Eco-Link@BKE looks like a bridge. Mathematically, it is more interesting than that. It is a connection inserted into a fragmented network.

Bukit Timah Nature Reserve and Central Catchment Nature Reserve once formed contiguous forest. The Bukit Timah Expressway, built through the landscape in 1986, separated the two. Eco-Link@BKE was completed in 2013 to reconnect them and give wildlife a safer route between the reserves. NParks says the bridge helps animals expand habitat range and genetic pools, and by 2021 around 100 fauna species had been recorded using it.

That is the ecological story.

The mathematical story begins with a different question:

What changes in a system when two important regions that were disconnected become connected again?

This article belongs to the What about Bukit Timah? hub. The ecological owner remains How Bukit Timah Nature Reserve Works. The statistical owner for biodiversity measurement is The Mathematics of Bukit Timah Rainforest. Here, our job is narrower: use Eco-Link@BKE to understand graph theory, barriers, connectivity and movement through a network.

1. Turn the Landscape Into a Graph

Graph theory begins with two simple objects:

nodes — the places or states in the system;
edges — the possible connections between them.

For a highly simplified first model, we could represent:

Node B = Bukit Timah Nature Reserve
Node C = Central Catchment Nature Reserve

Before the expressway barrier, we might represent movement between them with an edge:

B — C

After a major barrier fragments the habitat, the useful connection is impaired. The simple graph becomes:

B C

Eco-Link@BKE restores a connection:

B — Eco-Link — C

This model is intentionally crude. A real forest is not two points connected by one line. But the abstraction is useful because it isolates the system property we care about: connectivity.

2. Fragmentation Is Not Merely “Less Forest”

Suppose a continuous habitat of area A is split into two patches whose areas add to the same total A.

A beginner might say: total area has not changed, therefore nothing important has changed.

But area is not the only variable.

Fragmentation changes:

which individuals can reach which habitats;
the number of possible movement paths;
the probability of successful dispersal;
exposure to dangerous boundaries;
the degree of population isolation;
the movement of seeds and animals;
the opportunity for genetic exchange.

So two landscapes can have similar habitat area and very different network structure.

Quantity of habitat and connectivity of habitat are different variables.

3. A Connection Is Not Always Binary

In the simplest graph, an edge either exists or does not.

Ecological movement is usually more complicated.

A route can be:

easy for one species and difficult for another;
safe at night and risky during the day;
usable during dry weather but less usable in another condition;
physically present but behaviourally avoided;
possible but associated with high mortality risk.

This suggests a weighted graph.

Instead of writing only whether an edge exists, we attach a weight representing cost, resistance, risk, travel time or another species-specific quantity.

edge = connection + weight

The weight is not universal. A barrier perceived by a pangolin may not be perceived the same way by a bird.

That immediately gives us a more mature model:

connectivity is species-dependent.

4. The Landscape Can Be Modelled as Resistance

Imagine dividing a map into cells.

Each cell receives a resistance value:

low resistance for suitable forest;
moderate resistance for less suitable vegetation;
high resistance for exposed or dangerous areas;
very high resistance for major barriers.

Now movement from one reserve to another becomes a path problem across a weighted surface.

One possible objective is to find a route with minimum total resistance.

For a path P containing cells or edges with costs w₁, w₂, …, wₙ:

cost(P) = Σ wᵢ

The least-cost path is the path whose total resistance is smallest under the model.

This is mathematically similar to route planning in transport networks, but the meaning of “cost” changes. Instead of travel minutes or dollars, the weight might represent ecological resistance.

The algorithm may be familiar. The world represented by the weights is different.

5. The Shortest Path Is Not Necessarily the Best Path

Suppose Route A is 200 metres long but crosses an extremely dangerous open area.

Route B is 350 metres long but remains under vegetation and avoids traffic.

If we optimise only geometric distance, Route A wins.

If we optimise ecological resistance, Route B may win.

This is an important distinction in Mathematics:

“Best” has no meaning until the objective function is declared.

Students often ask, “What is the shortest route?”

Real systems ask harder questions:

shortest by distance?
fastest by time?
safest by risk?
cheapest by cost?
lowest ecological resistance?
most robust if one connection fails?

