Growth of Oxygen Minimum Zones May Indicate Approach of Global Anoxia

Abstract

The dynamics of large-scale components of the Earth climate system (tipping elements), particularly the identification of their possible critical transitions and the proximity to the corresponding tipping points, has been attracting considerable attention recently. In this paper, we focus on one specific tipping element, namely ocean anoxia. It has been shown previously that a sufficiently large, ‘over-critical’ increase in the average water temperature can disrupt oxygen production by phytoplankton photosynthesis, hence crossing the tipping point, which would lead to global anoxia. Here, using a conceptual mathematical model of the plankton–oxygen dynamics, we show that this tipping point of global oxygen depletion is going to be preceded by an additional, second tipping point when the Oxygen Minimum Zones (OMZs) start growing. The OMZ growth can, therefore, be regarded as a spatially explicit early warning signal of the global oxygen catastrophe. Interestingly, there is growing empirical evidence that the OMZs have indeed been growing in different parts of the ocean over the last few decades. Thus, this observed OMZ growth may indicate that the second tipping point has already been crossed, and hence, the first tipping point of global ocean anoxia may now be very close.

1. Introduction

1.1. General Context

Tipping points and critical transitions in the Earth climate system, also known as abrupt climate changes, have been attracting considerable and ever-increasing attention over the last two decades [1,2,3,4,5,6]. A climate tipping point is a phenomenon where a small increase in climate forcing, e.g., a further increase in CO₂ emission, triggers a large change in parts of the global climate system. From a more formal, mathematical perspective, a tipping point is a threshold or a bifurcation beyond which the state or the dynamical regime of the system exhibits qualitative changes [7,8]. Note that a change can already be happening during the approach to the tipping point but once the system crosses the threshold, the rate of change can be expected to significantly accelerate. Also, and perhaps even more importantly, it is believed that once an environment system crosses the tipping point, the changes may become irreversible.

Since an increase in the average Earth temperature (usually referred to as global warming) is a commonly used proxy of global climate change, the climate tipping point is often thought of as a threshold in the temperature increase. The tipping point paradigm has developed significantly in the course of time; instead of a unique global temperature threshold, several tipping elements and regional tipping points were identified [2], each of them having its own critical temperature threshold. Importantly, the tipping elements are not independent in the context of global climate change as crossing one of them may lead to a cascade of other critical transitions [6].

However, the precision of the currently existing estimate of the thresholds’ value remains rather low (cf. the error bars in Figure 2 in [6]). In addition, it tends to change along with advances in theory, methodology, and relevant mathematical models (for example, the temperature tipping point for melting the Greenland ice sheet, which used to be estimated as 3.1 °C [9], was later reestimated as just 1.6 °C [10]). Altogether, it creates considerable uncertainty with regard to how close the Earth’s climate system or its elements may be to the tipping point. This uncertainty is one reason why much attention has been paid to early warning signals for possible approaching tipping points [11,12,13,14].

While some generic early warning signals have been identified—e.g., see the references above—it seems reasonable to expect that, given the different natures of different tipping elements, some early warning signals can be element-specific. Here, we focus on one particular tipping element in the Earth’s climate system, and it is the distribution of dissolved oxygen in the ocean [15,16,17,18]. Using a conceptual mathematical model, we show that the growth of the Oxygen Minimum Zones (OMZ) [19,20], which was observed over the last two decades in different parts of the world, may be indicating that the dissolved oxygen dynamics has already passed its tipping point and the ocean is currently evolving into the state of global anoxia.

1.2. Oxygen in the Ocean

The spatial distribution of dissolved oxygen in the ocean is known to be distinctly heterogeneous [21,22]. In particular, in many parts of the ocean, the vertical distribution has a special profile, where well-oxygenated waters in the subsurface layer (0–200 m) and in the depth (1000 m or more) are separated by poorly oxygenated water with anoxic or even hypoxic conditions at the intermediate depth (e.g., 300–800 m or so). (More generally, depending to the local conditions of the region, the low-oxygen zone can be located at depths ranging from 200 to 1500 m [23], although Stramma et al. [24] showed that the OMZ in the eastern tropical Pacific and the northern boundaries of the tropical Indian Ocean can be found at lower depths, presumably because of higher water temperatures.) This vertical structure is called the Oxygen Minimum Zone (OMZ) [19,25]. In the horizontal direction, an OMZ can extend over hundreds or even thousands of kilometres, e.g., see Figure 1. Some studies indicated [19,26] that the overall volume occupied by the OMZs can be as large as ${10}^8{\mathrm{km}}^3$ and that about $10\%$ of the volume has anoxic conditions, with the oxygen concentration falling below 20 μmol·kg⁻¹. Moreover, it was shown in [26] that there is an expansion of the anoxic zones in both the horizontal and vertical directions.

