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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Shape optimization approach for solving the Bernoulli problem by tracking the Neumann data: A Lagrangian formulation
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 |
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.
Mathematics Subject Classification:
92D25, 34C23, 34K60.
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
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References:
| [1] |
L. Chang, G. Q. Sun, Z. Jin and Z. Wang,
Rich dynamics in a spatial predator-prey model with delay, Appl. Math. Comput, 256(C) (2015), 540-550.
|
| [2] |
T. Das, R. N. Mukherjee and K. S. Chaudhari, Bioeconomic harvesting of a prey-predator fishery, J. Biol. Dyn., 3 (2009), 447-462. Google Scholar |
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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. Gupta, P. 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. Krishna, P. D. N. Srinivasu and B. Prasad Kaymakcalan, Conservation of an ecosystem through optimal taxation, Bull. Math. Biol., 60 (1998), 569-584. Google Scholar |
| [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. Google Scholar |
| [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. Song, Y. 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. Chang, G. Q. Sun, Z. Jin and Z. Wang,
Rich dynamics in a spatial predator-prey model with delay, Appl. Math. Comput, 256(C) (2015), 540-550.
|
| [2] |
T. Das, R. N. Mukherjee and K. S. Chaudhari, Bioeconomic harvesting of a prey-predator fishery, J. Biol. Dyn., 3 (2009), 447-462. Google Scholar |
| [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. Gupta, P. 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. Krishna, P. D. N. Srinivasu and B. Prasad Kaymakcalan, Conservation of an ecosystem through optimal taxation, Bull. Math. Biol., 60 (1998), 569-584. Google Scholar |
| [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. Google Scholar |
| [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. Song, Y. 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 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 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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