Population dynamics between a prey and a predator using spectral collocation method. (English) Zbl 1422.65291
Summary: The struggle for the existence of the biological species is a well-known prey-predator model study in the literature. In this study, we present an improved model of A. J. Jerri [Introduction to integral equations with applications. 2nd ed. New York, NY: Wiley (1999; Zbl 0938.45001)] by introducing the intra-species competition term between the same species in addition to the existing environmental changes and few other factors in the model. The demand from the existing (limited) resources and other requirements induces competition between the same species which may alter the survival tactics among themselves. This intra-species term provides strength to the model as it makes the model more realistic. The governing equations are a system of two nonlinear delay integro differential equations, which are solved using spectral collocation method. The role of intra-species coefficients denoting the logistic growth/decay of the two species and two other parameters affecting the population dynamics are analyzed with the three basis functions such as Chebyshev, Legendre and Jacobi polynomials. With the help of simple matrix analysis, the governing equations are converted into a system of nonlinear algebraic equations. Detailed error estimation is computed to compare our results with the existing methods. It is shown with the help of tables and figures that the present method is very efficient, has better accuracy and has least computational cost.
MSC:
| 65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |
| 45J05 | Integro-ordinary differential equations |
| 97M60 | Biology, chemistry, medicine (aspects of mathematics education) |
| 76M22 | Spectral methods applied to problems in fluid mechanics |
| 92D25 | Population dynamics (general) |
| 35R09 | Integro-partial differential equations |
| 41A50 | Best approximation, Chebyshev systems |
| 42C10 | Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.) |
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
Citations:
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