Conservation Solutions

Derivation: Optimal Reserve Design from Island Biogeography

MacArthur & Wilson's (1967) theory of island biogeography provides the foundation for reserve design. The equilibrium number of species on an island is determined by the balance between immigration and extinction:

$\frac{dS}{dt} = I(S) - E(S) = I_0\left(1 - \frac{S}{P}\right) - E_0 \frac{S}{P}$

where $P$ is the mainland species pool, $I_0$ is the maximum immigration rate, and $E_0$ is the maximum extinction rate. At equilibrium ($dS/dt = 0$):

$S^* = \frac{I_0 \cdot P}{I_0 + E_0}$

Immigration decreases with distance ($I_0 \propto e^{-d/D}$) and extinction decreases with area ($E_0 \propto A^{-z}$). This leads to the classic SLOSS debate (Single Large Or Several Small reserves):

• • Single Large: lower extinction rate ($E_0 \propto A^{-z}$), supports more species, maintains large-bodied species with big home ranges, avoids edge effects.
• • Several Small: captures more habitat heterogeneity and beta diversity; if reserves contain different species assemblages, total diversity may be higher (Tjorve, 2010).

Modern reserve design resolves this debate using systematic conservation planning (Margules & Pressey, 2000), which optimizes the reserve network to maximize species representation while minimizing cost, using integer linear programming.

Corridor Connectivity: Graph Theory

Corridors connecting reserves allow species to move between habitat patches, critical for climate adaptation. The connectivity of a reserve network is quantified using graph theory, where reserves are nodes and corridors are edges.

Betweenness centrality identifies the most important nodes/corridors for maintaining network connectivity:

$C_B(v) = \sum_{s \neq v \neq t} \frac{\sigma_{st}(v)}{\sigma_{st}}$

where $\sigma_{st}$ is the total number of shortest paths from node $s$to node $t$, and $\sigma_{st}(v)$ is the number passing through$v$. Nodes with high betweenness centrality are critical “stepping stones” — their loss would fragment the network.

The probability of connectivity (Saura & Pascual-Hortal, 2007) integrates habitat area and dispersal:

$PC = \frac{1}{A_L^2} \sum_{i=1}^{n} \sum_{j=1}^{n} a_i \cdot a_j \cdot p_{ij}^*$

where $a_i$ is the area of patch $i$, $p_{ij}^*$ is the maximum product probability of dispersal between patches, and $A_L$ is the total landscape area.

Derivation: Net CO$_2$ Removal

The net carbon benefit of a nature-based intervention is:

$\text{Net CO}_2 = \underbrace{C_{\text{seq}}}_{\text{sequestration}} - \underbrace{E_{\text{project}}}_{\text{project emissions}} - \underbrace{C_{\text{opp}}}_{\text{opportunity cost}}$

Key pathways and their potential (Griscom et al., 2017):

• • Reforestation: ~3.0 Gt CO$_2$/yr. Tropical forests sequester ~4.4 t C/ha/yr (Lewis et al., 2009). Global potential: 0.9 billion hectares of degraded land suitable for restoration (Bastin et al., 2019), yielding up to ~0.9 Gt C/yr = 3.3 Gt CO$_2$/yr.
• • Avoided deforestation: ~3.6 Gt CO$_2$/yr. Tropical deforestation releases ~4.4 Gt CO$_2$/yr; avoiding 80% would yield this figure.
• • Wetland restoration: ~0.7 Gt CO$_2$/yr. Peatlands store 30% of soil carbon on 3% of land area.
• • Blue carbon (mangroves, seagrasses, salt marshes): ~0.6 Gt CO$_2$/yr. Mangroves sequester 6–8 t CO$_2$/ha/yr, 3–5x more than terrestrial forests per unit area (Mcleod et al., 2011).

Blue Carbon: Coastal Ecosystems

Coastal vegetated ecosystems — mangroves, seagrasses, and salt marshes — are disproportionately important for carbon storage. The carbon accumulation rate is modelled as:

$C_{\text{acc}}(t) = C_{\max}\left(1 - e^{-\lambda t}\right)$

where $C_{\max}$ is the long-term carbon stock capacity and $\lambda$is the accumulation rate constant. For mangroves, $C_{\max} \approx 1,000$ t C/ha (including soil carbon to 1 m depth) and $\lambda \approx 0.03$ yr$^{-1}$, so reaching half capacity takes $t_{1/2} = \ln 2 / \lambda \approx 23$ years.

