Periodic solutions of a periodic delay predator-prey system

• DOI: 10.1090/s0002-9939-99-05210-7
• Publication Date: 2002-07-26T22:14:44Z
• Authors: Li Yongkun
• Keywords: 0101 mathematics; 01 natural sciences

The existence of a positive periodic solution for <disp-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout Enlarged left-brace 1st Row StartFraction normal d upper H left-parenthesis t right-parenthesis Over EndFraction equals r left-bracket 1 minus tau K right-bracket alpha P comma b plus beta sigma EndLayout"> <mml:semantics> <mml:mrow> <mml:mo>{</mml:mo> <mml:mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mml:mtr> <mml:mtd> <mml:mfrac> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">d</mml:mi> </mml:mrow> <mml:mi>H</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>t</mml:mi> stretchy="false">)</mml:mo> </mml:mfrac> <mml:mo>=</mml:mo> <mml:mi>r</mml:mi> <mml:mo>[</mml:mo> <mml:mn>1</mml:mn> <mml:mo>−</mml:mo> <mml:mi>τ</mml:mi> <mml:mi>K</mml:mi> <mml:mo>]</mml:mo> <mml:mi>α</mml:mi> <mml:mi>P</mml:mi> <mml:mo>,</mml:mo> <mml:mtext> </mml:mtext> <mml:mi>b</mml:mi> <mml:mo>+</mml:mo> <mml:mi>β</mml:mi> <mml:mi>σ</mml:mi> </mml:mtd> </mml:mtr> </mml:mtable> fence="true" stretchy="true" symmetric="true"/> <mml:annotation encoding="application/x-tex">\begin{equation*} \begin {cases} \frac {\mathrm {d}H(t)}{\mathrm {d}t}=r(t)H(t) \left [1-\frac {H(t-\tau (t))}{K(t)}\right ] -\alpha (t)H(t) P(t),\ {d}P(t)}{\mathrm {d}t}=-b(t)P(t)+\beta (t)P(t)H(t-\sigma (t)) \end{cases} \end{equation*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> is established, where <inline-formula alttext="r"> encoding="application/x-tex">r</mml:annotation> </inline-formula>, alttext="upper K"> encoding="application/x-tex">K</mml:annotation> alttext="alpha"> encoding="application/x-tex">\alpha</mml:annotation> alttext="b"> encoding="application/x-tex">b</mml:annotation> alttext="beta"> encoding="application/x-tex">\beta</mml:annotation> </inline-formula> are continuous functions with period alttext="omega greater-than 0"> <mml:mi>ω</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> encoding="application/x-tex">\omega &gt;0</mml:annotation> and alttext="tau"> encoding="application/x-tex">\tau</mml:annotation> alttext="sigma"> encoding="application/x-tex">\sigma</mml:annotation> alttext="omega"> encoding="application/x-tex">\omega</mml:annotation> </inline-formula>.