Highly Accurate Numerical Technique for Population Models via Rational Chebyshev Collocation Method

Ramadan Ramadan,Nassar Nassar,Baleanu Baleanu

doi:10.3390/math7100913

Ramadan Ramadan, Nassar Nassar + Show 1 more

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https://doi.org/10.3390/math7100913

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Abstract

The present work introduces the application of rational Chebyshev collocation technique for approximating bio-mathematical problems of continuous population models for single and interacting species (C.P.M.). We study systematically the logistic growth model in a population, prey-predator model: Lotka-Volterra system (L.V.M.), the simple two-species Lotka-Volterra competition model (L.V.C.M.) and the prey-predator model with limit cycle periodic behavior (P.P.M.). For testing the accuracy, the numerical results for our method and others existing methods as well as the exact solution are compared. The obtained numerical results indicate the ability, the reliability and the accuracy of the present method.

Highlights

IntroductionThe nonlinear differential equations and their system play a crucial role due to their applications in applied mathematics and science, for example, in real life phenomena modeling and in many other fields of science, such as the epidemic model [1,2], kinetic model [3,4], ozone decomposition model [5,6], dynamical models of happiness [7], modeling of mosquito dispersal [8], modeling a thermal explosion [9] and Volterra population model [10].The purpose of this investigation is applying rational Chebyshev (RC) collocation method to solve four nonlinear biological problems
The nonlinear system is in the rational Chebyshev (RC) coefficients, where one can use a suitable numerical method to solve this system
We can see the rational Chebyshev functions are better bases to deal with problems such as the four models, as we see the analytical solution of the first model C.P.M. in fraction form for this proposed technique gives high efficiency along the domain

Summary

Introduction

The nonlinear differential equations and their system play a crucial role due to their applications in applied mathematics and science, for example, in real life phenomena modeling and in many other fields of science, such as the epidemic model [1,2], kinetic model [3,4], ozone decomposition model [5,6], dynamical models of happiness [7], modeling of mosquito dispersal [8], modeling a thermal explosion [9] and Volterra population model [10].The purpose of this investigation is applying rational Chebyshev (RC) collocation method to solve four nonlinear biological problems. Four problems are investigated; the first problem is the continuous population model (C.P.M.) represented as a nonlinear first order ordinary differential equation, whereas the other models are systems of non-linear differential equations. They are represented, respectively, as the Lotka-Volterra system (L.V.M.), Lotka-Volterra competition model (L.V.C.M.) and prey-predator model (P.P.M.) [11]

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Results

Conclusion