Numerous empirical studies have addressed the problem of ocean de-oxygenation, e.g., see [16,26,27,28,29,30,31] and references therein. In particular, it has been observed that, between 1956 and 2006, there was a considerable decrease in the concentration of oxygen in the Pacific Ocean at depths between 100 and 400 m [32], arguably due to the increase in the ocean temperature [33,34]. Importantly, the OMZs were shown to expand both horizontally and vertically [26]. Several other studies also suggested that an increase in the extent of OMZs is due to climate change [35,36,37,38]. Additionally, with regard to the effect of temperature, there are some other factors that affect the expansion of the OMZ, e.g., an increase in some mineral elements such as iron and phosphorous in the ocean waters [26,37].

1.3. Methodology, Goals, and an Outline of the Main Result

In this study, we analyse the spatiotemporal dynamics of an OMZ—in particular, its tendency to grow—by means of mathematical modelling. Our main assumption is that the area where the gradient of oxygen concentration reaches its maximum (‘oxycline’) can be regarded as a front, which separates two spatial domains, where the ocean water is at two qualitatively different states, i.e., oxygen-rich and oxygen-poor. We then further assume (and quantify it using a relevant mathematical model that accounts for the oxygen production in phytoplankton photosynthesis) that these states are the equilibria of a certain dynamical system. Any change in the OMZ size has apparently resulted from a change in the position of its boundaries (i.e., the locations of the oxycline). Therefore, the analysis of the OMZ spatial dynamics can be performed by analysing the movement of the corresponding travelling fronts: the OMZ will grow if the two fronts (i.e., its boundaries) travel away from each other, and the OMZ will shrink if the two fronts travel towards each other.

As a specific mathematical framework to describe and analyse the dynamics of the corresponding travelling fronts, we use reaction–diffusion equations. We mention here that reaction–diffusion systems or their immediate generalisations are frequently used as appropriate modelling frameworks in biological and ecological systems of various origins [39,40]; in particular, they have often been used as a generic model of plankton dynamics [41,42,43,44,45]. In the context of the OMZ dynamics, reaction terms account for oxygen production by phytoplankton and plankton growth (see Section 2 for details), and the diffusion terms account for the spatial mixing due to marine turbulence [41] (see Section 3). The relevance of reaction–diffusion systems to the OMZ dynamics is further justified by the observation that the existence of travelling fronts is their generic property [46,47]. Importantly, the direction of front propagation depends on the parameters of the reaction terms, and hence, a change in the direction can be readily linked to a change in relevant environmental conditions.

The overarching goal of this study is to estimate how close the oceanic dissolved oxygen—as a tipping element of the global Earth system—may be to its tipping point. It was previously shown that a sufficiently large increase in the average ocean water temperature may disrupt phytoplankton photosythesis, which in turn would lead to global anoxia [48,49,50]. However, the previous analysis did not allow for drawing any conclusion with regard to how close the plankton–oxygen system is to the tipping point or whether it has already crossed it. In this paper, we show that the critical transition to the state of global anoxia follows a succession of two tipping points. More specifically, our analysis suggests that the (first) tipping point leading to global anoxia, when the phytoplankton oxygen production stops because of photosynthesis disruption, is preceded by another (second) tipping point when the OMZ starts growing. Thus, the spatial extension of OMZs, observed during the last few decades [19,24], may indicate that the first tipping point has already been crossed.

2. Mathematical Model of Oxygen–Phytoplankton Dynamics: Nonspatial Case

Our main goal in this paper is to model the spatiotemporal dynamics of dissolved oxygen. For the sake of clarity, before proceeding to the ‘full’ spatially explicit model, it is helpful to consider its nonspatial counterpart. Indeed, it is well known that the properties of a nonspatial model make a certain skeleton that, to a large extent, shapes or predetermines the dynamics of the spatial system. This is particularly true for reaction–diffusion systems [39,46,51].

