Publication: Dynamical systems approach to a bioeconomic differential algebraic predator–prey model with harvesting
Abstract
Bu çalışmada, fonksiyonel tepkinin Holling 2 türü olduğu, hem av hem de avcı hasadına dayanan, biyoekonomik diferensiyel-cebirsel bir av-avcı modeli sunulmuştur. Ekonomik faktörü de göz önünde bulundurarak biyoekonomik diferansiyel-cebirsel bir denklem sistemi elde edilmiştir.Yukarıda verilen sistemin bir iç denge noktasına sahip olduğu gösterilmiştir. Bu denge noktası, ekonomik kar küçükken kararlı olduğu bulunmuştur. Ancak, ekonomik kar arttırıldığında denge noktasının kararlılığını kaybettiği gözlenmiştir. Bu fenomene çatallanma denilir. Biz bu tezde denge noktasının yerel çatallanmasını hem tekillik kaynaklı çatallanma hem de Hopf çatallanması paradigmalarıyla inceledik. Matematiksel bir kesinlikle, net ekonomik karın sıfır olduğu noktada tekillik kaynaklı bir çatallanma olduğunu gösterdik. Ayrıca denge noktasının bir Hopf çatallanması gösterdiğini normal form analizi sayesinde gösterdik. Son olarak da teorik sonuçlarımızın geçerliliğini nümerik simülasyonlar kullanarak inceledik. Nümerik simülasyonlarımızın teorik sonuçlarla uyumlu olduğunu gözlemledik.
In this work, we offer a differential algebraic population model which describe the dynamics of predator and prey species with Holling Type II functional response and a harvesting of both the prey and the predator populations. Considering the economic factor, a bioeconomic differential algebraic equation is obtained.We show that the above system displays an interior equilibrium point. This equilibrium is stable when the economic profit is negative. However, as the economic profit is increased this equilibrium loses stability. This phenomenon is known as bifurcation. We investigate this bifurcation both from the view point of singularity induced bifurcation as well as Hopf bifurcation. We rigorously prove that a singularity induced bifurcation occurs when the net economic profit is zero. We also study the existence of Hopf bifurcation by means of normal form analysis. Finally we investigate the theoretical findings by means of numerical simulations. We find that our theoretical results match well with the numerical simulations.
In this work, we offer a differential algebraic population model which describe the dynamics of predator and prey species with Holling Type II functional response and a harvesting of both the prey and the predator populations. Considering the economic factor, a bioeconomic differential algebraic equation is obtained.We show that the above system displays an interior equilibrium point. This equilibrium is stable when the economic profit is negative. However, as the economic profit is increased this equilibrium loses stability. This phenomenon is known as bifurcation. We investigate this bifurcation both from the view point of singularity induced bifurcation as well as Hopf bifurcation. We rigorously prove that a singularity induced bifurcation occurs when the net economic profit is zero. We also study the existence of Hopf bifurcation by means of normal form analysis. Finally we investigate the theoretical findings by means of numerical simulations. We find that our theoretical results match well with the numerical simulations.