbl

[6] Hastings, A.; Wollkind, D. Age structure in predator-prey systems. I. A general model and a specific example, Theoret. Population Biol., Volume 21 (1982), pp. 44-56 | DOI | Zbl

[7] Kuznetsov, Y. A. Elements of applied Bifurcation Theory, Springer-Verlag, New York, 2004 | DOI | Zbl

[8] Kuznetsov, Y. A. Andronov-Hopf bifurcation, Scholarpedia, Volume 1 (2006), pp. 1-6

[9] LeCaven, E. D.; Kipling, C.; McCormack, J. C. A study of the numbers, biomass and year class strengths of pereh (Perca fluviatillis L) in Winderemere from 1941--1966, J. Anim. Ecol., Volume 46 (1977), pp. 281-307

[10] LLoyd, M.; Dybas, H. S. The periodical cicada problem. I. Population ecology, Evolution, Volume 20 (1966), pp. 133-149 | DOI

[11] LLoyd, M.; Dybas, H. S. The periodical cicada problem. II. Evolution, Evolution, Volume 20 (1966), pp. 466-505 | DOI

[12] Marsden, J. E.; McCracken, M. The Hopf Bifurcation and Its Applications, Springer-Verlag, New York, 1976 | DOI | Zbl

[13] Murray, J. D. Mathematical biology. I: An introduction, Springer, New York, 2004

[14] Perko, L. Differential Equations and Dynamical Systems, Springer-Verlag, New York, 2001 | DOI

[15] Promrak, J.; Wake, G. C.; Rattanakul, C. Predator-prey model with age structure, ANZIAM J., Volume 59 (2017), pp. 155-166 | DOI | Zbl

[16] Sigal, R. Algorithms for the Routh-Hurwitz Stability Test, Math. Comput. Modelling, Volume 13 (1990), pp. 69-77 | DOI | Zbl

[17] Skalski, G. T.; Gilliam, J. F. Functional responses with predator interference: Viable alternatives to the Holling type II model, Ecology, Volume 82 (2001), pp. 3083-3092 | DOI

[18] Tang, H.; Liu, Z. Hopf bifurcation for a predator-prey model with agestructure, Appl. Math. Model., Volume 40 (2016), pp. 726-737 | DOI

[19] Toth, D. J. A. Bifurcation structure of a chemostat model for an age-structured predator and its prey, J. Biol. Dyn., Volume 2 (2008), pp. 428-448 | DOI | Zbl

[20] Xi, H. Y.; Huang, L. H.; Qiao, Y. C.; Li, H. Y.; Huang, C. X. Permanence and partial extinction in a delayed three-species food chain model with stage structure and time-varying coefficients, J. Nonlinear Sci. Appl., Volume 10 (2017), pp. 6177-6191 | Zbl

Cité par Sources :