We are motivated by [48,49] to consider the following conceptual oxygen–phytoplankton system as a baseline (nonspatial) model of oxygen production in phytoplankton photosynthesis:

$$\begin{array}{ccc}\hfill {\displaystyle \frac{dc\left(t\right)}{dt}}& =& {\displaystyle \frac{Au\left(t\right)}{c\left(t\right)+1}}-{\displaystyle \frac{\delta u\left(t\right)c\left(t\right)}{c\left(t\right)+c_2}}-mc\left(t\right),\hfill \end{array}$$

(1)

$$\begin{array}{ccc}\hfill {\displaystyle \frac{du\left(t\right)}{dt}}& =& \left({\displaystyle \frac{Bc\left(t\right)}{c\left(t\right)+c_1}}-u\left(t\right)\right)u\left(t\right)-\sigma u\left(t\right),\hfill \end{array}$$

(2)

(in dimensionless variables) where $c\left(t\right)$ is the concentration of dissolved oxygen and $u\left(t\right)$ is the phytoplankton density at time t. Here, A is the rate of oxygen production in photosynthesis, $\delta$ is the maximum per capita rate of the oxygen consumption (‘respiration’) by phytoplankton (Note that phytoplankton produces oxygen in photosynthesis during the day but consumes it during the night, as required for its metabolic processes [52,53]; the difference between the two rates is called the rate of net oxygen production), $\sigma$ is the natural mortality rate of phytoplankton, B is the maximum phytoplankton per capita growth rate in the large oxygen limit (i.e., when oxygen is not a limiting factor), $c_{1,2}$ are the half-saturation constants of the corresponding processes. The parameter m is the rate of oxygen ‘natural depletion’, i.e., its consumption in other processes not explicitly included in the model (e.g., biochemical reactions in the water related to detritus decay, etc.). Note that, in a special case where the oxygen concentration is maintained constant, the right-hand side of Equation (2) coincides with the logistic growth where the square term $(-u^2)$ accounts for competition, e.g., see [54,55].

The necessary details of the derivation of the model (1) and (2) can be found in [48] (see also [49,50] for extra details and further discussion). In brief, in Equation (1), the term ${\displaystyle \frac{Au}{c+1}}$ describes the per capita rate of oxygen production (per unit of phytoplankton density) in photosynthetic activity, the term ${\displaystyle \frac{\delta uc}{c+c_2}}$ quantifies the phytoplankton respiration, and $mc$ describes the loss of oxygen due to natural depletion. In Equation (2), the term ${\displaystyle \frac{Bc}{c+c_1}}$ describes the growth of phytoplankton (which can depend on the oxygen concentration, especially under anoxic conditions [52]) and $\sigma u$ is the phytoplankton natural mortality rate.

System (1) and (2) contain as many as seven parameters. A full investigation of the model properties in the 7-dimensional parameter space does not seem possible. Instead, we chose parameters A and B (as, arguably, the two ‘most important’ ones for biological reasons) and varied them in a representative range but kept other parameters fixed at some hypothetical values.

Note that, in order to have a unique solution, system (1) and (2) have to be complemented with the initial conditions, i.e., $c\left(0\right)$ and $u\left(0\right)$. However, here we are not interested in the detailed properties of a specific solution. Instead, for the purposes of this study, we are interested in the solution’s properties in the large-time limit (e.g., the existence of steady states) rather than in details of its transient behaviour at a short and intermediate time. Correspondingly, in order to study the system (1) and (2), we use the technique known as the phase plane analysis [56,57]. Therefore, we now seek to discuss the existence and stability of each steady state for the system. Steady states are defined by the conditions $dc\left(t\right)/dt=0$ and $du\left(t\right)/dt=0$; hence, the steady-state values of the oxygen concentration and the phytoplankton density are given by the non-negative (due to the physical meaning of variables c and u) solutions of the following nonlinear algebraic system:

$$\begin{array}{ccc}\hfill {\displaystyle \frac{Au}{c+1}}-{\displaystyle \frac{\delta uc}{c+c_2}}-mc& =& 0,\hfill \end{array}$$

(3)

$$\begin{array}{ccc}\hfill \left({\displaystyle \frac{Bc}{c+c_1}}-u\right)u-\sigma u& =& 0.\hfill \end{array}$$

(4)

It is readily seen that the system (3) and (4) has up to three non-negative steady states, which we denote as $S_1=(0,0)$, $S_2^1=(\widehat{c_1},\widehat{u_1})$ and $S_2^2=(\widehat{c_2},\widehat{u_2})$. In particular, the following holds:

