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Multi-Parameter Reaction–Diffusion Systems with Quadratic Nonlinearity and Delays: New Exact Solutions in Elementary Functions

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  • Andrei D. Polyanin

    (Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences, 101 Vernadsky Avenue, Bldg. 1, 119526 Moscow, Russia
    Chemical Engineering and Biotechnology Department, Moscow Polytechnic University, 38 Bolshaya Semenovskaya St., 107023 Moscow, Russia)

  • Alexei I. Zhurov

    (Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences, 101 Vernadsky Avenue, Bldg. 1, 119526 Moscow, Russia
    College of Biomedical & Life Sciences, Cardiff University, Heath Park, Cardiff CF14 4XY, UK)

Abstract

The study considers a nonlinear multi-parameter reaction–diffusion system of two Lotka–Volterra-type equations with several delays. It treats both cases of different diffusion coefficients and identical diffusion coefficients. The study describes a few different techniques to solve the system of interest, including (i) reduction to a single second-order linear ODE without delay, (ii) reduction to a system of three second-order ODEs without delay, (iii) reduction to a system of three first-order ODEs with delay, (iv) reduction to a system of two second-order ODEs without delay and a linear Schrödinger-type PDE, and (v) reduction to a system of two first-order ODEs with delay and a linear heat-type PDE. The study presents many new exact solutions to a Lotka–Volterra-type reaction–diffusion system with several arbitrary delay times, including over 50 solutions in terms of elementary functions. All of these are generalized or incomplete separable solutions that involve several free parameters (constants of integration). A special case is studied where a solution contains infinitely many free parameters. Along with that, some new exact solutions are obtained for a simpler nonlinear reaction–diffusion system of PDEs without delays that represents a special case of the original multi-parameter delay system. Several generalizations to systems with variable coefficients, systems with more complex nonlinearities, and hyperbolic type systems with delay are discussed. The solutions obtained can be used to model delay processes in biology, ecology, biochemistry and medicine and test approximate analytical and numerical methods for reaction–diffusion and other nonlinear PDEs with delays.

Suggested Citation

  • Andrei D. Polyanin & Alexei I. Zhurov, 2022. "Multi-Parameter Reaction–Diffusion Systems with Quadratic Nonlinearity and Delays: New Exact Solutions in Elementary Functions," Mathematics, MDPI, vol. 10(9), pages 1-28, May.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:9:p:1529-:d:807723
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    References listed on IDEAS

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    1. Kudryashov, Nikolay A. & Zakharchenko, Anastasia S., 2015. "Analytical properties and exact solutions of the Lotka–Volterra competition system," Applied Mathematics and Computation, Elsevier, vol. 254(C), pages 219-228.
    2. Li, Wan-Tong & Yan, Xiang-Ping & Zhang, Cun-Hua, 2008. "Stability and Hopf bifurcation for a delayed cooperation diffusion system with Dirichlet boundary conditions," Chaos, Solitons & Fractals, Elsevier, vol. 38(1), pages 227-237.
    3. Andrei D. Polyanin & Vsevolod G. Sorokin, 2021. "Nonlinear Pantograph-Type Diffusion PDEs: Exact Solutions and the Principle of Analogy," Mathematics, MDPI, vol. 9(5), pages 1-22, March.
    4. Polyanin, Andrei D., 2019. "Functional separable solutions of nonlinear reaction–diffusion equations with variable coefficients," Applied Mathematics and Computation, Elsevier, vol. 347(C), pages 282-292.
    5. Roman Cherniha & Vasyl’ Davydovych, 2021. "New Conditional Symmetries and Exact Solutions of the Diffusive Two-Component Lotka–Volterra System," Mathematics, MDPI, vol. 9(16), pages 1-17, August.
    6. Alexander V. Aksenov & Andrei D. Polyanin, 2021. "Methods for Constructing Complex Solutions of Nonlinear PDEs Using Simpler Solutions," Mathematics, MDPI, vol. 9(4), pages 1-31, February.
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    Cited by:

    1. Vsevolod G. Sorokin & Andrei V. Vyazmin, 2022. "Nonlinear Reaction–Diffusion Equations with Delay: Partial Survey, Exact Solutions, Test Problems, and Numerical Integration," Mathematics, MDPI, vol. 10(11), pages 1-39, May.
    2. Andrei D. Polyanin & Vsevolod G. Sorokin, 2023. "Exact Solutions of Reaction–Diffusion PDEs with Anisotropic Time Delay," Mathematics, MDPI, vol. 11(14), pages 1-19, July.
    3. Andrei D. Polyanin & Vsevolod G. Sorokin, 2023. "Reductions and Exact Solutions of Nonlinear Wave-Type PDEs with Proportional and More Complex Delays," Mathematics, MDPI, vol. 11(3), pages 1-25, January.

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