Bifurcations of a Leslie‐Gower prey‐predator system with ratio‐dependent Holling IV functional response and prey harvesting

Abstract

The dynamics of a Leslie‐Gower prey‐predator system with ratio‐dependent Holling IV functional response and constant harvesting rate of prey are taken into account. The results developed in this article reveal far richer dynamics compared with the system without harvesting. We first make qualitative and bifurcation analysis of the system without harvesting and show that the system has a weak focus of multiplicity at most 2, at which a Hopf bifurcation occurs. However, the system with harvesting has four nonhyperbolic equilibria for some parameter values, such as two saddle‐node, a cusp, and a weak focus of multiplicity at most 4, and exhibits two saddle‐node bifurcations, a Bogdanov‐Takens bifurcation of codimension 2, and a Hopf bifurcation. It reveals that there exist some critical harvesting values such that the species are in danger of extinction when the harvesting rate is greater than the critical values, which indicates that the dynamics of the system are sensitive to the constant prey harvesting. Moreover, numerical simulations are presented to illustrate our theoretical results.

Global dynamics and bifurcations in a SIRS epidemic model with a nonmonotone incidence rate and a piecewise-smooth treatment rate

“…In the next theorem, we express the bifurcation curves within Bogdanov-Takens bifurcation by the original parameters of system (9) (see [25]).…”

Analisis dinamik model predator-prey tipe Gause dengan wabah penyakit pada prey

“…Interaksi antara dua atau lebih spesies dalam suatu ekosistem merupakan topik yang populer dalam pemodelan matematika karena menyangkut eksistensi dari populasi spesies-spesies tersebut. Salah satu model interaksi antara dua spesies yang memiliki perkembangan pesat dalam pemodelan matematika yaitu model predator-prey, sebagai contoh lihat [1][2][3][4][5][6][7][8]. Selain karena adanya interaksi antara populasi, eksistensi dari suatu populasi juga dipengaruhi oleh adanya infeksi penyakit baik yang mewabah secara tertutup dalam suatu populasi ataupun menyebar lintas populasi.…”

Mathematical Study on a Dynamical Predator-Prey Model with Constant Prey Harvesting and Proportional Harvesting in Predator

“…To tackle that type of situation, many researchers worked on a predator-prey system with harvesting effects (15) (16) (17) (18). Basically researchers used three types of harvesting and the classification is constant harvesting (19), proportional harvesting (20) and non-linear harvesting (21). In this paper we consider harvested predator and prey population.…”