Multidimensional Stability of Planar Traveling Waves for Competitive–Cooperative Lotka–Volterra System of Three Species
We investigate the multidimensional stability of planar traveling waves in competitive–cooperative Lotka–Volterra system of three species in <i>n</i>-dimensional space. For planar traveling waves with speed <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML&quo...
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2025-01-01
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| author | Na Shi Xin Wu Zhaohai Ma |
| author_facet | Na Shi Xin Wu Zhaohai Ma |
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| description | We investigate the multidimensional stability of planar traveling waves in competitive–cooperative Lotka–Volterra system of three species in <i>n</i>-dimensional space. For planar traveling waves with speed <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>c</mi><mo>></mo><msup><mi>c</mi><mo>*</mo></msup></mrow></semantics></math></inline-formula>, we establish their exponential stability in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>L</mi><mo>∞</mo></msup><mrow><mo>(</mo><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, which is expressed as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>t</mi><mrow><mo>−</mo><mstyle><mfrac><mi>n</mi><mn>2</mn></mfrac></mstyle></mrow></msup><msup><mi mathvariant="normal">e</mi><mrow><mo>−</mo><msub><mi>ε</mi><mi>τ</mi></msub><mi>σ</mi><mi>t</mi></mrow></msup></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>σ</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula> is a constant and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ε</mi><mi>τ</mi></msub><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula> depends on the time delay <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>τ</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula> as a decreasing function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ε</mi><mi>τ</mi></msub><mo>=</mo><mi>ε</mi><mrow><mo>(</mo><mi>τ</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. The time delay is shown to significantly reduce the decay rate of the solution. Additionally, for planar traveling waves with speed <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>c</mi><mo>=</mo><msup><mi>c</mi><mo>*</mo></msup></mrow></semantics></math></inline-formula>, we demonstrate their algebraic stability in the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>t</mi><mrow><mo>−</mo><mstyle><mfrac><mi>n</mi><mn>2</mn></mfrac></mstyle></mrow></msup></semantics></math></inline-formula>. Our analysis employs the Fourier transform and a weighted energy method with a carefully chosen weight function. |
| format | Article |
| id | doaj-art-b4e428d30eec48a09819b82d83afb669 |
| institution | Kabale University |
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| language | English |
| publishDate | 2025-01-01 |
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| spelling | doaj-art-b4e428d30eec48a09819b82d83afb6692025-01-24T13:39:42ZengMDPI AGMathematics2227-73902025-01-0113219710.3390/math13020197Multidimensional Stability of Planar Traveling Waves for Competitive–Cooperative Lotka–Volterra System of Three SpeciesNa Shi0Xin Wu1Zhaohai Ma2School of Science, China University of Geosciences, Beijing 100083, ChinaSchool of Sciences, East China JiaoTong University, Nanchang 330013, ChinaSchool of Science, China University of Geosciences, Beijing 100083, ChinaWe investigate the multidimensional stability of planar traveling waves in competitive–cooperative Lotka–Volterra system of three species in <i>n</i>-dimensional space. For planar traveling waves with speed <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>c</mi><mo>></mo><msup><mi>c</mi><mo>*</mo></msup></mrow></semantics></math></inline-formula>, we establish their exponential stability in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>L</mi><mo>∞</mo></msup><mrow><mo>(</mo><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, which is expressed as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>t</mi><mrow><mo>−</mo><mstyle><mfrac><mi>n</mi><mn>2</mn></mfrac></mstyle></mrow></msup><msup><mi mathvariant="normal">e</mi><mrow><mo>−</mo><msub><mi>ε</mi><mi>τ</mi></msub><mi>σ</mi><mi>t</mi></mrow></msup></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>σ</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula> is a constant and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ε</mi><mi>τ</mi></msub><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula> depends on the time delay <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>τ</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula> as a decreasing function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ε</mi><mi>τ</mi></msub><mo>=</mo><mi>ε</mi><mrow><mo>(</mo><mi>τ</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. The time delay is shown to significantly reduce the decay rate of the solution. Additionally, for planar traveling waves with speed <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>c</mi><mo>=</mo><msup><mi>c</mi><mo>*</mo></msup></mrow></semantics></math></inline-formula>, we demonstrate their algebraic stability in the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>t</mi><mrow><mo>−</mo><mstyle><mfrac><mi>n</mi><mn>2</mn></mfrac></mstyle></mrow></msup></semantics></math></inline-formula>. Our analysis employs the Fourier transform and a weighted energy method with a carefully chosen weight function.https://www.mdpi.com/2227-7390/13/2/197multidimensional stabilitycompetitive–cooperative systemweighted energy methodplanar traveling wavesFourier transform |
| spellingShingle | Na Shi Xin Wu Zhaohai Ma Multidimensional Stability of Planar Traveling Waves for Competitive–Cooperative Lotka–Volterra System of Three Species Mathematics multidimensional stability competitive–cooperative system weighted energy method planar traveling waves Fourier transform |
| title | Multidimensional Stability of Planar Traveling Waves for Competitive–Cooperative Lotka–Volterra System of Three Species |
| title_full | Multidimensional Stability of Planar Traveling Waves for Competitive–Cooperative Lotka–Volterra System of Three Species |
| title_fullStr | Multidimensional Stability of Planar Traveling Waves for Competitive–Cooperative Lotka–Volterra System of Three Species |
| title_full_unstemmed | Multidimensional Stability of Planar Traveling Waves for Competitive–Cooperative Lotka–Volterra System of Three Species |
| title_short | Multidimensional Stability of Planar Traveling Waves for Competitive–Cooperative Lotka–Volterra System of Three Species |
| title_sort | multidimensional stability of planar traveling waves for competitive cooperative lotka volterra system of three species |
| topic | multidimensional stability competitive–cooperative system weighted energy method planar traveling waves Fourier transform |
| url | https://www.mdpi.com/2227-7390/13/2/197 |
| work_keys_str_mv | AT nashi multidimensionalstabilityofplanartravelingwavesforcompetitivecooperativelotkavolterrasystemofthreespecies AT xinwu multidimensionalstabilityofplanartravelingwavesforcompetitivecooperativelotkavolterrasystemofthreespecies AT zhaohaima multidimensionalstabilityofplanartravelingwavesforcompetitivecooperativelotkavolterrasystemofthreespecies |