<p>In this paper, a single population model with impulsive state feedback control and nonlinear harvest is established. The mathematical model is converted into a two-dimensional system by letting <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math> <mi>y</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mi>t</mi> </msubsup> <mspace width="0.25em" /> <mi>α</mi> <mo>exp</mo> <mo stretchy="false">[</mo> <mo>−</mo> <mi>α</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mi mathvariant="normal">d</mi> <mi>s</mi> </math></EquationSource> <EquationSource Format="TEX">$y=\int ^{t}_{-\infty}\ \alpha \exp [-\alpha (t-s)]x(s)\mathrm{d}s$</EquationSource> </InlineEquation>. Sufficient conditions for the existence and stability of order-1 periodic solution of the system are obtained by differential equation geometry theory and successor function theory. In the case of <InlineEquation ID="IEq2"> <EquationSource Format="MATHML"><math> <mi>h</mi> <mo>&gt;</mo> <msup> <mi>x</mi> <mo>∗</mo> </msup> </math></EquationSource> <EquationSource Format="TEX">$h&gt;x^{*}$</EquationSource> </InlineEquation> (here <i>h</i> is the threshold of population density and <InlineEquation ID="IEq3"> <EquationSource Format="MATHML"><math> <msup> <mi>x</mi> <mo>∗</mo> </msup> </math></EquationSource> <EquationSource Format="TEX">$x^{*}$</EquationSource> </InlineEquation> represents the equilibrium point), we first discuss the sub-case where the stability of the unique positive equilibrium of the system is stable, and then discuss the sub-case where the unique positive equilibrium is unstable and has a unique stable limit cycle. Numerical simulations are provided to support evidences of our analytical findings and explore biological significance in the end of this paper.</p>