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Too me it appears the problem is not at the tree, but some distance above the tree, where the water is coming from. The reason I say this is there is a lot of water passing by the tree, moving fairly rapidly fast enough to cause the erosion.

I would look at the area where the water is coming from and determine what is funneling the water to the tree.

Once you understand the water flow in the area re-contour the ground to move the water down the slow in several places, and divert it down across and down the slope so it is not running so fast to erode the ground it is moving over.

I would like to revisit this topic I posted. Since this rut is underneath a pear tree in my front yard, and it is sloping down in this section, do you think if I fill the rut with dirt, and then lay some grass sod on it, with a more shade tolerant grass, like zoysia, that it might help fix my issue? Most of my grass is Bermuda.....not sure how it would like having Zoysia mixed in with Bermuda. Or should I just maybe put plants that aid in soil stabilization all around the tree in the shady areas?

That netting looks like the leftovers of a grass mat meant to hold grass in place for erosion control. Obviously it failed and failed miserably.

What the ideal would be is for tree roots to grow into the eroded area to hold the soil and the grass be the surface's first line of defense.

I would discern the reason for the pear tree is not spreading out as it should, correct that problem and place sod down that is dug up with a shovel. I fill in those type of problems with sod from an area where the roots are 6" deep or more and relocate it by the shovel full.

I'm guessing your pear tree roots are either drying out or are bound up by artificial means.
Let us know what you find. :)

"do you think if I fill the rut with dirt, and then lay some grass sod on it, with a more shade tolerant grass, like zoysia, that it might help fix my issue? "

No. Your issue is not the hole by the tree ... it's the water that is causing the hole, even though the water is not currently visible.

What will fix it is to go up-slope to where the water is coming from and start there with diverting it, creating a good drainage channel, slowing it down to soak in ... whatever it takes to stop the water that is flowing past the tree and eroding the dirt.

Then you can fill the hole and plant some grass. Pruning the tree up a bit might let the bermuda spread in there.

"do you think if I fill the rut with dirt, and then lay some grass sod on it, with a more shade tolerant grass, like zoysia, that it might help fix my issue? "

No. Your issue is not the hole by the tree ... it's the water that is causing the hole, even though the water is not currently visible.

What will fix it is to go up-slope to where the water is coming from and start there with diverting it, creating a good drainage channel, slowing it down to soak in ... whatever it takes to stop the water that is flowing past the tree and eroding the dirt.

Then you can fill the hole and plant some grass. Pruning the tree up a bit might let the bermuda spread in there.

I agree with others here that you need to fix the water flowing under and through the tree first. But I would warn against trying to plant grass under the tree unless you are getting at least 6-8 hours of sun under the tree which is very unlikely as the grass will get thin over time and not have much of a root system to hold anything. Your better bet would to make a mulch bed under and around the tree and plant either some ground cover or some plants that require light sun or shade to help hold the soil.

We had a golf course greens keeper come talk to us at a local organic meeting back in the 90s. He had some ditches where water flowed pretty hard during storms. He did not want to resort to the normal plastic sheeting solution used around here, so he dumped chipped wood they had around from clearing the golf course area. It worked perfectly. They just dumped a pile in the center of the ditch. None of it washed away, and it worked perfectly to slow the flow of water down the ditch.

I'm not quite sure the best way to stop this water flow towards this spot. Would one of these catch basins with a top grate that I see on Lowe's web site placed further up the hill in line with the erosion problem help? I see it has connectors to be able to connect corrugated pipe....but I'm not sure in my application where I would run my pipe to.

You can not stop the water. Diverting away fro the area is the only thing to do. If you want to build hardware into your landscape like a wild drain sewer, then there is no need to divert before the water gets there.
I would go up hill and build mini berms that move the flow sideways. Do you have pix above the spot?

It will take a little work, but this is what I would do. If this is a large area I may consider renting specialized equipment, ie sod cutter

I would go to the area where the water is coming from an sit down in several places on the grass and look at the area, studying exactly how water runs down the slopes.

I would then lay out swales create by low mini berms, Once I have it laid out I would cut the sod from the area where the swale is to be. I would dig down a couple of inches on the up hill side, and use the dirt to build the berm on the down hill side. When building the berm, you want the depth of the swale and the height of the berm to be such that you can run the mower over them with out scalping the grass. A four foot wide area, with the up hill side dug out a couple of inches and the dirt pile on the down hill side creates a 8 inch swale that is easily mowed over.

One I raked the area to the contour I wanted, I would replace the sod. In a couple of weeks the newly sodded area of your yard will blend in and you will never see the work you put into the area.

A variation would be to make the mine berms raised flower beds, again creating a swale on the up hill side.

Just curious as to why you don't want to use mulch???