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Existence of periodic solutions for a periodic mutualism model on time scales. (English) Zbl 1146.34326

Summary: By using Mawhin’s continuation theorem of coincidence degree theory, sufficient criteria are obtained for the existence of periodic solutions of the mutualism model

$\left\{\begin{array}{c}{x}^{{\Delta }}\left(t\right)={r}_{1}\left(t\right)\left[\frac{{K}_{1}\left(t\right)+{\alpha }_{1}\left(t\right)exp\left\{y\left(t-{\tau }_{2}\left(t,y\left(t\right)\right)\right)\right\}}{1+exp\left\{y\left(t-{\tau }_{2}\left(t,y\left(t\right)\right)\right)\right\}}-exp\left\{x\left(t-{\sigma }_{1}\left(t,x\left(t\right)\right)\right)\right\}\right],\hfill \\ {y}^{{\Delta }}\left(t\right)={r}_{2}\left(t\right)\left[\frac{{K}_{2}\left(t\right)+{\alpha }_{2}\left(t\right)exp\left\{x\left(t-{\tau }_{1}\left(t,x\left(t\right)\right)\right)\right\}}{1+exp\left\{x\left(t-{\tau }_{1}\left(t,x\left(t\right)\right)\right)\right\}}-exp\left\{y\left(t-{\sigma }_{2}\left(t,y\left(t\right)\right)\right)\right\}\right],\hfill \end{array}\right\$

where ${r}_{i}$, ${K}_{i}$, ${\alpha }_{i}\in C\left(𝕋,{ℝ}^{+}\right)$, ${\alpha }_{i}>{K}_{i}$, $i=1,2,{\tau }_{i}$, ${\sigma }_{i}\in C\left(𝕋×ℝ,{𝕋}^{+}\right)$, $i=1,2,{r}_{i},{K}_{i},{\alpha }_{i},{\tau }_{i},{\sigma }_{i}$ $\left(i=1,2\right)$ are functions of period $\omega >0$.

##### MSC:
 34K13 Periodic solutions of functional differential equations 92D25 Population dynamics (general) 39A10 Additive difference equations
##### References:
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