Ecosystem Dynamics Formula Sheet: Population Ecology

Ecosystem dynamics involve complex interactions between populations, communities, and their environment. This comprehensive formula sheet provides essential equations for understanding population ecology, community dynamics, and ecosystem stability.

Population Growth Models

Exponential Growth

Exponential growth occurs when resources are unlimited and population growth rate is constant.

Formula: $N_t = N_0 e^{rt}$

Where:

• (N_t) = Population size at time t
• (N_0) = Initial population size
• (r) = Intrinsic growth rate
• (t) = Time
• (e) = Natural logarithm base (≈ 2.718)

Example: A bacterial population starts with 1000 cells and grows exponentially with r = 0.5 per hour. What is the population after 3 hours?

Solution: (N_3 = 1000 \times e^{0.5 \times 3} = 1000 \times e^{1.5} = 1000 \times 4.482 = 4,482) cells

Logistic Growth

Logistic growth occurs when population growth is limited by carrying capacity.

Formula: $\frac{dN}{dt} = rN\left(1 - \frac{N}{K}\right)$

Where:

• (\frac{dN}{dt}) = Rate of population change
• (r) = Intrinsic growth rate
• (N) = Current population size
• (K) = Carrying capacity

Population size over time: $N_t = \frac{K}{1 + \left(\frac{K - N_0}{N_0}\right)e^{-rt}}$

Example: A population has K = 10,000, r = 0.3, and N₀ = 1,000. What is the population after 5 time units?

Solution: (N_5 = \frac{10,000}{1 + \left(\frac{10,000 - 1,000}{1,000}\right)e^{-0.3 \times 5}} = \frac{10,000}{1 + 9e^{-1.5}} = \frac{10,000}{1 + 9 \times 0.223} = \frac{10,000}{3.007} = 3,325)

Population Parameters

Growth Rate

Formula: $r = b - d$

Where:

• (r) = Net growth rate
• (b) = Birth rate
• (d) = Death rate

Per Capita Growth Rate

Formula: $r = \frac{1}{N}\frac{dN}{dt}$

Doubling Time

Formula: $T_d = \frac{\ln(2)}{r} = \frac{0.693}{r}$

Example: If r = 0.1 per year, doubling time = (\frac{0.693}{0.1} = 6.93) years

Predator-Prey Dynamics

Lotka-Volterra Model

The Lotka-Volterra model describes predator-prey interactions.

Prey population: $\frac{dN}{dt} = rN - aNP$

Predator population: $\frac{dP}{dt} = faNP - mP$

Where:

• (N) = Prey population
• (P) = Predator population
• (r) = Prey growth rate
• (a) = Attack rate
• (f) = Conversion efficiency
• (m) = Predator mortality rate

Functional Response

Type I (Linear): $F = aN$

Type II (Saturating): $F = \frac{aN}{1 + ahN}$

Type III (Sigmoidal): $F = \frac{aN^2}{1 + ahN^2}$

Where:

• (F) = Feeding rate
• (a) = Attack rate
• (h) = Handling time

Community Ecology

Species Diversity Indices

Shannon Diversity Index: $H' = -\sum_{i=1}^{S} p_i \ln(p_i)$

Where:

• (H') = Shannon diversity
• (p_i) = Proportion of species i
• (S) = Total number of species

Simpson's Diversity Index: $D = 1 - \sum_{i=1}^{S} p_i^2$

Example: A community has 3 species with proportions 0.5, 0.3, and 0.2. Calculate Shannon diversity.

Solution: (H' = -(0.5 \ln(0.5) + 0.3 \ln(0.3) + 0.2 \ln(0.2)) = -(0.5 \times -0.693 + 0.3 \times -1.204 + 0.2 \times -1.609) = 0.347 + 0.361 + 0.322 = 1.030)

Species Richness and Evenness

Species Richness (S): Total number of species

Pielou's Evenness: $J' = \frac{H'}{H'_{max}} = \frac{H'}{\ln(S)}$

Ecosystem Stability

Resilience

Formula: $R = \frac{1}{\tau}$

Where:

• (R) = Resilience
• (\tau) = Return time to equilibrium

Resistance

Formula: $Resistance = 1 - \frac{\Delta X}{X_0}$

Where:

• (\Delta X) = Change in ecosystem property
• (X_0) = Initial value

Energy Flow

Ecological Efficiency

Formula: $E = \frac{I_{n+1}}{I_n} \times 100\%$

Where:

• (E) = Ecological efficiency
• (I_{n+1}) = Energy at trophic level n+1
• (I_n) = Energy at trophic level n

Typical values:

• Plant to herbivore: 10-20%
• Herbivore to carnivore: 10-15%
• Overall food chain: 5-15%

Net Primary Productivity (NPP)

Formula: $NPP = GPP - R$

Where:

• (NPP) = Net primary productivity
• (GPP) = Gross primary productivity
• (R) = Respiration

Nutrient Cycling

Nutrient Turnover Rate

Formula: $k = \frac{L}{X}$

Where:

• (k) = Turnover rate
• (L) = Loss rate
• (X) = Standing stock

Residence Time

Formula: $T = \frac{X}{L} = \frac{1}{k}$

Metapopulation Dynamics

Levins Model

Formula: $\frac{dp}{dt} = cp(1-p) - ep$

Where:

• (p) = Proportion of occupied patches
• (c) = Colonization rate
• (e) = Extinction rate

Equilibrium: $p^* = 1 - \frac{e}{c}$

Island Biogeography

Species-Area Relationship

Formula: $S = cA^z$

Where:

• (S) = Number of species
• (A) = Area
• (c) = Constant
• (z) = Slope (typically 0.2-0.35)

Log form: $\log(S) = \log(c) + z\log(A)$

Climate Change Impacts

Temperature Response

Q₁₀ Rule: $Q_{10} = \left(\frac{R_2}{R_1}\right)^{\frac{10}{T_2-T_1}}$

Where:

• (Q_{10}) = Temperature coefficient
• (R_1, R_2) = Rates at temperatures T₁, T₂

Applications

Conservation Biology

• Minimum Viable Population (MVP): Population size needed for 95% survival over 100 years
• Effective Population Size: (N_e = \frac{4N_m N_f}{N_m + N_f})
• Genetic Drift: (\Delta p = \sqrt{\frac{p(1-p)}{2N_e}})

Fisheries Management

• Maximum Sustainable Yield: (MSY = \frac{rK}{4})
• Optimal Harvest Rate: (h = \frac{r}{2})

Study Tips

1. Understand the assumptions behind each model
2. Practice with real data from ecological studies
3. Consider limitations of mathematical models
4. Connect formulas to biological processes
5. Use units consistently in calculations

Common Calculations

Population Density

Formula: $Density = \frac{Number\ of\ individuals}{Area}$

Population Growth Rate

Formula: $Growth\ Rate = \frac{N_{t+1} - N_t}{N_t} \times 100\%$

Carrying Capacity Estimation

Formula: $K = \frac{r}{\alpha}$

Where (\alpha) is the density-dependent mortality rate.

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Keywords: ecosystem dynamics, population ecology, growth models, predator-prey relationships, community ecology, biodiversity indices, energy flow, nutrient cycling.

Last Updated: July 12, 2025, 05:24 PM +04