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This work examines the limits of the principal spectrum point, $\lambda_p$,of a nonlocal dispersal cooperative system with respect to the dispersal rates.In particular, we provide precise information on the sign of $\lambda_p$ as oneof the dispersal rates is : (i) small while the other dispersal rate isarbitrary, and (ii) large while the other is either also large or fixed. Wethen apply our results to study the effects of dispersal rates on a two-stagestructured nonlocal dispersal population model whose linearized system at thetrivial solution results in a nonlocal dispersal cooperative system. Theasymptotic profiles of the steady-state solutions with respect to the dispersalrates of the two-stage nonlocal dispersal population model are also obtained.Some biological interpretations of our results are discussed.

Asymptotic limits of the principal spectrum point of a nonlocal  dispersal cooperative system and application to a two-stage structured  population model