Article Contents
Article Contents

On a predator prey model with nonlinear harvesting and distributed delay

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.

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

 Citation:

• 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

•  [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. [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. [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. 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.

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