| Citation: |
Tiancai Liao, Hengguo Yu, Chuanjun Dai, Min Zhao. IMPACT OF NOISE IN A PHYTOPLANKTON-ZOOPLANKTON SYSTEM[J]. Journal of Applied Analysis & Computation, 2020, 10(5): 1878-1896. doi: 10.11948/20190272
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IMPACT OF NOISE IN A PHYTOPLANKTON-ZOOPLANKTON SYSTEM
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Abstract
In this paper, we investigate the dynamics of a delayed toxic phytoplankton-two zooplankton system incorporating the effects of L$ \acute{e} $vy noise and white noise. The value of this study lies in two aspects: Mathematically, we first prove the existence of a unique global positive solution of the system, and then we investigate the sufficient conditions that guarantee the stochastic extinction and persistence in the mean of each population. Ecologically, via numerical simulations, we find that the effect of white noise or L$ \acute{e} $vy noise on the stochastic extinction and persistence of phytoplankton and zooplankton are similar, but the synergistic effects of the two noises on the stochastic extinction and persistence of these plankton are stronger than that of single noise. In addition, an increase in the toxin liberation rate or the intraspecific competition rate of zooplankton was found to be capable to increase the biomass of the phytoplankton but decrease the biomass of zooplankton. These results may help us to better understand the phytoplankton-zooplankton dynamics in the fluctuating environments.-
Keywords:
- Noise /
- phytoplankton /
- zooplankton /
- extinction /
- persistence
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References
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Tiancai Liao, Hengguo Yu, Chuanjun Dai, Min Zhao. IMPACT OF NOISE IN A PHYTOPLANKTON-ZOOPLANKTON SYSTEM[J]. Journal of Applied Analysis & Computation, 2020, 10(5): 1878-1896. doi: 10.11948/20190272
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Figure 1. The solutions of stochastic system (1.2) with time step
$ \Delta t = 0.001 $ and different white noise intensities.a for$ (\delta_{1}, \delta_{2}, \delta_{3}, \delta_{4}, \delta_{5}) = (0.40, 0.41, 0.42, 0.43, 0.44) $ . b for$ (\delta_{1}, \delta_{2}, \delta_{3}, \delta_{4}, \delta_{5}) = (0.80, 0.81, 0.82, 0.83, 0.84) $ . c for$ (\delta_{1}, \delta_{2}, \delta_{3}, \delta_{4}, \delta_{5}) = (1.80, 1.81, 1.82, 1.83, 1.84) $ . -
Figure 2. The solutions of stochastic system (1.2) and its deterministic system (1.1) with time step
$ \Delta t = 0.001 $ . a for phytoplankton$ P(t) $ . b for zooplankton$ Z_{1}(t) $ . c for zooplankton$ Z_{2}(t) $ . -
Figure 3. The solutions of the stochastic system (1.2) and its deterministic system (1.1) with time step
$ \Delta t = 0.001 $ . a for phytoplankton$ P(t) $ . b for zooplankton$ Z_{1}(t) $ . c for zooplankton$ Z_{2}(t) $ . -
Figure 4. The sensitivity analysis of the threshold for phytoplankton
$ P(t) $ in system (1.2) with respect to its growth rate and noise intensities. a for$ r $ and$ \delta_{1} $ . b for$ \delta_{1} $ and$ \gamma_{1}(u) $ . The space region$ \rm{I} $ denotes the persistence of phytoplankton, the space region$ \rm{II} $ indicates the extinction of phytoplankton, and the planes (pink and green) represent the critical plane between the persistence and extinction of phytoplankton. -
Figure 5. The effects of toxin liberation rate
$ \rho_{i} $ on the dynamics of plankton in the stochastic system (1.2). -
Figure 6. The effects of intraspecific coefficient
$ g_{i}(i = 1, 2) $ on the dynamics of plankton in the stochastic system (1.2).