Effects of Random Environmental Perturbation on the Dynamics of a Nutrient–Phytoplankton–Zooplankton Model with Nutrient Recycling
A stochastic nutrient–phytoplankton–zooplankton model with instantaneous nutrient recycling is proposed and analyzed in this paper. When the nutrient uptake function and the grazing function are linear and the ingested phytoplankton is completely absorbed by the zooplankton, we establish two stochas...
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MDPI AG
2022-10-01
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| Online Access: | https://www.mdpi.com/2227-7390/10/20/3783 |
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| author | Lifan Chen Xingwang Yu Sanling Yuan |
| author_facet | Lifan Chen Xingwang Yu Sanling Yuan |
| author_sort | Lifan Chen |
| collection | DOAJ |
| description | A stochastic nutrient–phytoplankton–zooplankton model with instantaneous nutrient recycling is proposed and analyzed in this paper. When the nutrient uptake function and the grazing function are linear and the ingested phytoplankton is completely absorbed by the zooplankton, we establish two stochastic thresholds <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi mathvariant="script">R</mi><mn>0</mn><mi>S</mi></msubsup></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi mathvariant="script">R</mi><mn>1</mn><mi>S</mi></msubsup></semantics></math></inline-formula>, which completely determine the persistence and extinction of the plankton. That is, if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi mathvariant="script">R</mi><mn>0</mn><mi>S</mi></msubsup><mo><</mo><mn>1</mn></mrow></semantics></math></inline-formula>, both the phytoplankton and the zooplankton eventually are eliminated; if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi mathvariant="script">R</mi><mn>0</mn><mi>S</mi></msubsup><mo>></mo><mn>1</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi mathvariant="script">R</mi><mn>1</mn><mi>S</mi></msubsup><mo><</mo><mn>1</mn></mrow></semantics></math></inline-formula>, the phytoplankton is persistent in mean but the zooplankton is extinct; while for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi mathvariant="script">R</mi><mn>1</mn><mi>S</mi></msubsup><mo>></mo><mn>1</mn></mrow></semantics></math></inline-formula>, the entire system is persistent in mean. Furthermore, sufficient criteria for the existence of ergodic stationary distribution of the model are obtained and the persistent levels of the plankton are estimated. Numeric simulations are carried out to illustrate the theoretical results and to conclude our study. Our results suggest that environmental noise may cause the local bloom of phytoplankton, which surprisingly can be used to explain the formation of algal blooms to some extent. Moreover, we find that the nonlinear nutrient uptake function and grazing function may take credit for the periodic succession of blooms regardless of whether they are in the absence or presence of the environmental noises. |
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| spelling | doaj.art-a7f837dc8e1c4072a71a463bf5749df32023-11-24T01:06:56ZengMDPI AGMathematics2227-73902022-10-011020378310.3390/math10203783Effects of Random Environmental Perturbation on the Dynamics of a Nutrient–Phytoplankton–Zooplankton Model with Nutrient RecyclingLifan Chen0Xingwang Yu1Sanling Yuan2School of Arts and Sciences, Shanghai University of Medicine and Health Sciences, Shanghai 201318, ChinaSchool of Management Engineering, Zhengzhou University of Aeronautics, Zhengzhou 450046, ChinaCollege of Science, University of Shanghai for Science and Technology, Shanghai 200093, ChinaA stochastic nutrient–phytoplankton–zooplankton model with instantaneous nutrient recycling is proposed and analyzed in this paper. When the nutrient uptake function and the grazing function are linear and the ingested phytoplankton is completely absorbed by the zooplankton, we establish two stochastic thresholds <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi mathvariant="script">R</mi><mn>0</mn><mi>S</mi></msubsup></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi mathvariant="script">R</mi><mn>1</mn><mi>S</mi></msubsup></semantics></math></inline-formula>, which completely determine the persistence and extinction of the plankton. That is, if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi mathvariant="script">R</mi><mn>0</mn><mi>S</mi></msubsup><mo><</mo><mn>1</mn></mrow></semantics></math></inline-formula>, both the phytoplankton and the zooplankton eventually are eliminated; if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi mathvariant="script">R</mi><mn>0</mn><mi>S</mi></msubsup><mo>></mo><mn>1</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi mathvariant="script">R</mi><mn>1</mn><mi>S</mi></msubsup><mo><</mo><mn>1</mn></mrow></semantics></math></inline-formula>, the phytoplankton is persistent in mean but the zooplankton is extinct; while for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi mathvariant="script">R</mi><mn>1</mn><mi>S</mi></msubsup><mo>></mo><mn>1</mn></mrow></semantics></math></inline-formula>, the entire system is persistent in mean. Furthermore, sufficient criteria for the existence of ergodic stationary distribution of the model are obtained and the persistent levels of the plankton are estimated. Numeric simulations are carried out to illustrate the theoretical results and to conclude our study. Our results suggest that environmental noise may cause the local bloom of phytoplankton, which surprisingly can be used to explain the formation of algal blooms to some extent. Moreover, we find that the nonlinear nutrient uptake function and grazing function may take credit for the periodic succession of blooms regardless of whether they are in the absence or presence of the environmental noises.https://www.mdpi.com/2227-7390/10/20/3783stochastic nutrient–plankton modelpersistent levelnutrient recyclingthreshold dynamicsstationary distribution |
| spellingShingle | Lifan Chen Xingwang Yu Sanling Yuan Effects of Random Environmental Perturbation on the Dynamics of a Nutrient–Phytoplankton–Zooplankton Model with Nutrient Recycling Mathematics stochastic nutrient–plankton model persistent level nutrient recycling threshold dynamics stationary distribution |
| title | Effects of Random Environmental Perturbation on the Dynamics of a Nutrient–Phytoplankton–Zooplankton Model with Nutrient Recycling |
| title_full | Effects of Random Environmental Perturbation on the Dynamics of a Nutrient–Phytoplankton–Zooplankton Model with Nutrient Recycling |
| title_fullStr | Effects of Random Environmental Perturbation on the Dynamics of a Nutrient–Phytoplankton–Zooplankton Model with Nutrient Recycling |
| title_full_unstemmed | Effects of Random Environmental Perturbation on the Dynamics of a Nutrient–Phytoplankton–Zooplankton Model with Nutrient Recycling |
| title_short | Effects of Random Environmental Perturbation on the Dynamics of a Nutrient–Phytoplankton–Zooplankton Model with Nutrient Recycling |
| title_sort | effects of random environmental perturbation on the dynamics of a nutrient phytoplankton zooplankton model with nutrient recycling |
| topic | stochastic nutrient–plankton model persistent level nutrient recycling threshold dynamics stationary distribution |
| url | https://www.mdpi.com/2227-7390/10/20/3783 |
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