| Citation: |
Dongpo Hu, Ying Zhang, Zhaowen Zheng, Ming Liu. DYNAMICS OF A DELAYED PREDATOR-PREY MODEL WITH CONSTANT-YIELD PREY HARVESTING[J]. Journal of Applied Analysis & Computation, 2022, 12(1): 302-335. doi: 10.11948/20210171
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DYNAMICS OF A DELAYED PREDATOR-PREY MODEL WITH CONSTANT-YIELD PREY HARVESTING
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Abstract
In this paper, we study a delayed predator-prey model of Holling and Leslie type with constant-yield prey harvesting, in which two types of delays caused by maturation time of prey and the gestation time of predator are considered. We mainly investigate the local dynamics of the model with emphasis on the impact of delays. The stability of equilibrium and the existence conditions of Hopf bifurcation are discussed. First, based on the different values of delays, five cases of Hopf bifurcation are studied in detail. The critical values of Hopf bifurcation for each case are presented. In addition, we explore the properties of Hopf bifurcation. The direction of Hopf bifurcation and the stability of periodic solutions by using the normal form theory and central manifold theorem are determined. The qualitative analyses have demonstrated that the values of time delays can affect the stability of equilibrium and induce small amplitude period oscillations of population densities. Numerical simulations are carried out for illustrating the theoretical results. Meanwhile, we further investigate the effects of delay on the period of periodic solutions and the influence of the harvesting term on the stability of the equilibrium with time delays.
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Keywords:
- Predator-prey model /
- time delay /
- stability /
- Hopf bifurcation
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References
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Dongpo Hu, Ying Zhang, Zhaowen Zheng, Ming Liu. DYNAMICS OF A DELAYED PREDATOR-PREY MODEL WITH CONSTANT-YIELD PREY HARVESTING[J]. Journal of Applied Analysis & Computation, 2022, 12(1): 302-335. doi: 10.11948/20210171
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Figure 1. The equilibria of model (2.1). The magenta and yellow lines represent prey and predator nullclines, respectively. The points where the two nullclines cross are equilibria and there are four of these points for (a)
$ a = 0.7 $ ,$ h = \frac{3}{25} $ ,$ \delta = \frac{3}{5} $ ,$ \beta = 1 $ ; (b)$ a = 0.2 $ ,$ h = \frac{1}{16} $ ,$ \delta = \frac{1}{5} $ ,$ \beta = 0.3 $ . -
Figure 2. Time series of
$ x(t) $ ,$ y(t) $ of model (2.1) with$ \tau_2 = 0 $ and$ \tau_1 = 0, 0.5, 5, 10, 15 $ , respectively. The positive equilibrium$ E_1(0.51219177, 0.30731719) $ is locally asymptotically stable. Here the initial value is$ (0.5, 0.2). $ . -
Figure 3. (a)-(b): Time series of
$ x(t) $ ,$ y(t) $ of model (2.1) with$ \tau_2 = 0 $ and$ \tau_1 = 0, 0.5, 1, 2.5 < \tau_{1_0} = 3.11596840 $ , respectively. The positive equilibrium$ E_1(0.39929087, 0.26619585) $ is locally asymptotically. The initial value is$ (0.5, 0.2) $ . (c)-(d): Time series of$ x(t) $ ,$ y(t) $ and phase portrait of model (2.1) with$ \tau_2 = 0 $ and$ \tau_1 = 3.2 > \tau_{1_0} = 3.11596840 $ . The positive equilibrium$ E_1 $ is unstable and the orbit from the initial value$ (0.39, 0.26) $ located in a sufficiently small neighborhood of$ E_1 $ converges to a periodic solution. -
