A Different Numerical Perspective for Prey-predator Problem with Harvesting

Özdoğan N., Yakıt Ongun M.

1st International Conference on Applied Engineering and Natural Sciences, Konya, Türkiye, 1 - 03 Kasım 2021, ss.450, (Özet Bildiri)

  • Yayın Türü: Bildiri / Özet Bildiri
  • Basıldığı Şehir: Konya
  • Basıldığı Ülke: Türkiye
  • Sayfa Sayıları: ss.450
  • Süleyman Demirel Üniversitesi Adresli: Evet

Özet

Mathematical modelling is one of the most effective methods used to explain the interactions between species in ecosystem. Mathematical models of prey-predator systems have been researched by many ecologists because they are related to human behaviours and ecological life. These systems cannot be solved analytically, so numerical techniques have been developed to find the approximate solutions. Prey-predator models can be seen as the building blocks of ecology. Therefore, we have been investigated a prey-predator problem that predator density is subjected to harvesting. Harvesting can be two fold in the prey-predator problem. The main purpose is optimal use of the harvested stock to increase the profit. In this work, some numerical methods such as Runge-Kutta method, Theta method, Nonstandard Finite Difference method, have been proposed to observe the dynamic properties and the stability of the equilibrium points in the system. Among the numerical methods that we have been compared, it was seen that the nonstandard finite difference discretization based on Micken’s rules, gave more accurate results with the use of arbitrary time step-size. One of the advantages of this method is that better results can be found with the appropriate denominator function. Also, this new scheme preserves the stability of the equilibrium points and the main features such as positivity of the continuous system. Therefore it provides reliable numerical results. In this study, numerical simulations have been presented to see the effectiveness of the methods. The effect of time step size on numerical methods were shown with the help of the tables. Also phase portraits were drawn. Keywords – Dynamic Consistency, Local Stability, Equilibrium Points, Nonstandard Finite Difference Scheme, Numerical Methods. .