Periodic solutions of a logistic type population model with harvesting. (English) Zbl 1210.34057

Summary: We consider a bifurcation problem arising from population biology
\[ \frac{du(t)}{dt} = f(u(t))-\varepsilon h(t), \]
where \(f(u)\) is a logistic type growth rate function, \(\varepsilon \geqslant 0\), \(h(t)\) is a continuous function of period \(T\) such that \(\int _0^Th(t)\,dt >0\). We prove that there exists an \(\varepsilon_0 >0\) such that the equation has exactly two \(T\)-periodic solutions when \(0< \varepsilon < \varepsilon _0\), exactly one \(T\)-periodic solution when \(\varepsilon = \varepsilon _0\), and no \(T\)-periodic solution when \(\varepsilon > \varepsilon _0\)


34C25 Periodic solutions to ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
92D25 Population dynamics (general)
Full Text: DOI


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