The optimisation problem depends on the goal.

6. Eco-Link@BKE Is a Bottleneck — and That Is Mathematically Important

When two large regions are connected by relatively few viable routes, those connections become disproportionately important.

In network language, a bridge-like connection can behave as a bottleneck.

If one edge lies on many useful paths, removing or degrading that edge can fragment the network again.

This can be formalised with ideas such as:

edge connectivity;
cut edges or bridges;
betweenness centrality;
minimum cuts;
network robustness.

For a toy graph, suppose regions B and C are connected only through edge e.

B — e — C

Remove e and the graph separates.

That edge is structurally critical.

Real ecological networks are more complex, but the principle helps explain why reconnecting fragmented habitats can matter far beyond the physical area occupied by the corridor itself.

7. Redundancy Makes Networks More Robust

Suppose two habitat regions have exactly one usable connection.

The system is connected, but fragile.

Now suppose there are three independent usable routes.

The network can remain connected even if one route becomes temporarily unusable.

This is redundancy.

Engineering systems use redundancy for reliability. Transport systems use alternate routes. Computer networks use multiple paths. Ecological networks also benefit when movement does not depend on a single narrow possibility.

This is why Eco-Link@BKE should not be misunderstood as proof that all fragmentation has been solved.

One connection can be extremely valuable and still leave a broader connectivity problem.

8. A Route Existing Is Different From a Route Being Used

The graph may contain an edge.

An animal still has to encounter, accept and successfully cross it.

This introduces probability.

For a simplified conceptual model, let:

p₁ = probability an animal reaches the corridor;
p₂ = probability it enters;
p₃ = probability it crosses successfully;
p₄ = probability it reaches suitable habitat on the other side.

If these stages were independent — a strong assumption used here only to show structure — overall movement probability could be written:

P(success) = p₁ × p₂ × p₃ × p₄

If each stage had probability 0.8, the overall probability would be:

0.8⁴ = 0.4096

Even apparently high stage-level success can compound into a much lower end-to-end probability.

The numbers are hypothetical. The lesson is general:

A chain is governed by the whole sequence, not the best individual step.

9. Observed Use Is Evidence — But It Is Not the Entire Connectivity Model

NParks reported that as of 2021 around 100 fauna species had been recorded on Eco-Link@BKE. Between 2018 and 2021, 31 additional fauna species were recorded using the bridge for the first time, including butterflies, birds, mammals and a reptile.

That is important observational evidence that the bridge is functioning as habitat and movement infrastructure.

But statistically, “recorded using” is not the same as:

every individual uses it;
every species uses it equally often;
movement is equally successful in both directions;
all genetic isolation has disappeared;
the bridge alone explains every population change.

The evidence answers a narrower question: species have been observed using the Eco-Link, and the list of recorded users increased over time.

That distinction follows the principle developed in The Mathematics of Bukit Timah Rainforest: observation and inference are connected, but they are not identical.

10. Centrality: Some Connections Matter More Than Others

In a network, a node or edge can be important because many paths pass through it.

One measure is betweenness centrality.

For an edge e, a conceptual form is:

BC(e) = Σ σₛₜ(e) / σₛₜ

where σₛₜ is the number of shortest paths between s and t, and σₛₜ(e) is the number of those paths using edge e.

The exact metric is not automatically the correct ecological measure, because wildlife does not necessarily choose mathematical shortest paths. But the idea is valuable: some connections sit in structurally strategic positions.

A small physical link can have large network importance if it joins otherwise separated regions.

11. Minimum Cut: How Little Must Fail Before the Network Breaks?

A minimum cut asks for the smallest set of edges or capacities whose removal disconnects specified parts of a network.

In infrastructure, this helps identify vulnerabilities.

In ecology, the analogy encourages a useful question:

Which small number of corridor failures would produce a disproportionately large loss of landscape connectivity?

The answer depends on the species and landscape model, but the reasoning matters because conservation resources are finite. If one corridor is structurally irreplaceable, maintaining it may have unusually high system value.

