Owl Carousel, navigation disabled after reaching first/last item

I had the same issue with Owl Carousel 2, My solution - with the native navigation buttons, after calling the slider:

             var elm = '.slider'; //your slider class
             function toggleArrows(){ 
                if($(elm).find(".owl-item").last().hasClass('active') && 
                   $(elm).find(".owl-item.active").index() == $(elm).find(".owl-item").first().index()){                       
                    $(elm).find('.owl-nav .owl-next').addClass("off");
                    $(elm).find('.owl-nav .owl-prev').addClass("off"); 
                }
                //disable next
                else if($(elm).find(".owl-item").last().hasClass('active')){
                    $(elm).find('.owl-nav .owl-next').addClass("off");
                    $(elm).find('.owl-nav .owl-prev').removeClass("off"); 
                }
                //disable previus
                else if($(elm).find(".owl-item.active").index() == $(elm).find(".owl-item").first().index()) {
                    $(elm).find('.owl-nav .owl-next').removeClass("off"); 
                    $(elm).find('.owl-nav .owl-prev').addClass("off");
                }
                else{
                    $(elm).find('.owl-nav .owl-next,.owl-nav .owl-prev').removeClass("off");  
                }
            }

            //turn off buttons if last or first - after change
            $(elm).on('initialized.owl.carousel', function (event) {
                toggleArrows();
            });
             $(elm).on('translated.owl.carousel', function (event) { toggleArrows(); });

As for the css - give the class "off" the properties you want for disabled button.

You can use callbak afterAction and with your custom controls

afterAction: function(){
  if ( this.itemsAmount > this.visibleItems.length ) {
    $('.next').show();
    $('.prev').show();

    $('.next').removeClass('disabled');
    $('.prev').removeClass('disabled');
    if ( this.currentItem == 0 ) {
      $('.prev').addClass('disabled');
    }
    if ( this.currentItem == this.maximumItem ) {
      $('.next').addClass('disabled');
    }

  } else {
    $('.next').hide();
    $('.prev').hide();
  }
}

Check the working demo - http://jsfiddle.net/p3d52z4n/9/

Simply use callbacks-

loop:false,
navRewind: false

You'll notice that a 'disabled' class is added to owl-next and owl-prev when the first/last item is reached. Adding CSS-

.owl-next.disabled, .owl-prev.disabled {
display: none !important;
}

will do the trick.

Simplest solution:

OwlCarousel 2 gives disabled class to nav elements when the first/last element is reached.

So you could just need something like that:

.owl-nav{
  .disabled{
    display: none;
  }
}