A numerical scheme for continuous population models for single and interacting species

Year 2017, Volume: 19 Issue: 1, 12 - 28, 08.06.2017

Yalçın Öztürk , Ayşe Anapalı Mustafa Gülsu

https://doi.org/10.25092/baunfbed.320025

Cited By: 1

Abstract

In this article, the dynamic of models such as logistic growth model,
prey-predator model and 2-species Lotka-Volterra competition model is
approximately solved by the Chebyshev collocation method.  These nonlinear mathematical models are
transformed into the matrix form by Chebyshev expansion method and converted
nonlinear algebraic equation system. Chebyshev coefficients are obtained by
solving nonlinear equation system. Results are compared with Homotopy
perturbation and Adomian decomposition method and then comparision numerical
result and exact solution are presented by graphics for logistic growth model.
Plots are showed the numbers of prey and predator versus time for various N
values on predaor prey model. In the 2 spices Lotka Volterra competition model
numerical results are presented by graphics. Matlab R2010a and Mapple14 are
used for all calculations and graphs. In the conclusion part, the CPU times of
the programs are given and the models are compared

Keywords

Logistic growth model, prey and predator model, Lotka-Volterra model, 2-species Lotka-Volterra model, system of nonlinear differential equations

Tek ve etkileşimli türlerin sürekli populasyon modelleri için bir sayısal yöntem

Abstract

Bu makalede, lojistik büyüme modeli, av avcı modeli ve 2-tür
Lotka-Volterra yaşama mücadelesi modeli gibi modeller Chebyshev sıralama metodu
ile çözülmüştür. Bu lineer olmayan matematiksel modeller Chebyshev açılımı
metodu ile matris formuna dönüştürülmüş ve lineer olmayan cebirsel denklem
sistemine indirgenmiştir. Lineer olmayan denklem sistemi çözülerek Chebyshev
katsayıları elde edilmiştir. Lojistik büyüme modeli için sonuçlar homotopy
perturbation metodu ve Adomian decomposition metodu ile karşılaştırılmış ve
elde edilen nümerik sonuçlar ile tam çözümün karşılaştırılması grafiklerle
sunulmuştur. Av-avcı modelinde grafikler yardımı ile av ve avcı sayılarının
zamana karşı olan durumları farklı N değerleri için gösterilmiştir. 2 tür Lotka
Volterra yaşama mücadelesi modelinde nümerik sonuçlar grafik ile ifade
edilmiştir. Yapılan tüm hesaplamalar ve grafik çizimlerinde Matlab R2010a ve
Maple14 kullanılmıştır. Ayrıca sonuç kısmında programların CPU zamanları
verilerek modeller arası karşılaştırmalar yapılmıştır. 

Keywords

Lojistik büyüme modeli, av-avcı modeli, Lotka-Volterra modeli, 2-tür Lotka-Volterra modeli, linear olmayan diferansiyel denklem sistemi

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