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

Öz

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

Anahtar Kelimeler

• 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

Öz

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. 

Anahtar Kelimeler

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

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