Systems Ecology: Modelling Populations & Ecosystems
CollegeChapter 16 described energy flow and cycles qualitatively. Ecology becomes a quantitative, predictive science once populations and ecosystems are described mathematically — the closing chapter of this course, and a fitting place to see all of biology working as one connected system.
Modelling population growth
Under unlimited resources, a population grows exponentially:
where N is population size, t is time, and r is the intrinsic growth rate. No real environment has unlimited resources, so growth is more realistically modelled as logistic, which incorporates a carrying capacity K — the maximum population the environment can sustain:
As N approaches K, the term (1 − N/K) approaches zero, so growth rate slows and the population levels off in a characteristic S-shaped (sigmoid) curve. Limiting factors can be density-dependent (their effect intensifies as population grows — disease, competition, predation) or density-independent (act regardless of population size — a wildfire, a flood).
Predator–prey dynamics and trophic cascades
Predator and prey populations often oscillate together, rising and falling out of phase (formalised by the Lotka–Volterra equations): prey increase, giving predators more food so predator numbers rise, which then drives prey down, causing predator numbers to fall, allowing prey to recover — a repeating cycle. Removing a top predator can trigger a trophic cascade that reorganises an entire ecosystem, as herbivore numbers surge unchecked and vegetation is over-grazed.
Quantifying productivity
Gross primary productivity (GPP) is the total rate at which producers capture energy via photosynthesis. Producers use some of that energy for their own respiration (R). What remains — available to be eaten, stored, or grown as biomass — is the net primary productivity (NPP):
Why simplified models still explain real ecosystems
Real populations are influenced by weather, disease, migration, genetics, and human activity all at once — far too complex to model exactly. The power of the logistic and Lotka–Volterra models is not that they capture every detail, but that they isolate the dominant mechanism (resource limitation; predator–prey feedback) and show that this alone is enough to reproduce the large-scale patterns seen in real data — an S-shaped growth curve, or oscillating predator and prey numbers. This is the core method of systems science: build the simplest model that captures the essential feedback loop, test it against real data, then add complexity only where the simple model demonstrably fails. It also explains the earlier "10% rule" (Chapter 16) in sharper terms: it is really a rough summary of the fact that R consumes a large share of GPP at every trophic level, leaving only a fraction of captured energy as NPP available to the next level up.
When wolves were eliminated from Yellowstone in the 1920s, elk populations grew unchecked and overgrazed young trees along streams. Reintroducing wolves in 1995 reduced elk numbers and changed their grazing behaviour; vegetation recovered, which stabilised riverbanks and even altered river channel patterns — a single apex predator's presence cascading through multiple trophic levels to reshape the physical landscape.
Worked example: logistic growth and productivity
Part A. A population has intrinsic growth rate r = 0.4/year and carrying capacity K = 10,000. Find the population growth rate (not size) when N = 5,000.
- Apply the logistic equation. dN/dt = rN(1 − N/K).
- Insert values. dN/dt = 0.4 × 5000 × (1 − 5000/10000) = 0.4 × 5000 × 0.5.
- Compute. = 1000 individuals/year.
- Notice the pattern. Because N = K/2 here, (1 − N/K) = 0.5, which is the maximum value of N(1−N/K) across the whole range — this is a general result: logistic growth rate is always greatest exactly at N = K/2, and slows on both sides of that point.
Part B. A forest's producers fix 50,000 kJ/m²/year as GPP, and use 30,000 kJ/m²/year for their own respiration.
- Apply the formula. NPP = GPP − R.
- Compute. NPP = 50,000 − 30,000 = 20,000 kJ/m²/year available to primary consumers and for the forest's own growth.
Test yourself
Q1 Contrast exponential and logistic population growth, and state the condition under which each is a reasonable model.
Exponential growth (dN/dt = rN) assumes unlimited resources, producing a J-shaped curve that grows ever faster — a reasonable model only briefly, such as early in colonisation of a new, resource-rich environment. Logistic growth (dN/dt = rN(1−N/K)) incorporates a carrying capacity K, so growth rate slows as resources become limiting, producing an S-shaped curve that levels off — a far more realistic model for populations in a stable, resource-limited environment over the longer term.
Q2 Prove, using the logistic equation, why the maximum population growth rate occurs at exactly N = K/2.
Growth rate is dN/dt = rN(1 − N/K), a function of N that is zero at N = 0 and again at N = K (since 1−N/K = 0), and positive in between — a downward parabola in N. A parabola's maximum lies exactly halfway between its two roots, so the maximum occurs at N = (0 + K)/2 = K/2. At that point (1 − N/K) = 0.5, so the "braking" term and the "more individuals reproducing" term are balanced at their most productive combination — biologically, this is why harvesting or conservation strategies often target keeping a population near K/2 for maximum sustainable growth.
Q3 Using the Yellowstone wolves example, explain what a trophic cascade is and why removing a single species can restructure an entire ecosystem.
A trophic cascade is a chain reaction of population and behavioural changes that ripples through multiple trophic levels after a change at one level, typically the removal or return of a predator. Removing wolves let elk populations grow unchecked and graze intensively on streamside vegetation; the loss of vegetation destabilised riverbanks and reduced habitat for other species. Because trophic levels are linked by feeding relationships, a change at the top is not contained to that level — it propagates downward through herbivores to plants and even to the physical landscape, showing that ecosystems are tightly coupled systems, not independent stacked layers.
Q4 Distinguish density-dependent from density-independent limiting factors, giving an example of each and explaining why the distinction matters for population regulation.
Density-dependent factors have an effect that intensifies as population density rises — e.g. disease spreads faster in crowded populations, and competition for food intensifies as more individuals compete for the same resources. Density-independent factors affect a population by roughly the same proportion regardless of its size — e.g. a wildfire or severe frost kills a similar fraction of individuals whether the population is large or small. The distinction matters because only density-dependent factors can act as a stabilising regulating mechanism that pushes a population back toward its carrying capacity; density-independent factors can cause sharp crashes unrelated to how "full" the environment currently is.
Q5 A field has GPP = 40,000 kJ/m²/year and NPP = 25,000 kJ/m²/year. Calculate the producers' respiration, and explain what NPP represents ecologically.
Rearranging NPP = GPP − R gives R = GPP − NPP = 40,000 − 25,000 = 15,000 kJ/m²/year. Ecologically, NPP represents the energy actually left over as new plant biomass and available to be eaten by primary consumers — it is the real "energy budget" that supports every other trophic level above the producers, which is why NPP, not GPP, is the figure ecologists use to compare how much life an ecosystem can ultimately support.
How the ideas connect
Every key idea in this chapter, branching from the core concept — use it to see the whole picture at a glance.
The key facts, visualised
Worked problems, step by step
Follow each solution line by line, then try to reproduce it on paper before moving on.
Example 1With r=0.5/yr, N=200, K=1000, find dN/dt by the logistic model.
- dN/dt = rN(K-N)/K
- 0.5 x 200 x (800/1000) = 0.5 x 200 x 0.8 = 80
Example 2GPP is 9000 kJ/m2/yr and plant respiration is 4000. Find NPP.
- NPP = GPP - respiration
- 9000 - 4000 = 5000
Now you try
Work each one out first, then tap to reveal the worked answer.