12. Connectivity Has Capacity, Not Just Presence

Transport students quickly learn that a road existing does not mean it can carry unlimited traffic.

Networks have capacity.

Ecological corridors also have functional limits. A route may support some species, some movement frequencies and some behaviours better than others.

In a flow network, each edge can have capacity c(e). The maximum-flow problem asks how much flow can move from source to sink without exceeding edge capacities.

Wildlife movement is not literal fluid flow, so a maximum-flow model should not be applied mechanically. But the analogy helps separate:

an edge existing;
an edge being usable;
an edge being used often;
an edge supporting enough movement to affect population connectivity.

Those are different levels of claim.

13. From Movement to Genetic Connectivity

NParks explains Eco-Link@BKE partly in terms of expanding habitat and genetic pools.

The logic is that if isolated populations can move and reproduce across previously separated habitat, genetic exchange may increase.

But movement connectivity and genetic connectivity are not identical.

An individual can cross without reproducing.
Crossing frequency may be low.
Some demographic groups may cross more than others.
Species differ in generation time and dispersal behaviour.

A responsible model therefore separates the chain:

physical connection → movement → successful dispersal → reproduction → genetic exchange

Each arrow is a mechanism that needs evidence.

14. Adjacency Matrices: Put the Landscape Into a Table

A graph can be represented as an adjacency matrix.

For a toy three-node system B, E and C:

	B	E	C
B	0	1	0
E	1	0	1
C	0	1	0

where E represents the Eco-Link connection.

A 1 indicates a modelled connection; 0 indicates no direct connection.

For weighted connectivity, the matrix entries could instead contain resistance, probability or another edge attribute.

This is the bridge between visual graph theory and linear algebra. A landscape can become a matrix, and matrix operations can reveal path structure that is difficult to see by inspection in a large network.

15. Network Laplacians: Connectivity Has Algebraic Structure

For more advanced students, a graph has a degree matrix D and adjacency matrix A. The graph Laplacian is:

L = D − A

The eigenvalues of L contain information about connectivity.

In particular, for a simple undirected graph, the number of zero eigenvalues equals the number of connected components. The second-smallest eigenvalue — often called algebraic connectivity — is positive when the graph is connected and can reflect how strongly connected the network is.

We should not mistake this abstract graph for the living forest. But it demonstrates something beautiful:

whether a network is joined or fragmented can appear inside its algebra.

16. Effective Resistance: Many Paths Can Act Together

Another network concept borrows from electrical circuits.

If edges are treated like resistors, multiple parallel paths lower effective resistance between nodes. One narrow route gives higher network vulnerability; several independent paths create lower effective resistance and greater redundancy.

Landscape ecology has used circuit-theory approaches because they can represent movement across many possible paths rather than assuming organisms always follow one least-cost route.

The core mathematical intuition is accessible even without advanced formulas:

one viable path = connection, but fragile;
many viable paths = connection with alternatives;
high-resistance areas discourage movement;
low-resistance corridors concentrate potential flow.

Again, the weights must be biologically meaningful for the target species.

17. Before-and-After Data Need More Than Two Numbers

Suppose we record 50 wildlife crossings before an intervention and 80 after.

Can we say the intervention caused a 60% increase?

The arithmetic is easy:

(80 − 50) ÷ 50 = 0.6 = 60%

The causal claim is harder.

Was monitoring effort identical?
Were cameras placed in the same way?
Did animal abundance change?
Did weather differ?
Were the same species present?
Was the observation period equally long?
Were there other habitat changes nearby?

This is why ecological monitoring must connect network theory with statistical design.

A graph tells us what connections are possible under the model.

Data tell us what we observed.

Inference tries to connect the two responsibly.

18. What Do “31 Additional Species Recorded” and “Around 100 Species” Mean?

NParks reported that 31 additional fauna species were recorded on Eco-Link@BKE for the first time between 2018 and 2021, bringing the total number recorded as of 2021 to around 100 species.

Mathematically, several statements must remain separate.

31 additional species were newly recorded by the monitoring programme in that period.
This increased the cumulative recorded-species list.
It does not mean those 31 species first began using the bridge only in that period.
It does not mean exactly 100 species use the bridge at every point in time.
It does not mean every recorded species uses it with equal frequency.

