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Minimal non-standard number in non-standard models of PA

Excuse me, if the question sounds too naive.
From godel's incompleteness theorem we know that there would be non-standard models where the godel sentence would be false. These models will have an initial segment isomorphic to standard natural numbers. Will there be a minimal non-standard number in such models such that every number smaller than it is a standard natural number and every number bigger than it would be non-standard ?
~~Since non-standard model would be a model of arithmetic then i think there should be a minimal non-standard number, but then maybe my concept is unclear about it. Any help ?~~

Excuse me, if the question sounds too naive.
From Gödel's incompleteness theorem we know that there would be non-standard models where the Gödel sentence would be false. These models will have an initial segment isomorphic to standard natural numbers. Will there be a minimal non-standard number in such models such that every number smaller than it is a standard natural number and every number bigger than it would be non-standard ?
Since non-standard model would be a model of arithmetic then I think there should be a minimal non-standard number, but then maybe my concept is unclear about it. Any help?

logic model peano-arithmetic nonstandard-models