A Fractional-Order Predator-Prey Model with Age Structure on Predator and Nonlinear Harvesting on Prey

Abstract

In this manuscript, the dynamics of a fractional-order predator-prey model with age structure on predator and nonlinear harvesting on prey are studied. The Caputo fractional-order derivative is used as the operator of the model by considering its capability to explain the present state as the impact of all of the previous conditions. Three biological equilibrium points are successfully identified including their existing properties. The local dynamical behaviors around each equilibrium point are investigated by utilizing the Matignon condition along with the linearization process. The numerical simulations are demonstrated not only to show the local stability which confirms all of the previous analytical results but also to show the existence of periodic signal as the impact of the occurrence of Hopf bifurcation.

Analisis Dinamik Model Predator-Prey dengan Struktur Usia dan Perilaku Anti-Predator

“…Riset tentang interaksi dua populasi yang berkaitan dengan model Lotka-Volterra terus dilakukan dengan berbagai jenis modifikasi. Diantaranya adalah modifikasi model Lotka-Volterra dengan mempertimbangkan adanya struktur usia pada predator dan fungsi respon Monod-Haldane [4,5]. Beberapa penelitian tentang model predator-prey juga ditemukan dengan modifikasi model berupa perlakukan khusus pada salah satu pepulasi, seperti adanya perlakuan berupa pemberian racun pada predator [6] dan perlakuan berupa proteksi yang diberikan pada populasi prey [7].…”

A Fractional-Order Leslie-Gower Model with Fear and Allee Effect

“…The main concern of this situation is how to maintain the availability of the ecosystem resources. In ecology, the presence of prey depends on how they can protect themselves, and the presence of predators depends on the availability of prey [1]. These relationships make researchers interested in forming predator-prey interactions into mathematical models.…”

“…That is, all current conditions of population density depend on all previous conditions [23]. Therefore, we also consider the memory effect which means the effect of all previous biological conditions to the present condition by replacing the first-order derivative with the fractional-order derivative [1], [9], [17], [24]. The memory effect shows that the population dynamics of present condition depend on all previous conditions stored in their memory system such as the experience in foraging, the best place to take shelter, the perfect time to migrate, and so on [1].…”

“…Therefore, we also consider the memory effect which means the effect of all previous biological conditions to the present condition by replacing the first-order derivative with the fractional-order derivative [1], [9], [17], [24]. The memory effect shows that the population dynamics of present condition depend on all previous conditions stored in their memory system such as the experience in foraging, the best place to take shelter, the perfect time to migrate, and so on [1]. To play the superlative form of model, ordinary calculus is less effective in describing complex phenomenon involving memory effect and hereditary biological properties [25].…”

“…There are several well-known fractional-order derivatives that are used as operators in the predator-prey model. By considering the availability of analytical tools, we choose the Caputo fractional-order for our model as done by [1], [9], [17], [24]. According to [22], the Caputo fractional-order can be used on the classic initial condition as in the integer order equations.…”