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:
[1]

L. ChangG. Q. SunZ. Jin and Z. Wang, Rich dynamics in a spatial predator-prey model with delay, Appl. Math. Comput, 256(C) (2015), 540-550.

[2]

T. DasR. N. Mukherjee and K. S. Chaudhari, Bioeconomic harvesting of a prey-predator fishery, J. Biol. Dyn., 3 (2009), 447-462.

[3]

J. Dieudonné, Foundations of Modern Analysis, Academic Press, New York, 1960.

[4]

R. P. Gupta and P. Chandra, Bifurcation analysis of modified Leslie-Gower predator-prey model with Michaelis-Menten type prey harvesting, J. Math. Anal. Appl., 398 (2013), 278-295.

[5]

R. P. GuptaP. Chandra and M. Banerjee, Dynamical complexity of a prey-predator model with nonlinear predator harvesting, Discrete and continuous dynamical systems series B, 20 (2015), 423-443.

[6]

J. K. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977.

[7]

S. V. KrishnaP. D. N. Srinivasu and B. Prasad Kaymakcalan, Conservation of an ecosystem through optimal taxation, Bull. Math. Biol., 60 (1998), 569-584.

[8]

J. Liu and L. Zhang, Bifurcation analysis in a prey-predator model with nonlinear predator harvesting, Journal of the Franklin Institute, 353 (2016), 4701-4714.

[9]

N. MacDonald, Time Lags in Biological Systems, Springer, New York, 1978.

[10]

T. Pradhan and K. S. Chaudhuri, Bioeconomic harvesting of a schooling fish species: A dynamic reaction model, Korean J. Comput. Appl. Math., 6 (1999), 127-141.

[11]

H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, AMS, Graduate Studies in Mathematics Volume: 118 (2011), 405 pp.

[12]

H. L. Smith and X. Q. Zhao, Global asymptotic stability of traveling waves in delayed reactiondiffusion equations, SIAM J. Math. Anal., 31 (2000), 514-534.

[13]

Y. SongY. Peng and M. Han, Travelling wave fronts in the diffusive single species model with Allee effect and distributed delay, Appl. Math. Comput., 152 (2004), 483-497.

[14]

P. D. N. Srinivasu, Bioeconomics of a renewable resource in presence of a predator, Nonlin. Anal. Real World Appl., 2 (2001), 497-506.

[15]

J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay, J.Dyn.Diff.Equ., 13 (2001), 651-687.

[16]

R. Xu and Z. Ma, An HBV model with diffusion and time delay, J. Theor. Biol., 257 (2009), 499-509.

show all references

References:
[1]

L. ChangG. Q. SunZ. Jin and Z. Wang, Rich dynamics in a spatial predator-prey model with delay, Appl. Math. Comput, 256(C) (2015), 540-550.

[2]

T. DasR. N. Mukherjee and K. S. Chaudhari, Bioeconomic harvesting of a prey-predator fishery, J. Biol. Dyn., 3 (2009), 447-462.

[3]

J. Dieudonné, Foundations of Modern Analysis, Academic Press, New York, 1960.

[4]

R. P. Gupta and P. Chandra, Bifurcation analysis of modified Leslie-Gower predator-prey model with Michaelis-Menten type prey harvesting, J. Math. Anal. Appl., 398 (2013), 278-295.

[5]

R. P. GuptaP. Chandra and M. Banerjee, Dynamical complexity of a prey-predator model with nonlinear predator harvesting, Discrete and continuous dynamical systems series B, 20 (2015), 423-443.

[6]

J. K. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977.

[7]

S. V. KrishnaP. D. N. Srinivasu and B. Prasad Kaymakcalan, Conservation of an ecosystem through optimal taxation, Bull. Math. Biol., 60 (1998), 569-584.

[8]

J. Liu and L. Zhang, Bifurcation analysis in a prey-predator model with nonlinear predator harvesting, Journal of the Franklin Institute, 353 (2016), 4701-4714.

[9]

N. MacDonald, Time Lags in Biological Systems, Springer, New York, 1978.

[10]

T. Pradhan and K. S. Chaudhuri, Bioeconomic harvesting of a schooling fish species: A dynamic reaction model, Korean J. Comput. Appl. Math., 6 (1999), 127-141.

[11]

H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, AMS, Graduate Studies in Mathematics Volume: 118 (2011), 405 pp.

[12]

H. L. Smith and X. Q. Zhao, Global asymptotic stability of traveling waves in delayed reactiondiffusion equations, SIAM J. Math. Anal., 31 (2000), 514-534.

[13]

Y. SongY. Peng and M. Han, Travelling wave fronts in the diffusive single species model with Allee effect and distributed delay, Appl. Math. Comput., 152 (2004), 483-497.

[14]

P. D. N. Srinivasu, Bioeconomics of a renewable resource in presence of a predator, Nonlin. Anal. Real World Appl., 2 (2001), 497-506.

[15]

J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay, J.Dyn.Diff.Equ., 13 (2001), 651-687.

[16]

R. Xu and Z. Ma, An HBV model with diffusion and time delay, J. Theor. Biol., 257 (2009), 499-509.

Figure 1.  The solution $u$ and $v$ for $\tau = 2$, the fixed point $(u^*, v^*)\approx(0.31, 0.68)$ is locally asymptotically stable
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
Figure 3.  The vector field for $v = 0$ and $u, x\geq0$
Figure 4.  The time series of $u$ and $v$ for $T = 40$. The solution converges slowly to the asymptotically stable fixed point $(u_*, x_*, v_*)$
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
Figure 6.  For $T = 1.5 < T_*$ the fixed point $(u_*, z_*, y_*, v_*)$ is locally asymptotically stable
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
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
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
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
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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