On a predator prey model with nonlinear harvesting and distributed delay

Tomás Caraballo; Renato Colucci; Luca Guerrini

doi:10.3934/cpaa.2018128

Article Contents

2018, Volume 17, Issue 6: 2703-2727. Doi: 10.3934/cpaa.2018128

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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: August 31, 2017

Revised: January 31, 2018

Published: June 2018

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.

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Abstract

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.

Keywords:

Mathematics Subject Classification: 92D25, 34C23, 34K60.

Citation:

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Figure 1. The solution $u$ and $v$ for $\tau = 2$, the fixed point $(u^*, v^*)\approx(0.31, 0.68)$ is locally asymptotically stable

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Figure 2. 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

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Figure 3. The vector field for $v = 0$ and $u, x\geq0$

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Figure 4. The time series of $u$ and $v$ for $T = 40$. The solution converges slowly to the asymptotically stable fixed point $(u_*, x_*, v_*)$

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Figure 5. 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

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Figure 6. For $T = 1.5 < T_*$ the fixed point $(u_*, z_*, y_*, v_*)$ is locally asymptotically stable

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Figure 7. 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

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Figure 8. 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

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Figure 9. 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

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Figure 10. 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

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Figure 11. For $T = 4>T_*$ we observe a possible chaotic attractor which is represented together with the time series of $u$ and $v$ respectively

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