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Existence of positive periodic solution of periodic time-dependent predator-prey system with impulsive effects. (English) Zbl 1058.34051

The following impulsive problem is considered

$\begin{array}{cc}\hfill {\stackrel{˙}{N}}_{1}& ={N}_{1}\left({b}_{1}-{c}_{11}{N}_{1}-{c}_{12}{N}_{2}\right),\hfill \\ \hfill {\stackrel{˙}{N}}_{2}& ={N}_{2}\left(-{b}_{2}+{c}_{21}{N}_{1}-{c}_{22}{N}_{2}\right),\hfill \\ \hfill {\Delta }{N}_{1}& ={c}_{k}{N}_{1},\phantom{\rule{1.em}{0ex}}{\Delta }{N}_{2}={d}_{k}{N}_{2},\phantom{\rule{1.em}{0ex}}t={\tau }_{k},\hfill \end{array}$

where ${b}_{i}\left(t\right),{c}_{ij}\left(t\right)$ are continuous, $T$-periodic positive functions,

${c}_{k+q}={c}_{k},\phantom{\rule{3.33333pt}{0ex}}{d}_{k+q}={d}_{k},\phantom{\rule{3.33333pt}{0ex}}{\tau }_{k+q}={\tau }_{k}+T,1+{c}_{k}>0,\phantom{\rule{3.33333pt}{0ex}}1+{d}_{k}>0·$

Under some additional assumptions, using bifurcation theory, the authors prove the existence of a positive periodic solution for this system and discuss some biological applications.

##### MSC:
 34C25 Periodic solutions of ODE 34A37 Differential equations with impulses 92D25 Population dynamics (general) 34C23 Bifurcation (ODE)
##### Keywords:
predator-prey system; impulsive effects; periodic solution
##### References:
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