• The extinction (trivial) steady state $S_1=(0,0)$ always exists without any dependence on the choice of parameter values.
• The second and third steady states $S_2^1=({\widehat{c}}_1,{\widehat{u}}_1)$ and $S_2^2=({\widehat{c}}_2,{\widehat{u}}_2)$ are the intersection points between the nontrivial parts of the corresponding isoclines (see Figure 2a and Figure 3a), which are given by the following equations:

  $$u={\displaystyle \frac{mc(c+1)(c+c_2)}{A{c}_2+(A-\delta )c-\delta c^2}},$$

  (5)

  $$u={\displaystyle \frac{Bc}{c+c_1}}-\sigma .$$

  (6)

Contrary to the trivial steady state, the two positive steady states only exist in a certain parameter range, as is summarised by the bifurcation diagrams shown in Figure 2b and Figure 3b. It is readily seen that the positive steady states only exist for A when it is not too small (for a fixed value of B) and for B when it is not too small (for a fixed value of A). It, therefore, points out the existence of some critical or threshold values, $A_{cr}$ and $B_{cr}$, respectively, so that the positive states only exist for $A\ge A_{cr}^{\left(1\right)}$ and $B\ge B_{cr}$, e.g., see Figure 2b where $A_{cr}^{\left(1\right)}\approx 0.94$ and Figure 3b where $B_{cr}\approx 0.3$.

In addition, Figure 4 (obtained for $A=3$ and $B=1.8$) visualise the phase flow of the system. Note that one of the positive steady states, $S_2^1$ (lower positive), always stays close to the extinction steady state, and the shape of the first isocline succession can be described as the convergence of the upper positive steady state $S_2^2$ with the lower positive steady state $S_2^1$. The convergence of the system’s steady states to $S_2^1$ (lower positive) brings with it a significant ecological problem called depletion of oxygen concentration in water. However, the existence of the two positive steady states depends on the choice of controlling parameters A and B, which exist only if A is not too small and if there is a decreasing B parameter.

The stability of the steady states can be revealed directly from the structure of the phase flow of the system, see Figure 4 (obtained with the help of function ‘quiver’ in Matlab ^® R2022b). The direction and shape of the flow in the vicinity of the steady states unambiguously indicate that the trivial (extinction) steady state $S_1^1$ is a stable node, the lower positive steady state $S_2^1$ is a saddle, and the upper positive steady state $S_2^2$ is also stable node. This is confirmed by a rigorous analysis of the eigenvalues of the corresponding Jacobian; see the Appendix A for details. Thus, the disappearance of the positive steady states when A or B cross their threshold values occurs due to the saddle-node bifurcation. In the range $A>A_{cr},B>B_{cr}$, the system exhibits bistability.

3. Spatiotemporal Dynamics of the OMZ

In order to apply the baseline model (1) and (2) to describe the dynamics of OMZ, we consider its spatially explicit extension, which is given by the following reaction–diffusion system:

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial c(\mathbf{r},t)}{\partial t}}& =& D{\nabla }^{2}c(\mathbf{r},t)+{\displaystyle \frac{Au(\mathbf{r},t)}{c(\mathbf{r},t)+1}}-{\displaystyle \frac{\delta u(\mathbf{r},t)c(\mathbf{r},t)}{c(\mathbf{r},t)+c_2}}-mc(\mathbf{r},t),\hfill \end{array}$$

(7)

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial u(\mathbf{r},t)}{\partial t}}& =& D{\nabla }^{2}u(\mathbf{r},t)+\left({\displaystyle \frac{Bc(\mathbf{r},t)}{c(\mathbf{r},t)+c_1}}-u(\mathbf{r},t)\right)u(\mathbf{r},t)-\sigma u(\mathbf{r},t),\hfill \end{array}$$

(8)

where $\mathbf{r}$ is the position in space. The Laplacian in the right-hand side of Equations (7) and (8) account for turbulent mixing, with D being the coefficient of turbulent diffusion. In appropriately chosen dimensionless variables [48,49] (recall that the reaction part of system (1) and (2) is already written in dimensionless variables), $D=1$. In order to ensure the mathematical correctness of the problem, Equations (7) and (8) must be supplemented with the initial and boundary conditions, which we will discuss below.

