Dynamics of a Single Species in a Fluctuating Environment under Periodic Yield Harvesting

Mustafa Hasanbulli,Svitlana P Rogovchenko,Yuriy V Rogovchenko

doi:10.1155/2013/167671

Abstract

We discuss the effect of a periodic yield harvesting on a single species population whose dynamics in a fluctuating environment is described by the logistic differential equation with periodic coefficients. This problem was studied by Brauer and Sánchez (2003) who attempted the proof of the existence of two positive periodic solutions; the flaw in their argument is corrected. We obtain estimates for positive attracting and repelling periodic solutions and describe behavior of other solutions. Extinction and blow-up times are evaluated for solutions with small and large initial data; dependence of the number of periodic solutions on the parameterσassociated with the intensity of harvesting is explored. Asσgrows, the number of periodic solutions drops from two to zero. We provide bounds for the bifurcation parameter whose value in practice can be efficiently approximated numerically.

Highlights

Environmental conditions like weather or food availability change significantly throughout the year and influence directly the growth of populations
Since in many cases environmental fluctuations have a clearly pronounced seasonal character, they can be efficiently modeled with the help of nonautonomous differential equations with periodic coefficients
A striking example of a positive effect of a periodically fluctuating environment on the dynamics of a species has been reported by Jillson [1] who observed that total population numbers in the flour beetle population in the periodically fluctuating environment were more than twice those in the constant environment

Summary

Introduction

Environmental conditions like weather or food availability change significantly throughout the year and influence directly the growth of populations. We investigate the effect of a periodic yield harvesting on the dynamics of a population in a fluctuating environment described by dx (t) dt. We obtain estimates for positive attracting and repelling periodic solutions to (1) in case of periodic yield harvesting, describe behavior of other solutions, and derive estimates for extinction and blowup times. For completeness of mathematical analysis of the problem, we investigate behavior of solutions that satisfy negative initial conditions or become negative at some instant t∗ In the former case, such analysis is completely irrelevant for applications, whereas in the latter case the phrase “solutions decay to −∞” should be interpreted in biological terms as “the population goes extinct”; we provide useful estimates for extinction times

Periodic Solutions and Harvesting

Saddle-Node Bifurcation

Conclusions