tl;dr- The cited study's completely unscientific, and its authors have denounced it in a new paper. Their new paper's junk, too, but at least they seem to have learned their lesson about making doomsday predictions.
The original study was
and their new study's
The new study has this to say of the original one:
Admittedly, the apocalyptic prediction of the global oxygen depletion remains hypothetical as it was made in a theoretical study based on a rather simple, albeit general, mathematical model. More research is required before this could be accepted as a well established scientific fact.
-"Regime shifts and ecological catastrophes in a model of plankton-oxygen dynamics under the climate change" (2017-07-07)
So, the original authors don't stand by this claim. They're being excessively generous in describing their prior paper as "theoretical", but at least they've conceded that it's "hypothetical".
The original study was indefensible. They literally picked vastly oversimplified equations, kinda like trying to do a large quantum mechanical model with a linear regression, and then literally guessed values for their constants based on what seemed easiest. Then, when the equations crashed, they described it as predicting doomsday.
As described in an observer's blog:
There is also the 'extreme-warmer' view, that the effects of CO2 will be so large as to 'fry the planet', leading to the extinction of humans, and perhaps all life, which is surprisingly common among the general public, despite being utterly implausible. Of course, they are encouraged in this belief by alarmist papers such as 'Mathematical Modelling of Plankton–Oxygen Dynamics Under the Climate Change' by Sekerci and Petrovskii, who apparently don’t understand that any arbitrary system of differential equations has a good chance of producing unstable behaviour, and that calling such a system a 'model of a coupled plankton–oxygen dynamics' does not make it a good model. It is very, very unlikely that life on earth would have lasted for over three billion years if the global ecosystem were really as unstable as is suggested in this paper.
-"What can global temperature data tell us?", Radford Neal's blog (2015-12-03) [backticks modified]
Regarding their new study
The new study's highlights are:
Highlights
A model of plankton-oxygen dynamics under the climate change is developed and studied.
Sustainable oxygen production is only possible within a relatively narrow parameter range.
The global warming can cause the oxygen production to stop.
The regime shift is shown to be preceded by an increased regularity in plankton spatial distribution and long term transient dynamics.
-"Regime shifts and ecological catastrophes in a model of plankton-oxygen dynamics under the climate change" (2017-07-07)
The "parameters" referred to in
Sustainable oxygen production is only possible within a relatively narrow parameter range.
are their guessed values. What this means is that they don't have real data or anything like that, but rather they're making up numbers, and most of their numbers result in the equations crashing (non-"[s]ustainable oxygen production").
So, their strategy's to keep guessing numbers 'til their equations stop crashing, the range of which constitutes the "relatively narrow parameter range".
Then, how does global warming affect that? They have no idea. In fact, any slight cooling or warming can throw off those parameters again. Alternatively, they could select parameters that're stable for extreme global warming, in which case their equations would predict that global warming is necessary to save the world.
Technical stuff
This section's to provide further explanation of the original study, and some of the content's a tad technical.

Fig. 1 Structure of our conceptual model describing the interactions between oxygen, phytoplankton, and zooplankton. Arrows show flows of matter through the system. Phytoplankton produces oxygen through photosynthesis during the daytime and consumes it during the night. Zooplankton feeds on phytoplankton and consumes oxygen through breathing; more details are given in the text
-"Mathematical Modelling of Plankton–Oxygen Dynamics Under the Climate Change" (2015-11-25)
This simple flow model's then described by a few differential equations as
Equations (13–15) then turn into the following:

-"Mathematical Modelling of Plankton–Oxygen Dynamics Under the Climate Change" (2015-11-25)
Of course, these equations are vastly too simplistic such that the results'll often be absurd. Still, the authors want the results to seem reasonable, so they do a steady-state analysis to try to get the equations to predict stability.
3.2 Oxygen–Phytoplankton–Zooplankton System: Steady States Analysis
We now proceed to the analysis of the equilibria of the full system (19–21). Obviously, the steady state values are the solutions of the following equations:

-"Mathematical Modelling of Plankton–Oxygen Dynamics Under the Climate Change" (2015-11-25)
Finally, after finding a few fake values that they like, they then just pick those values as-if they were real.
We fix most of the parameters at some hypothetical values as B = 1.8, σ = 0.1, c1 = 0.7, c2 = 1, c3 = 1, c4 = 1, η = 0.7, δ = 1, ν = 0.01, μ = 0.1, and h = 0.1, but vary A in a broad range.
-"Mathematical Modelling of Plankton–Oxygen Dynamics Under the Climate Change" (2015-11-25)
Anyway, since their equations usually blow up and they made-up fake values that just barely hold together, then whenever they change things a bit, the equations blow up again. And so, the authors report doomsday.