3.1. 1D Case

In order to better explain the relation of model (7) and (8) to the dynamics of OMZ and to highlight our main findings, we begin with the 1D case. Equations (7) and (8) then take the following somewhat more specific form:

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial c(x,t)}{\partial t}}& =& {\displaystyle \frac{{\partial }^{2}c(x,t)}{\partial x^2}}+{\displaystyle \frac{Au(x,t)}{c(x,t)+1}}-{\displaystyle \frac{\delta u(x,t)c(x,t)}{c(x,t)+c_2}}-mc(x,t),\hfill \end{array}$$

(9)

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial u(x,t)}{\partial t}}& =& {\displaystyle \frac{{\partial }^{2}u(x,t)}{\partial x^2}}+\left({\displaystyle \frac{Bc(x,t)}{c(x,t)+c_1}}-u(x,t)\right)u(x,t)-\sigma u(x,t).\hfill \end{array}$$

(10)

For a broad class of initial conditions, the large-time asymptotic solution of the system (9) and (10) is given by a travelling front connecting the lower steady state on one side of the front to the upper steady state on the other side of the front [46,58,59], i.e., in terms of our model, connecting $(c,u)=(0,0)$ to $(c,u)=({\widehat{c}}_2,{\widehat{u}}_2)$. In case the initial conditions are compact (e.g., as given by choosing $(c,u)=(0,0)$ inside a certain finite spatial domain but keeping $(c,u)=({\widehat{c}}_2,{\widehat{u}}_2)$ outside of the domain), the generic large-time solution of (9) and (10) is given by a set of two travelling fronts emerging, respectively, on both sides of the domain [60,61]; see the thick black curve in Figure 5.

Importantly, depending on parameter values, the direction of the front(s) propagation can be different and have crucial implications for the OMZ dynamics. Fronts travelling towards each other (see the black arrows in Figure 5) would mean that the OMZ is shrinking and may eventually disappear in the course of time; fronts travelling away from each other (cf. the red arrows) would mean that the OMZ is growing, which in turn may lead to an onset of anoxia on a large (possibly, global) scale.

We therefore need to understand how the speed of the travelling front depends on the problem parameters, in particular on A and B. However, an analytical calculation of the travelling front speed in a system of two reaction–diffusion equations is a highly nontrivial problem [51,62]. Note that the well-known method developed by Okubo et al. [63] (see also Section 1.3 in [64]) based on linearisation of the equations, at the leading edge of the front, does not apply to our case; as in our system, the travelling front connects the trivial steady state on one side of the front to the positive steady state on the other side (instead of connecting it to a boundary state as in [63]), and linearisation at the positive steady state does not appear to be helpful.

In order to investigate whether the direction of travelling front(s) in our models (9) and (10) can indeed change with a change in parameter values, Equations (9) and (10) are solved numerically. In performing that, and for the sake of simplicity, we assume that the distribution of oxygen is approximately symmetric with respect to the centre of the OMZ (cf. the midpoint in Figure 5). Correspondingly, we focus on the propagation of only one of the travelling fronts–namely, the one in the left-hand side of the domain in Figure 5—as the other one (in the right-hand side) will exhibit the same properties. Therefore, we perform simulations in a spatial domain $0<x<L$ for $t>0$ using the following initial conditions:

$$\begin{array}{ccc}\hfill c(x,0)={\widehat{c}}_{2}for0<x<d,& & c(x,0)=0ford\le x<L,\hfill \end{array}$$

(11)

$$\begin{array}{ccc}\hfill u(x,0)={\widehat{u}}_{2}for0<x<d,& & u(x,0)=0ford\le x<L,\hfill \end{array}$$

(12)

where parameter d can be interpreted as (one half of) the initial size of the OMZ. In order to minimise the effect of the domain boundaries on the front propagation, we chose d to be sufficiently smaller than L. At the boundaries of the domain, i.e., at $x=0$ and $x=L$, the zero-flux Neumann boundary condition is used.

Equations (9) and (10) with initial conditions (11) and (12) were solved numerically by the finite differences method using Matlab® as the coding platform. Figure 6 and Figure 7 show the solution of system (9) and (10) obtained for several different values of A (having other parameters fixed as $D=1$, $B=1.8$, $\sigma =0.1$, $c_1=0.7$, $c_2=1$, $\delta =1$, $m=1$, $L=1000$ and $d=500$). We readily observe in Figure 6 that the initial conditions (11) and (12) indeed converge to a travelling front which propagates to the right, i.e., towards the area with low oxygen concentration. Thus, in this case, the OMZ is shrinking. However, by comparing Figure 6a,b, we notice that the speed of the front propagation decreases with decreases in A. It gives a reason to anticipate that a further decrease in A may stop or even reverse the travelling front, and, as we show below, this is indeed the case.

