| Citation: |
R. Sivasamy, K. Sathiyanathan, K. Balachandran. DYNAMICS OF A MODIFIED LESLIE-GOWER MODEL WITH GESTATION EFFECT AND NONLINEAR HARVESTIN[J]. Journal of Applied Analysis & Computation, 2019, 9(2): 747-764. doi: 10.11948/2156-907X.20180165
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DYNAMICS OF A MODIFIED LESLIE-GOWER MODEL WITH GESTATION EFFECT AND NONLINEAR HARVESTIN
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Abstract
This study focuses on the dynamics of a modified Leslie-Gower predator-prey model where the intake rate of prey is by per capita predator according to Crowley-Martin functional response and prey is harvested through nonlinear harvesting strategy. Further the time-delay $(\tau)$ is imposed to utilize gestation period of predations. We investigate the permanence analysis of proposed system. The local stability of non-delayed model at all possible equilibrium points is studied. It is shown that the given model undergoes Hopf bifurcation around positive equilibrium point with respect to delay parameter $\tau$. Subsequently the stability of Hopf bifurcation and its direction are explored through normal and center manifold theories. The derived theoretical results are justified with the help of numerical simulations. -
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References
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R. Sivasamy, K. Sathiyanathan, K. Balachandran. DYNAMICS OF A MODIFIED LESLIE-GOWER MODEL WITH GESTATION EFFECT AND NONLINEAR HARVESTIN[J]. Journal of Applied Analysis & Computation, 2019, 9(2): 747-764. doi: 10.11948/2156-907X.20180165
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- Figure 1. The existence of all possible equilibrium points (red bullets)
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Figure 2. (a) The phase space diagram of (5.1) for
$ \gamma = 0.1 $ . (b) The time trajectories of system (5.1) for$ \gamma = 0.1 $ . Thus$ E_* = (0.114545, 0.982954) $ is locally asymptotically stable -
Figure 3. (a) The phase space diagram of (5.1) for
$ \gamma=0.07 $ . (b) The time trajectories of system (5.1) for$ \gamma=0.07 $ . Thus the model (5.1) undergoes Hopf bifurcation around$ E_*. $ -
Figure 4. (a) The phase space diagram of (5.2) for
$ \tau = 3.0 $ . (b) The time trajectories of system (5.2) for$ \tau = 3.0 $ . Thus$ E_* $ is locally asymptotically stable -
Figure 5. (a) The phase space diagram of (5.2) for
$ \tau = 3.27 $ . (b) The periodic solutions of system (5.2) for$ \tau = 3.27 $ . Thus the model (5.2) undergoes Hopf bifurcation around$ E_*. $