Derivation: Trophic Cascade Strength

A trophic cascade occurs when predators suppress herbivores, indirectly benefiting primary producers. The cascade strength is the ratio of the indirect effect to the direct effect. For a three-level food chain (predator $P$ → herbivore $H$ → vegetation $V$):

$\frac{dV}{dt} = rV\left(1 - \frac{V}{K_V}\right) - a_{HV} H V$

$\frac{dH}{dt} = e_{HV} a_{HV} H V - a_{PH} P H - d_H H$

$\frac{dP}{dt} = e_{PH} a_{PH} P H - d_P P$

The cascade strength is quantified by the log ratio of vegetation biomass with vs. without predators (Shurin et al., 2002):

$\text{CS} = \ln\!\left(\frac{V^*_{\text{with } P}}{V^*_{\text{without } P}}\right)$

Yellowstone wolves: The reintroduction of 31 grey wolves in 1995–1996 triggered a cascade: wolf predation reduced elk density by ~50%, releasing willows and aspens from browsing pressure. Riparian vegetation recovery stabilized streambanks, reducing erosion and increasing beaver habitat. Ripple & Beschta (2012) measured a 5x increase in willow height in areas with high wolf activity (a “landscape of fear” effect).

Pleistocene Rewilding Debate

Donlan et al. (2006) proposed Pleistocene rewilding— introducing ecological proxies for extinct megafauna (e.g., African elephants as proxies for mammoths in North America). Arguments:

• • For: restores ecological functions lost 13,000 years ago; megaherbivores maintained open landscapes, seed dispersal networks, and nutrient cycling. Many plants evolved with megafauna and have “anachronistic” fruits.
• • Against: ecosystems have reorganized in 13,000 years; invasion risk is substantial; public opposition; unpredictable ecological outcomes.

Derivation: Social Cost of Carbon

The social cost of carbon (SCC) is the present value of all future damages caused by emitting one additional tonne of CO$_2$ today. It is derived by integrating the marginal damage function over the future:

$\text{SCC} = \int_0^{\infty} D'(T(t)) \cdot \frac{dT}{dE} \cdot e^{-\delta t} \, dt$

where $D'(T)$ is the marginal damage from an incremental temperature increase,$dT/dE$ is the temperature response to emissions, and $\delta$ is the social discount rate. Key sensitivities:

• • Discount rate $\delta$: Nordhaus uses $\delta \approx 5\%$ (SCC ~$50/t CO$_2$); Stern uses$\delta \approx 1.4\%$ (SCC ~$200/t CO$_2$). The Ramsey equation relates: $\delta = \rho + \eta \cdot g$, where $\rho$ is pure time preference, $\eta$ is elasticity of marginal utility, $g$ is consumption growth.
• • Damage function: The most common form is quadratic: $D(T) = \alpha_1 T + \alpha_2 T^2$. Weitzman (2012) argued for higher-order terms to capture catastrophic damages: $D(T) = 1 - 1/(1 + (T/20.46)^2 + (T/6.081)^{6.754})$.
• • Climate sensitivity: uncertainty in equilibrium climate sensitivity ($\Delta T_{2\times}$) contributes ~30% of SCC variance.

REDD+ and Biodiversity Credits

REDD+ (Reducing Emissions from Deforestation and Degradation) pays developing countries to preserve forests. The carbon credit calculation:

$\text{Credits} = (E_{\text{baseline}} - E_{\text{actual}}) \times A \times t$

where $E_{\text{baseline}}$ is the projected deforestation emission rate (t CO$_2$/ha/yr),$E_{\text{actual}}$ is the realized emission rate, $A$ is the project area, and $t$ is the crediting period.

Biodiversity credits are an emerging mechanism that values biodiversity independently of carbon. Metrics include species richness, functional diversity, and ecosystem intactness. The challenge is fungibility— unlike carbon (where 1 tonne is equivalent regardless of source), biodiversity cannot be interchanged between sites.