Since the spatial profiles of functions $c(x,t)$ and $u(x,t)$ have a similar shape (in particular, the corresponding fronts of oxygen concentration and phytoplankton density propagate with the same speed), we only show the distribution of oxygen in Figure 7. It is readily seen that, for $A=A_{cr}^{\left(2\right)}=0.98$ (Figure 7b), the front speed is zero. The initial conditions (11) and (12) converge to a stationary distribution; the initial OMZ is neither grows not shrinks. For a smaller value of A, the direction of front propagation changes from positive to negative. Figure 7a shows the solution of system (9) and (10) obtained for $A=0.941$; obviously, the front now propagates to the left, i.e., towards the area where the oxygen concentration is high. It means that the OMZ is growing, which will eventually result in the onset of global anoxia. We emphasise that this happens for $A>A_{cr}^{\left(1\right)}$, i.e., in the parameter range where the nonspatial system possesses the positive ‘safe’ steady state, see Figure 2b and the lines after Equations (5) and (6).

Having performed numerous simulations similar to those shown in Figure 6 and Figure 7 but with different values of the parameter A, we obtain the speed of the front as a function of A; see Figure 8. We, therefore, obtain that, for a succession of decreasing values of A, the system undergoes two critical transitions. For $A>A_{cr}^{\left(2\right)}$, the system is in a safe, ‘healthy’ state: there exists a stable positive equilibrium (a stable node, see Figure 2b and Figure 4), any initial OMZ (that may emerge as a result of factors not included into our model, e.g., random perturbations such as singular/extreme weather events) will shrink and eventually disappear. When crossing the critical threshold $A_{cr}^{\left(2\right)}$, the spatiotemporal dynamics of the OMZ changes. For $A_{cr}^{\left(1\right)}<A<A_{cr}^{\left(2\right)}$, the corresponding nonspatial system still possesses a stable positive equilibrium, but the initial OMZ grows with time, its boundaries (oxyclines) propagating as travelling fronts, eventually resulting in the onset of anoxia over the whole available space. When crossing the other threshold $A_{cr}^{\left(1\right)}$, the dynamics change again. For $A<A_{cr}^{\left(1\right)}$, the system only possesses a trivial steady state $(0,0)$, so that any initial oxygen distribution converges to zero uniformly over the space.

An important question arises here as to what can be the effect of the parameter B on the speed of the travelling front. The results shown in Figure 8 indicate that the observed reverse of the front propagation is not limited to a particular value of B. However, since B is a parameter of high biological significance (i.e., the phytoplankton growth rate), this issue requires a more systematic approach. We, therefore, addressed this question by performing numerous numerical simulations, varying parameters A and B over a sufficiently broad, representative range. The results are shown in Figure 9. We observe that the transition of the marine ecosystem from a ‘healthy’, well-oxygenated state (Domain III—large A, large B) to the deadly anoxic state (Domain I—either small A and/or small B) always occurs through a succession of two tipping points (shown by the two thick red curves in Figure 9). The OMZ shrinks in Domain III but grows in Domain II because of the change in the direction of the front propagation.

We also mention here that, as is obvious from the structure of the parameter plane $(A,B)$, for any fixed value of A, the transition from a healthy well-oxygenated state to the anoxic state occurs through the succession of two critical values of parameter B, say, $B_{cr}^{\left(1\right)}$ and $B_{cr}^{\left(2\right)}$, respectively.

3.2. OMZ Dynamics in the 2D Case

In this section, we attempt to show the effect of the controlling parameters A and B on the travelling front wave of the oxygen–phytoplankton system (9) and (10), extended onto the 2D case. Equations (7) and (8) take the following form:

