Potential contribution of fish restocking to the recovery of deteriorated coral reefs: an alternative restoration method?

The model

We examine the potential outcomes of fish restocking using modifications of mathematical models, consisting of differential equations, which have been used to examine reef resilience without intervention (Blackwood, Hastings & Mumby, 2011; Blackwood, Hastings & Mumby, 2012; Fung, Seymour & Johnson, 2011; Mumby, Hastings & Edwards, 2007).

The dynamic model we use here has been adapted from Blackwood, Hastings & Mumby (2011), the most recent version of the model first presented in Mumby, Hastings & Edwards (2007). This model enables us to follow the dynamics of coral coverage, macroalgae, algal turfs, grazing fish, and terrain rugosity; denoted, respectively, by the variables C, M, T, P and R. The algal turf coverage refers to the proportion of seabed either covered by scant and small algae, such as might occur due to grazing of macroalge, or exposed hard substrate. We assume that the corals, macroalgae, and algal turfs are competing for seabed in a constant size location, and define the algal turf coverage to be 1-M-C. The variable P describes the abundance of grazing fish relative to the maximum capacity of grazing fish possible in the habitat, so that 0 < P < 1. Rugosity is defined as the ratio between the horizontal distance of two points in the reef and the length of a chain laid on the reef surface between those points, usually satisfying 1 < R < 3 (Alvarez-Filip et al., 2009). The rest of the dynamics are given by the following set of ordinary differential equations: (E1)${\displaystyle \begin{array}{c}\frac{dM}{dt}=aMC-\frac{g\left(P\right)M}{M+T}+\gamma MT\hfill \\ \frac{dC}{dt}=rTC-dC-aMC\hfill \\ \frac{dT}{dt}=\frac{g\left(P\right)M}{M+T}-\gamma MT-rTC+dC\hfill \\ \frac{dP}{dt}=sP\left(1-\frac{P}{K\left(M+T,R\right)}\right)-mP-fP\hfill \\ \frac{dR}{dt}=h_{G}C\left(3-R\right)-H_e\left(1-C\right)\left(R-1\right).\hfill \end{array}}$ We assume that corals grow on a seabed covered with algal turfs at rate r, die of natural causes at rate d, and are covered by macroalgae at rate a. Macroalgae too grow over algal turfs, at rate γ. Grazing fish grow according to the logistic growth equation, with growth rate s, and a maximal carrying capacity function K(M + T, R), which is the product of a linear, increasing, function of rugosity and a Hill-Langmuir function of M + T (for details see Blackwood, Hastings & Mumby, 2011). The grazing fish graze macroalgae into algal turfs at a rate $\frac{g\left(P\right)M}{M+T}$, where g(P) is taken to be P. Thus, the grazing fish reduce the rate of macroalgae growth over corals, while simultaneously expanding algal turfs, which can provide seabed for coral growth (Jompa & McCook, 2002). Grazing fish are fished at rate f.

Terrain rugosity increases due to growth of corals, h_G, and decreases due to bioerosion at a rate h_E, where both h_G and h_E depend on current rugosity and coral coverage (functions were estimated from data by Blackwood, Hastings & Mumby, 2011). Grazing fish migrate from the reef at rate m. The parameter values and their meanings are given at Table 1. All parameter values used are taken from Blackwood, Hastings & Mumby (2011), except for spillover estimates (m). We used spillover rates estimated in Kaunda-Arara & Rose (2004) (acquired by the tag and release method) and converted them to yearly migration rates assuming an exponential decrease, in order for them to fit the parametrization of our equations (see Text S1). We first consider the effect of restocking on a single reef, and then extend the model to two reefs, each represented by the system of differential equations presented above. The reefs are coupled by the migration parameter. We assume that all fish from one reef, denoted Reef I, migrate to another reef, denoted Reef II; while fish from Reef II migrate as in the one reef system, and are effectively lost in our model. This is a conservative assumption, as we examine the worst case in terms of restocking benefit in which none of the fish from Reef II migrate to Reef I. Additionally, we assume that both reefs are relatively close, so that climatic or anthropogenic perturbations will affect both reefs similarly and bring them to the same initial conditions (a distance of ∼10 km might be an estimate for such conditions; Hughes et al., 1999). We introduce restocking by adding an amount δ_P to the initial value of P, namely P₀. Since P is normalized to be between 0 (no grazers) and 1 (maximal capacity of grazers), δ_P is given as a fraction of the maximal abundance of grazing fish possible in the modeled habitat.