Derivation: Robust Decision-Making Under Uncertainty

Traditional conservation planning assumes a stationary environment. Under climate change, planners must consider multiple future scenarios. Robust decision-making (Lempert et al., 2006) seeks strategies that perform acceptably across all plausible futures:

$\max_a \min_s \; U(a, s)$

where $a$ is the conservation action (e.g., reserve placement), $s$ is a climate scenario, and $U(a, s)$ is the biodiversity outcome. This maximin criterion chooses the action with the best worst-case outcome.

Climate refugia are areas where species can persist under climate change due to topographic buffering, cold-air pooling, or stable microclimates. Identifying refugia uses the climate velocity framework:

$R_{\text{refugia}} = \{x : v_{\text{climate}}(x) < v_{\text{dispersal,min}}\}$

Keppel et al. (2012) identified key refugia types: topographic (mountains, canyons), edaphic (unique soils), hydrological (springs, groundwater-fed systems), and oceanic (upwelling zones).

TEK as Complementary to Scientific Monitoring

Traditional Ecological Knowledge (TEK) encompasses the knowledge, practices, and beliefs of indigenous peoples about ecological relationships, accumulated over centuries to millennia. The IPBES (2019) Global Assessment highlighted that indigenous lands contain ~80% of the world's remaining biodiversity.

• • Aboriginal fire management: Indigenous Australians practiced “cool burning” for >50,000 years, creating mosaic landscapes that reduced catastrophic wildfire risk while promoting biodiversity. The 2019–2020 Australian megafires, which burned ~18.6 million hectares, occurred partly because traditional burning had ceased. Modern programs like the Arnhem Land Fire Abatement (ALFA) project have reinstated Indigenous burning, reducing emissions by ~37,500 t CO$_2$e/yr while generating carbon credits (Russell-Smith et al., 2013).
• • Marine management: Traditional ráhui (Polynesian temporary fishing closures) and taboo areas function as marine protected areas. Cinner et al. (2006) found that customary closures in Papua New Guinea maintained reef fish biomass at levels comparable to fully protected marine reserves.
• • Biodiversity monitoring: Indigenous community-based monitoring programs often detect ecological changes (e.g., phenological shifts, species disappearances) earlier than formal scientific surveys, because of continuous, fine-grained observation across large landscapes (Berkes et al., 2000).

Key References

• • Bastin, J.-F. et al. (2019). The global tree restoration potential. Science, 365(6448), 76–79.
• • Berkes, F. et al. (2000). Rediscovery of traditional ecological knowledge as adaptive management. Ecol. Appl., 10(5), 1251–1262.
• • Cinner, J. et al. (2006). Socioeconomic factors that affect artisanal fishers' readiness to exit a declining fishery. Conservation Biology, 20(5), 1368–1379.
• • Donlan, C.J. et al. (2006). Pleistocene rewilding: an optimistic agenda for twenty-first century conservation. Am. Nat., 168(5), 660–681.
• • Griscom, B.W. et al. (2017). Natural climate solutions. PNAS, 114(44), 11645–11650.
• • Keppel, G. et al. (2012). Refugia: identifying and understanding safe havens for biodiversity under climate change. Global Ecol. Biogeogr., 21(4), 393–404.
• • Lempert, R.J. et al. (2006). A general analytic method for generating robust strategies. Management Science, 52(4), 514–528.
• • MacArthur, R.H. & Wilson, E.O. (1967). The Theory of Island Biogeography. Princeton University Press.
• • Margules, C.R. & Pressey, R.L. (2000). Systematic conservation planning. Nature, 405(6783), 243–253.
• • Mcleod, E. et al. (2011). A blueprint for blue carbon. Front. Ecol. Environ., 9(10), 552–560.
• • Ripple, W.J. & Beschta, R.L. (2012). Trophic cascades in Yellowstone. Biol. Conserv., 145(1), 205–213.
• • Russell-Smith, J. et al. (2013). Managing fire regimes in north Australian savannas. J. Appl. Ecol., 50(3), 645–656.
• • Saura, S. & Pascual-Hortal, L. (2007). A new habitat availability index to integrate connectivity in landscape conservation planning. Landscape Ecology, 22(9), 1393–1406.
• • Shurin, J.B. et al. (2002). A cross-ecosystem comparison of the strength of trophic cascades. Ecology Letters, 5(6), 785–791.
• • Weitzman, M.L. (2012). GHG targets as insurance against catastrophic climate damages. J. Public Econ. Theory, 14(2), 221–244.