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\partial c(x,y,t)}{\partial t}}=\left({\displaystyle \frac{{\partial }^2}{\partial x^2}}+{\displaystyle \frac{{\partial }^2}{\partial y^2}}\right)c(x,y,t)+{\displaystyle \frac{Au(x,y,t)}{c(x,y,t)+1}}-{\displaystyle \frac{\delta u(x,y,t)c(x,y,t)}{c(x,y,t)+c_2}}-mc(x,y,t),\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\partial u(x,y,t)}{\partial t}}=\left({\displaystyle \frac{{\partial }^2}{\partial x^2}}+{\displaystyle \frac{{\partial }^2}{\partial y^2}}\right)u(x,y,t)+\left({\displaystyle \frac{Bc(x,y,t)}{c(x,y,t)+c_1}}-u(x,y,t)\right)u(x,y,t)-\sigma u(x,y,t),\hfill \end{array}$$

(14)

(in dimensionless variables) where $\left|x\right|\le L,\left|y\right|\le L$, and all notations have their previous meaning. At the domain boundary, we use the zero-flux Neumann boundary conditions. For the initial conditions, we consider them slightly differently from the 1D case, namely in the form of a large oxygen patch surrounded by the OMZ: both oxygen and phytoplankton are present at their steady state values inside a square domain of size $2d$ and are absent outside of it. This is described by the following equations:

$$\begin{array}{c}\hfill c(x,y,0)={\widehat{c}}_{2}for\left|x\right|<dand\left|y\right|<d,otherwisec(x,y,0)=0,\end{array}$$

(15)

$$\begin{array}{c}\hfill u(x,y,0)={\widehat{u}}_{2}for\left|x\right|<dand\left|y\right|<d,otherwiseu(x,y,0)=0.\end{array}$$

(16)

Equations (13) and (14) with initial conditions (15) and (16) were solved numerically by the finite differences method (using Matlab^® as the coding platform). Simulations were performed for various values of A while keeping other parameters fixed as $B=1.8$, $\sigma =0.1$, $c_1=0.7$, $c_2=1$, $\delta =1$ and $m=1$. Several typical snapshots of the oxygen spatial distribution at different times are shown in Figure 10 and Figure 11. Note that the phytoplankton spatial distribution and the oxygen distribution show essentially the same features; therefore, for the sake of brevity, here we only show the oxygen spatial distribution. Also, since the mathematical problem (13) and (16) is symmetrical with respect to reflection $x\to -x$, $y\to -y$, only the first quarter of $(x,y)$ plane is shown.

Figure 10 (obtained for $A=1.1$) shows snapshots of the oxygen concentration at different times. It is readily seen that the initial oxygen patch grows, its boundary propagating as a travelling front towards the area where oxygen is at a very low concentration. Thus, in this case the OMZ is shrinking and eventually the system’s state with high oxygen concentration prevails over the whole space. However, for a smaller A the situation may change to the opposite. Figure 11 shows snapshots of the oxygen concentration obtained for $A=0.941$. It is readily seen that the front separating well-oxygenated water from anoxic water now propagates towards the area with high oxygen concentration. Thus, the size of the area with anoxic conditions grows. The OMZ is growing and will eventually occupy the whole domain, hence resulting in the onset of global anoxia.

Therefore, a change in the direction of travelling front propagation in response to a change in parameter A is possible in the 2D case as well as in the 1D case. In order to gain more quantitative insight into this phenomenon, we calculated the speed of the front in the direction of axis y for different values of A. The results are presented in Figure 12. Thus, our simulation results obtained in the 2D case are in very good agreement with those obtained in the 1D case (cf. Figure 8a), namely, they reveal the existence of the second tipping point $A_{cr}^{\left(2\right)}$, where the propagation of the travelling front (i.e., the OMZ boundary) changes its direction.

In the conclusion of this section, we mention that the growth and shrinking of the square-shaped initial plankton–oxygen patch (cf. Equations (15) and (16)) follow somewhat different scenarios. In the case of growth (Figure 10), the initial square shape promptly evolves into a round-shaped patch, which can be intuitively expected as the effect of diffusion is stronger in the vicinity of the corners [46,61]. However, in the case of shrinking (Figure 11), and rather counter-intuitively, the near-square shape is preserved at all times, so that the shrinking of the oxygen patch occurs via the propagation of two plane fronts towards the origin, i.e., along axes x and y, respectively.

4. Discussion and Concluding Remarks

Identification of early warning signals of critical transitions in ecological and environmental systems—in particular, tipping points in the large-scale components of the Earth climate system [2]—has been a focus of intense research recently [11,12,13,14]. Great progress has been made in revealing and understanding early warning signals, e.g., see [65,66], also [13] and further references therein. However, the estimation of the proximity of a system to its critical threshold (tipping point) remains a considerable challenge [14,67]. For many of Earth’s subsystems, the critical temperature increase has been estimated, but the accuracy of the estimates remains low, so that, in some cases, the estimated threshold value spans over almost an order of magnitude [6].

