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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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-01-01
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| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/13/2/197 |
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| Summary: | 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. |
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| ISSN: | 2227-7390 |