Figure 4. Time series of
$ x(t) $ ,$ y(t) $ of model (2.1) with$ \tau_1 = 0 $ and$ \tau_2 = 0, 0.5, 5, 10, 15 $ , respectively. The positive equilibrium$ E_1(0.56316894, 0.21118835) $ is locally asymptotically stable. Here initial value is$ (0.5, 0.2) $ . -
Figure 5. (a)-(b): Time series of
$ x(t) $ ,$ y(t) $ of model (2.1) with$ \tau_1 = 0 $ and$ \tau_2 = 0, 0.5, 1.2, 1.5 < \tau_{2_0} = 1.76144387 $ , respectively. The positive equilibrium$ E_1(0.42147248, 0.50576827) $ is locally asymptotically. The initial value is$ (0.5, 0.2) $ . (c)-(d): Time series of$ x(t) $ ,$ y(t) $ and phase portrait of model (2.1) with$ \tau_1 = 0 $ and$ \tau_2 = 1.77 > \tau_{2_0} = 1.76144387 $ . The positive equilibrium$ E_1 $ is unstable and the orbit from the initial value$ (0.5, 0.2) $ located in a sufficiently small neighborhood of$ E_1 $ converges to a periodic solution. -
Figure 6. Time series of
$ x(t) $ ,$ y(t) $ of model (2.1) with$ \tau_1 = \tau_2 = 0, 0.5, 5, 10, 15 $ , respectively. The positive equilibrium$ E_1(0.62234655, 0.20744885) $ is locally asymptotically stable. Here initial value is$ (0.5, 0.2) $ . -
Figure 7. (a)-(b): Time series of
$ x(t) $ ,$ y(t) $ of model (2.1) with$ \tau_1 = \tau_2 = 0, 1, 1.5, 2 < \tau_{3_0} = 2.12332287 $ , respectively. The positive equilibrium$ E_1(0.65299477, 0.39179686) $ is locally asymptotically. The initial value is$ (0.5, 0.2) $ . (c)-(d): Time series of$ x(t) $ ,$ y(t) $ and phase portrait of model (2.1) with$ \tau_1 = \tau_2 = 2.35 > \tau_{3_0} = 2.12332287 $ . The positive equilibrium$ E_1 $ is unstable and the orbit from the initial value$ (0.5, 0.2) $ located in a sufficiently small neighborhood of$ E_1 $ converges to a periodic solution. -
Figure 8. Time series of
$ x(t) $ ,$ y(t) $ of model (2.1) with$ \tau_2 = 1 $ and$ \tau_1 = 0, 0.5, 5, 10, 15 $ , respectively. The positive equilibrium$ E_1(0.60848685, 0.17385342) $ is locally asymptotically stable. Here initial value is$ (0.5, 0.2) $ . -
Figure 9. (a)-(b): Time series of
$ x(t) $ ,$ y(t) $ of model (2.1) with$ \tau_2 = 1.5 $ and$ \tau_1 = 0, 0.1, 0.5, 0.8 < \tau_{4_0} = 1.08245692 $ , respectively. The positive equilibrium$ E_1(0.42147248, 0.50576827) $ is locally asymptotically. The initial value is$ (0.5, 0.2) $ . (c)-(d): Time series of$ x(t) $ ,$ y(t) $ and phase portrait of model (2.1) with$ \tau_2 = 1.5 $ and$ \tau_1 = 1.15 > \tau_{4_0} = 1.08245692 $ . The positive equilibrium$ E_1 $ is unstable and the orbit from the initial value$ (0.5, 0.2) $ located in a sufficiently small neighborhood of$ E_1 $ converges to a periodic solution. -
Figure 10. (a): The effect of
$ \tau_1 $ on the period$ T_2 $ with$ \tau_2 = 1.5\in[0, \tau_{2_0}) $ . The stars stand for$ \tau_1 = 1.15, 1.3, 1.5, 1.8, 2 > \tau_{4_0} = 1.08245692 $ , respectively. (b): Phase portraits without showing the transient of model (2.1) with$ \tau_2 = 1.5\in[0, \tau_{2_0}) $ and$ \tau_1 = 1.15, 1.3, 1.5, 1.8, 2 > \tau_{4_0} = 1.08245692 $ , respectively. The parameters are the same as Fig. 9. -
Figure 11. (a): The relationship between the location of the positive equilibrium
$ E_1(x^*, y^*) $ and$ h $ . The blue and red stars stand for$ (\frac{1}{80}, 0.42147248) $ and$ (\frac{1}{80}, 0.50576827) $ , respectively. (b): The relationship between the critical value$ \tau_{4_0} $ and$ h $ . The red star is$ (\frac{1}{80}, 1.08245692) $ .