^{1}

^{2}

^{3,4}

^{1}

^{2}

^{3}

^{4}

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 parameter

Environmental conditions like weather or food availability change significantly throughout the year and influence directly the growth of populations. Responding to seasonal environmental fluctuations, population density can alter quite fast during relatively brief periods, reflecting changes in the living conditions that become less favorable or converse. 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 [

Although importance of the systematic study of the effect of environmental changes on the dynamics of populations has been emphasized in the monographs of MacArtur and Wilson [

In this paper, we investigate the effect of a periodic yield harvesting on the dynamics of a population in a fluctuating environment described by

The dynamics of harvested populations in a fluctuating environment has been addressed by several authors. We mention papers by Benardete et al. [

A bifurcation problem for a differential equation

As Brauer and Sánchez [

For obvious reasons, in population biology, only solutions that take on positive values should be taken into consideration. However, for completeness of mathematical analysis of the problem, we also investigate behavior of solutions that satisfy negative initial conditions or become negative at some instant

We start by providing an introductory information regarding harvesting of a single species. In general case, harvesting of a population can be modeled by a differential equation

The type of harvesting where members of the population are removed at the constant rate per unit time, that is,

In addition to theoretical importance of the study undertaken in this paper, we also stress its practical importance. In fact, a logistic growth model with periodic harvesting

In what follows, we employ concepts of lower and upper fences, also termed lower and upper solutions (subsolutions and supersolutions). Basic facts regarding fences and funnels can be found in Hubbard and West [

The case of periodic proportional harvesting,

Compared to proportional harvesting, the case of periodic yield harvesting, (

Contrary to (

Sánchez [

In the sequel, we use notation

Passing to the case when (

Assume that condition

Observe first that [

Keeping in mind that a pullback attractor is a forward repeller, see Rasmussen [

Rough preliminary estimates (

Assume that (

By (

Consider now the functions

Numerical values in the following example, as well as in the rest of the paper are truncated to four decimal places.

Consider a 1-periodic differential equation

Tighter estimates for both periodic solutions of (

Exact solutions

Unstable and stable periodic solutions embraced by upper and lower solutions.

To describe behavior of solutions

We start with an upper bound for the extinction time

For solutions with initial data

Consider a 1-periodic differential equation

The upper solution (blue) and solution (orange) to (

The upper solution (blue) and solution (orange) to (

In this section, we assume that (

To estimate a forward blow-up time for the solution to (

The upper solution (blue), lower (purple), and solution (orange) to (

To estimate backward blow-up time for solutions to (

To find a backward blow-up time for a solution to (

The upper (blue), lower (purple), and solution to (

Red dashed lines represent two-sided estimates for backward blow-up time for a solution

In a similar manner, one can obtain two-sided estimates for extinction times derived in Section

Nkashama [

Consider now a

We know from Section

For

Let

Condition (

Theorem

Consider a 1-periodic differential equation

Nullclines and constant fences for (

Increasing the value of parameter

Consider a 1-periodic differential equation

Pinched together pieces of upper (blue) and lower (purple) nullclines and one of solutions to (

There exists a bifurcation value

Define the Poincaré map

We stress that Benardete et al. [

Consider a 1-periodic differential equation

Several solutions to (

Several solutions to (

Differential equation (

Qualitative properties of (

This research started when S. P. Rogovchenko and Yu. V. Rogovchenko visited the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy, whose warm hospitality, excellent research facilities, and financial support are gratefully acknowledged. Yu. V. Rogovchenko also acknowledges the research grant from the Faculty of Science and Technology of Umeå University. The authors thank an anonymous referee for useful remarks that helped us to accentuate the importance of the study undertaken in this paper.