In this paper, we have discovered an alternative approach to the understanding and quantification of the proximity to the tipping point for a particular tipping element of the Earth climate system, such as the amount of dissolved oxygen in the world ocean [2]. By using a conceptual mathematical model of the coupled dynamics of oxygen and marine phytoplankton as its main producer (in photosynthesis), we have shown that the critical transition to the state of global ocean anoxia occurs through a succession of two tipping points rather than a single one (see Figure 13). Both tipping points are quantified by the corresponding threshold values of the net oxygen production, $A_{cr}^{\left(1\right)}$ and $A_{cr}^{\left(2\right)}$. Since phytoplankton’s net oxygen production rate is known to depend on water temperature [68,69,70], these two critical values of A determine two corresponding threshold values of the temperature increase, say $\Delta T_{cr}^{\left(1\right)}$ and $\Delta T_{cr}^{\left(2\right)}$.

There are some field data [68] showing that, for a sufficiently large temperature increase (roughly estimated as $\Delta T^{\ast }\approx 6 °\mathrm{C}$, see [68]), the net oxygen production rate (A) may turn to zero or even become negative. Based on existing global warming scenarios [71], the increase of 6 °C may look unrealistically high. However, there is considerable theoretical evidence [48,49,50] showing that phytoplankton photosynthesis may actually be significantly disrupted, leading to global anoxia, for a much smaller temperature increase $\Delta T_{cr}^{\left(1\right)}<\Delta T^{\ast }$, i.e., as soon as the oxygen net production rate crosses its critical value $A_{cr}^{\left(1\right)}$.

Unfortunately, our study has to remain largely theoretical as it does not seem possible to relate our findings to real oceanic oxygen dynamics in a more quantitative way due to the lack of relevant data. To the best of our knowledge, such data simply do not exist at present; hopefully, they may become available in the future. We can only hope that our findings, due to, arguably, their great potential importance, might incentivise new empirical research in that direction.

The above conclusion was made in earlier work, which disregarded some specific yet important spatial aspects of the plankton–oxygen dynamics. The key point of this study is the observation that the spatial distribution of dissolved oxygen in the ocean is remarkably heterogeneous, often creating large-scale spatial patterns that are persistent in space and time. While the main part of the ocean contains well-oxygenated water, there are also many areas (called Oxygen Minimum Zones: OMZs) where the oxygen level is very low, often close to zero [17,19]. Moreover, there is considerable empirical evidence that the OMZs tended to grow over the last few decades [24,26,35]. This growth was generically attributed to global warming, although no specific mechanism was suggested. Here, we have shown that the OMZ growth may indicate the existence of another tipping point, the second critical value $A_{cr}^{\left(2\right)}$, which can be interpreted mathematically as the reverse of the travelling fronts in the corresponding reaction–diffusion-type model of plankton–oxygen dynamics. Therefore, our results suggest that the real-ocean plankton–oxygen system has already crossed the second tipping point and, hence, is very close to the next, ultimate critical transition, which will lead to the onset of global anoxia uniformly over space.

Note that in our study, the spatial position, x or $(x,y)$, has the meaning of horizontal coordinate(s). Thus, our analysis of the OMZ growth readily applies to the growth of its horizontal extent. Meanwhile, OMZs are also known to grow in the vertical direction, e.g., see [24], which can be modelled using a similar approach [72]. However, the spatiotemporal dynamics in the vertical direction is strongly constrained by the ocean stratification, so that the position of oxyclines is primarily controlled by different factors (e.g., by the rate of decay of sunlight intensity with the water depth).

In conclusion, we mention that our study has a number of limitations. Firstly, a more advanced model should account for more details of ocean transport, especially on the mesoscale, which is known to be a factor contributing to the dynamics of OMZ [73,74,75]. Secondly, a more realistic model should include more details about the plankton–oxygen interactions, e.g., the effect of zooplankton (which is known to be a factor controlling phytoplankton growth), the existence of multiple timescales, etc. (An OMZ growth in the horizontal direction was observed using a somewhat more advanced model in a recent study by Chowdhury et al. [76], but the possibility of reversing the plankton–oxygen fronts was not considered.) These issues should become a focus of future research.