# American Institute of Mathematical Sciences

November  2018, 17(6): 2703-2727. doi: 10.3934/cpaa.2018128

## On a predator prey model with nonlinear harvesting and distributed delay

 1 Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, c/ Tarfia s/n, 41012 Sevilla, Spain 2 Department of Engineering, Niccolò Cusano University, via Don Carlo Gnocchi 3, 00166 Roma, Italy 3 Department of Management, Università Politecnica delle Marche, Piazza Martelli 8, 60121 Ancona, Italy

Received  September 2017 Revised  February 2018 Published  June 2018

Fund Project: This work has been supported by grant MTM2015-63723-P (MINECO/FEDER, EU) and Consejería de Innovación, Ciencia y Empresa (Junta de Andalucía) under grant 2010/FQM314, and Proyecto de Excelencia P12-FQM-1492

A predator prey model with nonlinear harvesting (Holling type-Ⅱ) with both constant and distributed delay is considered. The boundeness of solutions is proved and some sufficient conditions ensuring the persistence of the two populations are established. Also, a detailed study of the bifurcation of positive equilibria is provided. All the results are illustrated by some numerical simulations.

Citation: Tomás Caraballo, Renato Colucci, Luca Guerrini. On a predator prey model with nonlinear harvesting and distributed delay. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2703-2727. doi: 10.3934/cpaa.2018128
##### References:

