| Citation: |
Salih Djilali. EFFECT OF HERD SHAPE IN A DIFFUSIVE PREDATOR-PREY MODEL WITH TIME DELAY[J]. Journal of Applied Analysis & Computation, 2019, 9(2): 638-654. doi: 10.11948/2156-907X.20180136
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EFFECT OF HERD SHAPE IN A DIFFUSIVE PREDATOR-PREY MODEL WITH TIME DELAY
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Abstract
In this paper, we deal with the effect of the shape of herd behavior on the interaction between predator and prey. The model analysis was studied in three parts. The first, The analysis of the system in the absence of spatial diffusion and the time delay, where the local stability of the equilibrium states, the existence of Hopf bifurcation have been investigated. For the second part, the spatiotemporal dynamics introduce by self diffusion was determined, where the existence of Hopf bifurcation, Turing driven instability, Turing-Hopf bifurcation point have been proved. Further, the order of Hopf bifurcation points and regions of the stability of the non trivial equilibrium state was given. In the last part of the paper, we studied the delay effect on the stability of the non trivial equilibrium, where we proved that the delay can lead to the instability of interior equilibrium state, and also the existence of Hopf bifurcation. A numerical simulation was carried out to insure the theoretical results. -
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References
[1] P. A. Braza, Predator-prey dynamics with square root functional responses, Nonlin Anal Real World Appl, 2012, 13, 1837-43. doi: 10.1016/j.nonrwa.2011.12.014 [2] M. Baurmanna, T. Gross and U. Feudel, Instabilities in spatially extended predator-prey systems: Spatio-temporal patterns in the neighborhood of Turing-Hopf bifurcations , Journal of Theoretical Biology, 2007, 245, 220-229. doi: 10.1016/j.jtbi.2006.09.036 [3] I. M. Bulai and E. Venturino, Shape effects on herd behavior in ecological interacting population models, Mathematics and Computers in Simulation, 2017, 141, 40-55. doi: 10.1016/j.matcom.2017.04.009 [4] I. Boudjema and S. Djilali, Turing-Hopf bifurcation in Gauss-type model with cross diffusion and its application, Nonlinear Studies, 2018, 25(3), 665-687. [5] E. Cagliero and E. Venturino, Ecoepidemics with infected prey in herd defense: the harmless and toxic cases, Int. J. Comput. Math., 2016, 93, 108-127. doi: 10.1080/00207160.2014.988614 [6] J. Carr, Applications of Center Manifold Theory, New York, SpringerVerlag, 1981. [7] M. Cavani and M. Farkas, Bifurcations in a predator-prey model with memory and diffusion. I: Andronov-hopf bifurcations, Acta Math Hungar, 1994, 63, 213-29. doi: 10.1007/BF01874129 [8] S. Djilali, Herd behavior in a predator-prey model with spatial diffusion: bifurcation analysis and Turing instability, Journal of Applied Mathematics and Computing, 2018, 58, 125-149. doi: 10.1007/s12190-017-1137-9 [9] S.Djilali, Impact of prey herd shape on the predator-prey interaction, Chaos, Solitons and Fractals, 2019, 120, 139-148. doi: 10.1016/j.chaos.2019.01.022 [10] S. Djilali, T. M. Touaoula and S. E-H.Miri, A heroin epidemic model: very general non linear incidence, treat-age, and global stability, Acta Applicandae Mathematicae, 2017, 152(1), 171-194. doi: 10.1007/s10440-017-0117-2 [11] T. Faria, Stability and Bifurcation for a Delayed Predator-Prey Model and the Effect of Diffusion, Applied Mathematics and Computation, 2001, 254, 433-463. [12] J. Luo and Y. Zhao, Stability and bifurcation analysis in a predator-prey system with constant harvesting and prey group defense, International Journal of Bifurcation and Chaos, 2017, 27, 1750179. doi: 