We model the economic impact of fish restocking using a cost-benefit analysis. We define X as the size of the coral reef in km²; B_C is the revenue, per km² per year, resulting from coral coverage (excluding revenue from fishing); $\hat{K}$ is the maximal carrying capacity of the restocked grazer fish per km², and r′ is the economic discount rate. For t years, the difference in revenue for a coral reef with restocking versus a reef with no intervention can be estimated by (also known as the net present value): (E2)${\displaystyle \text{Revenue}\left(t\right)=X\cdot B_C\cdot \sum _{i=0}^t{\left(1+r^{\prime }\right)}^{-i}\left(C^{\text{re}}\left(i\right)-C^{\text{no}}\left(i\right)\right)-\left(\tilde{c}\cdot \hat{K}\cdot X\cdot {\delta }_p\right)}$ where C^re(i) and C^no(i) are the coral coverages of reefs with and without restocking, at year i, respectively. Simply put, we calculate the difference in coral coverage between a reef with and without the restocking intervention. This difference is multiplied by the size of the reef and the financial benefit for each squared kilometer of the reef. The term is discounted with regard to inflation. Finally, the cost of restocking is subtracted. This is a conservative estimate, since in this model restocking increases coral coverage, and it is assumed that the benefit from coral reefs declines relative to the cost of the fish, due to discounting. The parameters and variables of the economic model describing the revenue are given in Table 2. The Matlab code for all results is given in File S3.

A single reef

First, we examine the long-term effects of restocking for various initial conditions of the dynamic system presented above, with a single coral reef. We assume that as part of the restoration treatments, fishing restrictions are implemented, so that f = 0 in all the following results. The state of a disturbed reef is represented by the initial conditions of the coral coverage (C) and macroalgae (M)(determining the amount of algal turfs, as T = 1 − M − C). The revenue of restocking (derived from (E2)) as well as the final outcomes of the restocking intervention (derived from (E1)), are presented as functions of the system’s initial conditions in Fig. 1.

Since this dynamic system has two attractors, one of high coral coverage and the other of high macroalgae coverage (Blackwood, Hastings & Mumby, 2011), the range of initial conditions can be divided into 3 areas: (I) initial conditions in which the system reaches a state with high coral coverage with or without restocking; (II) areas wherein the system would reach a high macroalgae state in the absence of intervention (but under fishing restrictions), but restocking would allow its return to the high coral coverage state; and (III) areas where the system will reach a state with high macroalgae coverage with or without restocking. These areas are denoted in Fig. 1 as (I), (II) and (III), respectively, and are separated by black borders. In addition, colors in Fig. 1 represent the expected revenue of restocking, 5 and 20 years after restocking has taken place, in log₁₀ scale, with negative revenue replaced by zeros (Figs. 1A and 1B). Note that the variables are normalized to represent the entire reef area, so that T = 1 − M − C and the state of the algal turfs (T) is defined by the other two variables. The parameters used in Fig. 1 for the dynamical system were P₀ = 0.1, δ_P = 0.1, R₀ = 1.6 (the rest of the dynamical system parameters were given the values estimated in Blackwood, Hastings & Mumby, 2011). The spillover was estimated from odds of tagged fish leaving and staying in the coral reefs (Kaunda-Arara & Rose, 2004), and was transformed to a rate term to yield m = 0.12. The reef size (X) was taken to be 15 km², the estimated grazing fish number per ${\mathrm{km}}^2\left(\hat{K}\right)$ was taken as 3,000, estimated from Gaudian, Medley & Ormond (1996), according to the density of the most common fish in the examined coral reef (accounting for 64% of all fish). Thus, when we enhance the number of grazing fish by δ_P = 0.1, we de facto add 300 fish per each km² of reef area. The financial benefit from the coral reef (B_C) was estimated from Cesar & Van Beukering (2004), as $200,000 per year per km², which is a very conservative estimate (see Spurgeon, 1999, for example). The average cost of each fish, $\tilde{c}$, was estimated to be $20 as an over-estimated price. This estimated cost is based on the recent average fish price for cultured fish (ca. $1.8/kg) taken from fish price trends in real terms during the last two decades (FAO, 2014). Estimating an average size of 500 gr of released fish results in a cost of $0.9 per fish. The actual fish cost, however, should be calculated based on the expected survival rates of the released fish. Estimating a survival rate of at least 10% of the released fish (Hervas et al., 2010), implies a release of ca.10 times the size of the desired population size. Therefore, the realistic (yet over-estimated) cost should be set at $10 per fish, and to account for variance of estimates we have multiplied this by a factor of 2. Therefore, using our estimates, the cost of restocking a reef spanning 15 km² will amount to approximately $\tilde{c}\cdot \hat{K}\cdot X\cdot {\delta }_p=20\cdot \text{3,000}\cdot 15\cdot 0.1=\text{90,000}$ USD.

