Comparison and analysis of two forms of harvesting functions in the two-prey and one-predator model

Xinxin Liu,Qingdao Huang

doi:10.1186/s13660-019-2260-y

Xinxin Liu, Qingdao Huang

Open Access

https://doi.org/10.1186/s13660-019-2260-y

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Abstract

A new way to study the harvested predator–prey system is presented by analyzing the dynamics of two-prey and one-predator model, in which two teams of prey are interacting with one team of predators and the harvesting functions for two prey species takes different forms. Firstly, we make a brief analysis of the dynamics of the two subsystems which include one predator and one prey, respectively. The positivity and boundedness of the solutions are verified. The existence and stability of seven equilibrium points of the three-species model are further studied. Specifically, the global stability analysis of the coexistence equilibrium point is investigated. The problem of maximum sustainable yield and dynamic optimal yield in finite time is studied. Numerical simulations are performed using MATLAB from four aspects: the role of the carrying capacity of prey, the simulation about the model equations around three biologically significant steady states, simulation for the yield problem of model system, and the comparison between the two forms of harvesting functions. We obtain that the new form of harvesting function is more realistic than the traditional form in the given model, which may be a better reflection of the role of human-made disturbance in the development of the biological system.

Highlights

The interaction between the predator and prey is one of basic relationships among biological species, which becomes one of the hot issues in ecology and biomathematics
From (3.5), the new form of harvesting function refines the effects of human intervention, which shows that harvesting prey affects the growth of the prey population, and the growth of the predator population
For drawing a precise comparison, we make two prey species have the same kind of growth function and functional response of the predator

Summary

Introduction

The interaction between the predator and prey is one of basic relationships among biological species, which becomes one of the hot issues in ecology and biomathematics. When we only harvest the prey species in a predator–prey system, model (1.1) becomes. 3.1 In the absence of prey x2 Model (2.1) in the absence of prey x2 is equivalent to the predator–prey model with traditional harvesting function given by. 3.2 In the absence of prey x1 Model (2.1) in the absence of prey x1 is equivalent to the predator–prey model with a new form harvesting function given by. Compared with the form of model (3.1), the terms δ1 and δ2 may be the correction terms for the traditional harvesting function given in (3.1) that interpret the schematic presented in Figure 1 better. As human harvesting is considered, the harvesting value Ei may influence the quantity of prey xi (i = 1, 2) in equilibrium point directly, the form of S2∗ is easier to be understood than the form of S1∗ in biological terms.

The predator-free boundary equilibrium point in the x1x2-plane is given by

Local stability of equilibria

Global stability analysis