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Abstract
The paper deals with the following Northeastern Puerto Rico Ecosystem%begin{equation*}left{begin{array}{l}frac{dN}{dt}=left( I-Nright) v-frac{alpha _{N}}{d}frac{[Chl]N}{%K_{N}+leftvert Nrightvert }, frac{d[Chl]}{dt}=[Chl]left[ mu left( frac{PAR^{gamma }}{PAR_{min}^{gamma }+PAR^{gamma }}frac{leftvert N_{B}rightvert ^{alpha -1}N_{B}%}{N_{Bmin }^{alpha }+leftvert N_{B}rightvert ^{alpha }}-xi right) -D%right] , frac{dN_{B}}{dt}=frac{alpha _{N}N}{K_{N}+leftvert Nrightvert }-muleft( frac{PAR^{gamma }}{PAR_{min }^{gamma }+PAR^{gamma }}frac{%leftvert N_{B}rightvert ^{alpha -1}N_{B}}{N_{Bmin }^{alpha}+leftvert N_{B}rightvert ^{alpha }}-xi right) N_{B}.%end{array}%right.end{equation*}%Our aim is to approach the existence of periodic solution undernonautonomous assumption by using the method of Leray-Schauder degree, andthe global asymptotically stable under autonomous assumption. Our globalasymptotically stable results partly generalized [textit{Patrick DeLeenheer, Simon A. Levin, Eduardo D. Sontag, Christopher A. Klausmeier,Global stability in a chemostat with multiple nutrients, J. Math. Biol. 52(2006), 419--438}] from $v=D$ to $vgeq D$.