This is the same discovery-date problem encountered in biodiversity sampling.

first recorded ≠ first existed

The monitoring system has its own observation boundary.

19. One Network Is Not Enough: Build a Species-Specific Graph

A common mistake is to draw one ecological network and assume it represents all wildlife.

Different species perceive landscapes differently.

For Species A:

dense forest may have low resistance;
open grass may have moderate resistance;
a road may have very high resistance.

For Species B:

forest edge may be acceptable;
open space may be easy;
water may be the dominant barrier.

The same physical map generates different weighted graphs.

This is an excellent example of model dependence:

the world is shared; the cost surface is species-specific.

20. Networks Change Through Time

Eco-Link@BKE was completed in 2013, but an ecological corridor does not become a mature forest connection instantly.

Vegetation grows. Species discover routes. Habitat structure changes. Nearby development changes. Populations change.

The correct mathematical object may therefore be a temporal network:

G(t)

where nodes, edges or weights can change with time.

This matters because a corridor’s effectiveness in Year 1 need not equal its effectiveness in Year 10.

NParks’ annual-report material notes that native vegetation on the Eco-Link has grown into a verdant forest and that wildlife records have accumulated over time. That is exactly the sort of system where the edge itself evolves.

21. The Edge Can Change Because It Is Used

Most classroom graphs assume an edge weight is fixed.

Living systems can create feedback.

As vegetation establishes, habitat quality may change. Seed dispersal can alter plant composition. Animal movement can influence ecological processes. Maintenance choices can affect cover and disturbance.

So a corridor is not always a static pipe through which animals move.

It can itself become habitat.

Mathematically, we may need edge weights that change:

w(e,t)

where the cost or suitability of edge e varies through time.

22. Robustness: What Happens When Conditions Change?

A good network should not be judged only under normal conditions.

Ask what happens when:

a corridor is temporarily disturbed;
a species avoids a new edge condition;
vegetation changes;
one neighbouring habitat patch degrades;
a movement route becomes unavailable.

Robustness can be explored by removing nodes or edges and recalculating connectivity.

For a graph G, let G − e be the graph after removing edge e.

If G is connected but G − e is disconnected, e is structurally critical.

This kind of counterfactual analysis is useful because resilience is often hidden until something fails.

23. What Primary Students Can Learn From Eco-Link@BKE

Young students do not need formal graph theory notation to understand connectivity.

Give them three islands drawn on paper and ask which are connected by bridges.

Then remove one bridge.

Ask:

Can every island still reach every other island?
How many bridges are needed?
Which bridge is most important?
Can there be two different routes?

That is graph theory before the vocabulary.

Eco-Link@BKE gives the child a real reason the question matters.

24. What Secondary Students Can Learn

Secondary students can formalise the network.

Represent habitat patches as nodes.
Represent corridors as edges.
Assign weights to movement costs.
Compare shortest-distance and least-resistance routes.
Build adjacency matrices.
Identify cut edges and bottlenecks.
Calculate simple path probabilities under stated assumptions.

The important habit is to explain what each mathematical object means in the real system.

A matrix entry is not “just a 1”. It stands for a declared relationship.

25. Additional Mathematics and JC: From Networks to Optimisation

At higher levels, the questions become richer.

How should resistance weights be transformed?
How sensitive is the optimal path to parameter changes?
What happens when there are multiple objectives?
Can corridor location be framed as constrained optimisation?
How does uncertainty in movement probability affect expected connectivity?
How should repeated monitoring update model parameters?

A simplified multi-objective problem might try to:

minimise construction cost + ecological resistance + disturbance

subject to:

geometric constraints;
safety constraints;
traffic constraints;
habitat requirements;
budget limits.

This is much closer to how real engineering and planning problems behave than the one-objective exercises students usually meet first.

26. A Corridor Has to Fit Several Systems at Once

Eco-Link@BKE crosses an expressway that still has to function as transport infrastructure.

The ecological connection cannot simply ignore road safety, structural engineering, construction constraints or maintenance.