10.1142/S0218127417501796 [13] X. Liu, T. Zhang, X. Meng and T. Zhang, Turing-Hopf bifurcations in a predator-prey model with herd behavior, quadratic mortality and prey-taxis, Physica A, 2018, 496, 446-460. doi: 10.1016/j.physa.2018.01.006 [14] B. Liu, R. Wu and L. Chen, Patterns induced by super cross-diffusion in a predator-prey system with Michaelis-Menten type harvesting, Mathematical Biosiences, 2018, 298, 71-79. doi: 10.1016/j.mbs.2018.02.002 [15] C. V. Pao, Convergence of solutions of reaction-diffusion systems whith time delays, Nonlinear Analysis, 2002, 48, 349-362. doi: 10.1016/S0362-546X(00)00189-9 [16] F. Rao, C. Chavez, Y. Kang, Dynamics of a diffusion reaction prey-predator model with delay in prey: Effecs of delay and spatial components, Journal of Mathematical Analysis and Applications, 2018, 461(2), 1177-1214. doi: 10.1016/j.jmaa.2018.01.046 [17] Y. Song and X. Zou, Bifurcation analysis of a diffusive ratio-dependent predator-prey model, Nonlinear Dynamics, 2017, 78, 49-70. [18] Y. Song and X. Zou, Spatiotemporal dynamics in a diffusive Ratio-dependent predator-prey model near a Hopf-Turing bifurcation point, Computers and Mathematics with Applications, 2014, 67, 1978-1967. doi: 10.1016/j.camwa.2014.04.015 [19] Y. Song, T. Zhang and Y. Peng, Turing-Hopf bifurcation in the reaction diffusion equations and its applications, Commun Nonlinear Sci NumerSimulat, 2015, 33, 229-258. [20] Y. Song, Y. Peng and X. Zou, Persisstence, stability and Hopf bifurcation in a diffusive Ratio-Dependent predator-prey model with delay, International Journal of Bifurcation and Chaos, 2014, 24, 1450093. doi: 10.1142/S021812741450093X [21] Y. Song, H. Jiang, Q. X. Liu and Y. Yuan, Spatiotemporal dynamics of the diffusive Mussel-Algae model near Turing-Hopf bifurcation, SIAM Journal of Applied Dynamical Systems, 2017, 16, 2030-2062. doi: 10.1137/16M1097560 [22] Y. Song and X. Tang, Stability, Steady-State Bifurcations, and Turing Patterns in a Predator- Prey Model with Herd Behavior and Prey-taxis, Stud. Appl. Math., 2017, 139, 371-404. doi: 10.1111/sapm.2017.139.issue-3 [23] M. Sambath, K. Balachandran and L. N. Guin, Spatiotemporal patterns in a predator-prey model with cross-diffusion effect, International Journal of Bifurcation and Chaos, 2018, 28, 1830004. doi: 10.1142/S0218127418300045 [24] X. Tang, H. Jiang, Z. Deng and T. Yu, Delay induced subcritical Hopf bifurcation in a diffusive predator-prey model with herd behavior and hyperbolic mortality, Journal of Applied Analysis and Computation, 2017, 7, 1385-1401. [25] X. Tang and Y. Song, Bifurcation analysis and Turing instability in a diffusive predator prey model with herd behavior and hyperbolic mortality, Chaos, Solitons and Fractals, 2015, 81, 303-3014. doi: 10.1016/j.chaos.2015.10.001 [26] X. Tang and Y. Song, Cross-diffusion induced spatiotemporal patterns in a predator-prey model with herd behavior, Nonlinear Analysis: Real World Applications, 2015, 24, 36-49. doi: 10.1016/j.nonrwa.2014.12.006 [27] X. Tang and Y. Song and T. Zhang, Turing-Hopf bifurcation analysis of a predator-prey model with herd behavior and cross diffusion, Nonlinear Dynamics, 2016, 86, 73-89. doi: 10.1007/s11071-016-2873-3 [28] X. Tang and Y. Song, Stability, Hopf bifurcations and spatial patterns in a delayed diffusive predator-prey model with herd behavior, Applied Mathematics and Computation, 2015, 254, 375-391. doi: 10.1016/j.amc.2014.12.143 [29] E. Venturino, A minimal model for ecoepidemics with group defense, J. Biol. Syst., 2011, 19, 763-785. doi: 10.1142/S0218339011004184 [30] E. Venturino, Modeling herd behavior in population systems, Nonlinear Analysis: Real World Applications, 2011, 12, 2319-2338. doi: 