The discount rate (r′) was set to 0.05, but our results remain robust when varying discount rates (Text S2).

From Fig. 1 we can see that restocking broadens the range of conditions under which the system will reach a high coral coverage state, but not to a very substantial extent. However, restocking increases the expected revenue of a disturbed reef under a wide range of initial conditions, especially in the long term (Fig. 1, compare A to B). This is the result of the relatively cheap cost of restocking (estimated at 6,000 USD per km²), combined with the high revenue of coral reef area (estimated at $200,000 per year per km²). Even if the reef does not restore to high coral coverage, the delay in its deterioration, enabled by restocking, will still be profitable for a large extent of the initial conditions. Similarly, even if the reef will eventually be restored without human intervention, restocking will shorten the period of time required to achieve this. This is shown in Fig. 2, where a time series of the values of the coral coverage (C), macroalgae coverage (M), and grazing fish (P) are plotted with and without restocking for two sets of initial conditions. Figures 2A and 2B presents the model variables simulated from initial values corresponding to area (I) in Fig. 1 (C₀ = 0.35, M₀ = 0.05), in which the reef will be restored without intervention. Although both scenarios lead to an eventual high coral coverage, we can see that restocking will shorten the time to equilibrium to about 65% of this time in a system without restocking (compare Figs. 2A and 2B). A change in initial conditions, to those corresponding to area II in Fig. 1, can change the dynamics entirely. Figures 2C and 2D represents initial conditions (C₀ = 0.35, M₀ = 0.07) in which the coral reef will deteriorate without intervention (Fig. 2C), but will return to high coral coverage when restocking is implemented (Fig. 2D).

In area III, the reef would remain in a high macroalgae state with or without restocking. In such a case, we could consider a combination of restoration methods. For instance, if feasible, eradication of macroalgae will be expressed as moving left on the phase plane presented in Fig. 1 in our model. We expect that when restocking is applied, the extent of eradication needed to bring the system to a point where restoration will be possible will be lower, and the restoration time from that point will be shorter.

Multiple reefs

We next generalize the notion of restocking to a system consisting of two reefs, in which the fish migrate from one reef to another. We define the direction of migration from Reef I (upstream) to Reef II (downstream). Under this range of initial conditions we note five possible scenarios: (I) initial conditions under which in both reefs the system reaches a state with high coral coverage without restocking; (II) areas wherein one reef will reach high coral coverage without intervention, while the other reef will only succeed if restocking is applied; (III) areas wherein one reef will reach high coral coverage without intervention, while the other will reach the macroalgae state even if restocking is applied; (IV) areas in which both reefs will deteriorate to the macroalgae state without intervention, but restocking will salvage one of them; and (V) areas in which both rates will deteriorate and restocking will not help either reef. These areas are marked accordingly on Fig. 3. Additionally, colors in Fig. 3 represent the expected revenue of restocking 5 and 20 years after the restocking has taken place, in log₁₀ scale, with negative revenue replaced by zeros. Parameters of Fig. 3 are as in Fig. 1, with the migration from Reef I is directed towards Reef II, and migration from Reef II is lost.

We can see that restocking broadens the range of conditions under which at least one of the reefs will reach a high coral coverage state. Moreover, restocking increases the expected revenue from the coral reefs under almost all conditions. This is due to the amplification of the effect seen in Figs. 1 and 2. Even when restocking is only performed for one of the reefs, it accelerates the return to a high coral coverage state, and delays deterioration of the reefs. Figure 4 presents time-series examples for these dynamics for the same parameters as in Fig. 2. C_I, M_I, P_I and C_II, M_II, P_II, are the coral coverage, macroalgae coverage and grazing fish, for the upstream and downstream reefs, respectively. We can see that restocking only the upstream reef can shorten the recovery time to about 60, for both the downstream and upstream reefs %, relative to the system without restocking (Fig. 4 compare A to B). In addition, for parameters that are within region II of Fig. 3, restocking can salvage the upstream reef from deterioration (Fig. 4 compare C to D).