Square-root models for the Volterra equations and the explicit solution of these models

Area of Science:

• Mathematical Biology
• Ecology
• Differential Equations

Background:

• Volterra's (1926) foundational equations describe ecological interactions.
• Original models present challenges in explicit analytical solutions.

Purpose of the Study:

• To modify Volterra's competition and predator-prey equations.
• To introduce root terms for enhanced analytical tractability.
• To compare the dynamic properties of modified and original models.

Main Methods:

• Introduction of root terms into Volterra's equations.
• Comparative analysis of dynamic properties.
• Derivation of explicit solutions for the modified differential equations.
• Numerical examples and detailed solution presentation.

Main Results:

• Modified equations retain essential dynamic properties of original Volterra models.
• The introduction of root terms facilitates explicit analytical solutions.
• Numerical examples demonstrate the application and validity of the solutions.

Conclusions:

• The modified Volterra equations offer a more analytically tractable framework for studying ecological dynamics.
• This approach preserves the core ecological insights of Volterra's original work.
• The explicit solutions provide valuable tools for ecological modeling and analysis.