This makes the design a fit problem across multiple systems:

ecology;
road transport;
structural engineering;
construction logistics;
vegetation establishment;
monitoring and research.

The corridor succeeds only if those systems can coexist.

That is a recurring mathematical idea throughout Bukit Timah: the best solution is often not the maximum of one variable. It is the feasible fit among several constraints.

27. What Would Make a Connectivity Model Fail?

A model can fail even when its equations are correct.

The resistance values may not match actual animal behaviour.
Species may respond differently from the assumed target species.
The model may ignore seasonal change.
The network may omit small but important habitat patches.
Movement data may be biased toward monitored locations.
A “shortest path” assumption may be biologically unrealistic.
Physical crossing may be confused with successful reproduction.

This is why ecological modelling requires repeated return to observation.

Model → observe → compare → revise.

Mathematics is not weakened when the model is revised.

That is Mathematics behaving properly in the world.

28. Why Eco-Link@BKE Is Not a Public Walking Bridge

NParks states that Eco-Link@BKE is not open to members of the public. Access is strictly supervised and only for research because the bridge is ecologically sensitive.

That is useful to include because it reveals another optimisation constraint.

A structure can have public educational value without maximising public physical access.

If the primary objective is wildlife connectivity, unrestricted human use could alter the very edge weight the system was designed to improve.

In other words, opening the bridge to more users could change the model.

29. The Teaching Lesson: A Line on a Map Is Not Yet a Connection

Students are often shown a diagram and taught to trust the line.

Eco-Link@BKE gives us a better habit.

Ask:

Who can use this edge?
At what cost?
How often?
Under what conditions?
What evidence shows it is being used?
What happens if the edge fails?
Are there alternate routes?
Does crossing produce the downstream outcome we care about?

Now graph theory stops being a drawing exercise.

It becomes systems reasoning.

30. What This Article Does Not Claim

It does not reduce the two nature reserves to literally two nodes in real ecological analysis.
It does not assume all wildlife species perceive the landscape with the same resistance values.
It does not claim the Eco-Link eliminates all habitat-fragmentation effects.
It does not treat “around 100 species recorded” as a complete population census.
It does not equate observed crossing with guaranteed genetic exchange.
It does not claim shortest-path algorithms reproduce animal behaviour automatically.
It does not treat network centrality metrics as substitutes for field ecology.

These boundaries matter because mathematical abstraction is powerful only when we remember which parts of reality it removed.

Evidence and Official Sources

The fixed facts in this article were checked against current NParks material. Graph-theory examples are explanatory models unless explicitly identified as Eco-Link monitoring data.

NParks — Eco-Link@BKE: connection between Bukit Timah and Central Catchment Nature Reserves, the 1986 separation, wildlife examples, habitat/genetic-pool purpose and research-only access.
NParks — Nature Areas and Nature Park Networks: the reserves’ former contiguity and Eco-Link completion in 2013.
NParks — Ecological Connectivity Initiatives: around 100 fauna species recorded on Eco-Link@BKE as of 2021 and 31 species recorded there for the first time between 2018 and 2021.
NParks Annual Report — 10 Years of Bridging Biodiversity: long-term vegetation development and wildlife use of the bridge.

So, What Is the Mathematics of Eco-Link@BKE?

It is the Mathematics of a system that changed state.

A once-contiguous forest became fragmented. A major barrier increased movement cost between two important habitat regions. A new ecological bridge restored a lower-resistance connection. Wildlife use was then observed over time.

In the language of Mathematics, we can study:

nodes and edges;
weighted graphs;
least-cost paths;
bottlenecks and cut edges;
redundancy and robustness;
movement probabilities;
adjacency matrices;
algebraic connectivity;
time-varying networks;
the difference between observed use and inferred system effect.

But the deepest lesson is simpler.

Two places can be geographically close and functionally far apart.
Mathematics helps us describe the difference.

Return to: What about Bukit Timah? — Live Hub · How Bukit Timah Nature Reserve Works · The Mathematics of Bukit Timah Rainforest · Singapore Mathematics Hub

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The Mathematics of Eco-Link@BKE | Fragmentation → Graphs → Connectivity → Wildlife Movement