10.1016/j.nonrwa.2011.02.002 [31] E. Venturino and S. Petrovskii, Spatiotemporal behavior of a prey-predator system with a group defense for prey, Ecological Complexity, 2013, 14, 37-47. doi: 10.1016/j.ecocom.2013.01.004 [32] R. Wu, M.Chen, B. Liu and L. Chen, Hopf bifurcation and Turing instability in a predator-prey model with Michaelis-Menten functional reponse, Nonlinear Dynamics, 2018, 91(3), 2033-2047. doi: 10.1007/s11071-017-4001-4 [33] C. Wang and S. Qi, Spatial dynamics of a predator-prey system with cross diffuusion, Chaos, Solitons and Fractals, 2018, 107, 55-60. doi: 10.1016/j.chaos.2017.12.020 [34] C. Xu, C. Yuan and T. Zhang, Global dynamics of a predator-prey model with defence mechanism for prey, Applied Mathematics Letters, 2016, 62, 42-48. doi: 10.1016/j.aml.2016.06.013 [35] Z. Xu and Y. Song, Bifurcation analysis of a diffusive predator-prey system with a herd behavior and quadratic mortality, Math. Meth. Appl. Sci., 2015, 38(4), 2994-3006. [36] R. Yang and Y. Song, Spatial resonance and Turing-Hopf bifurcations in the Gierer-Meinhardt model, Nonlinear Analysis: Real World Applications, 2016, 31, 356-387. doi: 10.1016/j.nonrwa.2016.02.006 [37] W. Yang, Analysis on existence of bifurcation solutions for a predator-prey model with herd behavior, Applied mathematical Modelling, 2017, 53, 433-446. [38] H. Zhu and X. Zhang, Dynamics and Patterns of a Diffusive Prey-Predator System with a Group Defense for Prey, Discrete Dynamics in Nature and Society, 2018. DIO: 10.1155/2018/6519696. doi: 10.1155/2018/6519696 -
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Salih Djilali. EFFECT OF HERD SHAPE IN A DIFFUSIVE PREDATOR-PREY MODEL WITH TIME DELAY[J]. Journal of Applied Analysis & Computation, 2019, 9(2): 638-654. doi: 10.11948/2156-907X.20180136
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Figure 1. the existence of the nontrivial equilibrium
$(R^{\ast }, F^{\ast })=(0.8112, 1.305)$ for the values k=1;r=1.1;a=0.15;$\mu=0.1$ ;$\alpha =\frac{2}{3}$ . -
Figure 2. The order of Hopf bifurcation points for the values
$ d_{1}=0.02$ ,$ d_{2}=0.03$ ,$k=10$ ,$R^{\ast}=3.72$ ,$r=1.1$ ,$a=0.15$ ,$\mu=0.05$ ,$\alpha=\frac{2}{3}$ and$T=-0.1267$ ,$N_{1}=1$ ,$R^{\ast}>R_{0}^{*}=2.5$ and$R_{1}^{*}=-0.25 < R^{\ast}$ . -
Figure 3. The existence of Turing-Hopf bifurcation point for the values the order of Hopf bifurcation points for the values
$d_{2}=0.03$ ,$k=10$ ,$R^{\ast}=0.44$ ,$r=1.1$ ,$a=1.15$ ,$\mu=0.5$ ,$\alpha=\frac{2}{3}$ ,$n_{*}=2$ and$T=-0.4201$ ,$D^{*}=0.15$ ,$n_{*}=2$ . The region$D_{1}$ ,$D_{2}$ the case when the non trivial equilibrium is unstable,$D_{3}$ represent the region of Turing instability,$D_{4}$ is the region of the stability of$E^{*}$ . -
Figure 4. Numerical simulation shows the dynamic near Hopf bifurcation point in system (1.2) and
$\tau =0$ for the values in Figure 2. For (A) and (B) we have$(d_{1}, \beta)=(0.01, 0.3)$ (which means$(d_{1}, \beta)$ in$D_{4}$ ) the non trivial equilibrium is asymptotically stable and the initial data$(\phi, \psi)=(R^{*}+0.3\cos2x, P^{*}+0.3\cos2x)$ . For (C) and (D) we have$(d_{1}, \beta)=(0.01, 0.2)$ (which means$(d_{1}, \beta)$ in$D_{1}$ ) the non trivial equilibrium is unstable and the initial data$(\phi, \psi)=(R^{*}+0.3\cos2x, P^{*}+0.3\cos2x)$ . -
Figure 5. Dynamics introduced by the presence of time delay for the system (1.2) where we assumed that
$(d_{1}, \beta)$ in$D_{4}$ for the same value of Figure 2 and 4 ((A) and (B)). For (A) and (B) we have$\tau=2 < \tau _{00}=2.8601$ and the initial conditions$(\phi, \psi)=(R^{*}+0.5, P^{*}+0.5)$ . For (C) and (D) we have$\tau=3 >\tau _{00}=2.8601$ and the initial conditions$(\phi, \psi)=(R^{*}+0.5, P^{*}+0.5)$ .