show all references

##### References:
The solution $u$ and $v$ for $\tau = 2$, the fixed point $(u^*, v^*)\approx(0.31, 0.68)$ is locally asymptotically stable
The solution $u$ and $v$ in the plane, for $\tau = 5$, the fixed point (in red) $(u^*, v^*)\approx(0.31, 0.68)$ is unstable. A stable limit cycle appears, the time series of $u$ and $v$ appears periodic
The vector field for $v = 0$ and $u, x\geq0$
The time series of $u$ and $v$ for $T = 40$. The solution converges slowly to the asymptotically stable fixed point $(u_*, x_*, v_*)$
The solution for $T = 40.5$, a stable limit cycle appears. The fixed point $(u_*, x_*, v_*)$ (in red) is unstable. The time series of $u$, , $v$ approach the limit cycle
For $T = 1.5 < T_*$ the fixed point $(u_*, z_*, y_*, v_*)$ is locally asymptotically stable
For $T = 2.5>T_*$ the fixed point $(u_*, z_*, y_*, v_*)$ is unstable and a stable limit cycle appears. In the figures it is represented the limit cycle together with the time series of $u$ and $v$ respectively
For $T = 3>T_*$ the fixed point $(u_*, z_*, y_*, v_*)$ is unstable and a stable limit cycle appears. In the figures it is represented the limit cycle together with the time series of $u$ and $v$ respectively
For $T = 3.132>T_*$ we observe a limit cycle with three periods. In the figures it is represented the limit cycle together with the time series of $u$ and $v$ respectively
For $T = 3.2>T_*$ we observe a limit cycle with four periods. In the figures it is represented the limit cycle together with the time series of $u$ and $v$ respectively
For $T = 4>T_*$ we observe a possible chaotic attractor which is represented together with the time series of $u$ and $v$ respectively
 [1] Alfonso Ruiz-Herrera. Chaos in delay differential equations with applications in population dynamics. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1633-1644. doi: 10.3934/dcds.2013.33.1633 [2] Hui Wan, Huaiping Zhu. A new model with delay for mosquito population dynamics. Mathematical Biosciences & Engineering, 2014, 11 (6) : 1395-1410. doi: 10.3934/mbe.2014.11.1395 [3] Robert Skiba, Nils Waterstraat. The index bundle and multiparameter bifurcation for discrete dynamical systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5603-5629. doi: 10.3934/dcds.2017243 [4] Zhihua Liu, Hui Tang, Pierre Magal. Hopf bifurcation for a spatially and age structured population dynamics model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1735-1757. doi: 10.3934/dcdsb.2015.20.1735 [5] Cecilia Cavaterra, M. Grasselli. Robust exponential attractors for population dynamics models with infinite time delay. Discrete & Continuous Dynamical Systems - B, 2006, 6 (5) : 1051-1076. doi: 10.3934/dcdsb.2006.6.1051 [6] Suqi Ma, Qishao Lu, Shuli Mei. Dynamics of a logistic population model with maturation delay and nonlinear birth rate. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 735-752. doi: 10.3934/dcdsb.2005.5.735 [7] Gang Huang, Yasuhiro Takeuchi, Rinko Miyazaki. Stability conditions for a class of delay differential equations in single species population dynamics. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2451-2464. doi: 10.3934/dcdsb.2012.17.2451 [8] Tomas Alarcon, Philipp Getto, Anna Marciniak-Czochra, Maria dM Vivanco. A model for stem cell population dynamics with regulated maturation delay. Conference Publications, 2011, 2011 (Special) : 32-43. doi: 10.3934/proc.2011.2011.32 [9] MirosŁaw Lachowicz, Tatiana Ryabukha. Equilibrium solutions for microscopic stochastic systems in population dynamics. Mathematical Biosciences & Engineering, 2013, 10 (3) : 777-786. doi: 10.3934/mbe.2013.10.777 [10] Bedr'Eddine Ainseba. Age-dependent population dynamics diffusive systems. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 1233-1247. doi: 10.3934/dcdsb.2004.4.1233 [11] Meng Liu, Ke Wang. Population dynamical behavior of Lotka-Volterra cooperative systems with random perturbations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2495-2522. doi: 10.3934/dcds.2013.33.2495 [12] Meng Liu, Chuanzhi Bai, Yi Jin. Population dynamical behavior of a two-predator one-prey stochastic model with time delay. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2513-2538. doi: 10.3934/dcds.2017108 [13] Julián López-Gómez, Marcela Molina-Meyer, Andrea Tellini. Intricate bifurcation diagrams for a class of one-dimensional superlinear indefinite problems of interest in population dynamics. Conference Publications, 2013, 2013 (special) : 515-524. doi: 10.3934/proc.2013.2013.515 [14] Tomás Caraballo, Francisco Morillas, José Valero. On differential equations with delay in Banach spaces and attractors for retarded lattice dynamical systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 51-77. doi: 10.3934/dcds.2014.34.51 [15] Silviu-Iulian Niculescu, Peter S. Kim, Keqin Gu, Peter P. Lee, Doron Levy. Stability crossing boundaries of delay systems modeling immune dynamics in leukemia. Discrete & Continuous Dynamical Systems - B, 2010, 13 (1) : 129-156. doi: 10.3934/dcdsb.2010.13.129 [16] Felix X.-F. Ye, Hong Qian. Stochastic dynamics Ⅱ: Finite random dynamical systems, linear representation, and entropy production. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 4341-4366. doi: 10.3934/dcdsb.2019122 [17] Mohammadreza Molaei. Hyperbolic dynamics of discrete dynamical systems on pseudo-riemannian manifolds. Electronic Research Announcements, 2018, 25: 8-15. doi: 10.3934/era.2018.25.002 [18] Bixiang Wang. Stochastic bifurcation of pathwise random almost periodic and almost automorphic solutions for random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3745-3769. doi: 10.3934/dcds.2015.35.3745 [19] J. C. Robinson. A topological time-delay embedding theorem for infinite-dimensional cocycle dynamical systems. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 731-741. doi: 10.3934/dcdsb.2008.9.731 [20] Yilei Tang. Global dynamics and bifurcation of planar piecewise smooth quadratic quasi-homogeneous differential systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2029-2046. doi: 10.3934/dcds.2018082

2018 Impact Factor: 0.925

## Metrics

• HTML views (134)
• Cited by (0)

• on AIMS