Geometric and Exponential Population Growth Models

Foundations of Population Ecology and Growth Modeling

• Population ecology investigates how population sizes change over time and space, relying on mathematical models to formalize, predict, and understand demographic dynamics.
• Ecological questions span multiple nested scales of biological organization:
  • Individuals: Singular organisms exhibiting specific physiological and behavioral traits.
  • Populations: Groups of individuals belonging to the same species that inhabit a defined geographic area and interbreed.
  • Communities: Assemblies of interacting populations of different species inhabiting a shared environment.
• Mathematical models of population growth serve critical roles across primary sectors:
  • Health: Modeling disease transmission vectors (e.g., mosquitoes) and pathogen proliferation rates (e.g., viruses like SARS-CoV-2).
  • Management: Establishing sustainable harvesting thresholds for commercially viable species (e.g., fisheries management).
  • Conservation: Assessing extinction risks and guiding recovery plans for endangered organisms (e.g., marine turtles).
  • Policy: Informing infrastructure planning, resource allocation, and environmental legislation based on human population trajectories.

Core Principles of Population Dynamics

• Population growth is represented mathematically as a temporal sequence of population sizes:
  • $N_t$: Population size at the present time step $t$.
  • $N_{t+1}$: Population size one time step into the future.
  • $N_{t+2}$, $N_{t+3}$: Population sizes two and three steps into the future.
  • $N_{t-1}$: Population size one time step in the past.
• The fundamental primary framework governing population size changes across time steps is the BIDE Model:

$N_{t+1} = N_t + B - D + I - E$

• Model variable definitions:
  • $N_t$: Initial population size at time $t$.
  • $B$: Total number of births occurring within the time interval $[t, t+1]$.
  • $D$: Total number of deaths occurring within the time interval $[t, t+1]$.
  • $I$: Total number of immigrants entering the population within the time interval $[t, t+1]$.
  • $E$: Total number of emigrants leaving the population within the time interval $[t, t+1]$.
• Simplified Closed Population Model:
  • Assuming zero dispersal across boundaries (a closed population where $I = 0$ and $E = 0$):

$N_{t+1} = N_t + B - D$

• Assuming no internal demographic structure (ignoring differences in age classes, sex ratios, or stage-specific survival rates).

Geometric Population Growth Model (Discrete Time)

• Geometric growth models population size changes at regular, discrete time intervals, assuming constant per capita rates and non-overlapping generations (e.g., species with synchronous annual breeding cycles).

Mathematical Derivation of Geometric Growth

• To express total births $B$ and deaths $D$ in terms of per capita (per individual) rates:
  • Per capita birth rate: $b = \frac{B}{N_t}$, giving $B = b N_t$
  • Per capita death rate: $d = \frac{D}{N_t}$, giving $D = d N_t$
  • Units for $b$ and $d$: births or deaths per individual per unit time.
• Substituting per capita terms into the closed population equation:

$N_{t+1} = N_t + (b N_t - d N_t)$

$N_{t+1} = N_t + (b - d)N_t$

• Define the discrete growth factor $r_d$ as the difference between per capita birth and death rates:

$r_d = b - d$

$N_{t+1} = N_t + r_d N_t = (1 + r_d)N_t$

• Redefining $(1 + r_d)$ as $\lambda$ (lambda), the finite rate of increase or finite growth rate:

$\lambda = 1 + r_d = 1 + (b - d)$

$N_{t+1} = \lambda N_t$

$\lambda = \frac{N_{t+1}}{N_t}$

• Parameter interpretation of $\lambda$:
  • $\lambda > 1$: Population grows ($N_{t+1} > N_t$).
  • $\lambda = 1$: Stable population size ($N_{t+1} = N_t$).
  • $\lambda < 1$: Population declines ($N_{t+1} < N_t$).

Projection Across Multiple Time Steps

• Projecting population size forward from an initial population $N_0$ across discrete steps assuming constant $\lambda$:
  • Step 1: $N_1 = \lambda N_0$
  • Step 2: $N_2 = \lambda N_1 = \lambda (\lambda N_0) = \lambda^2 N_0$
  • Step 3: $N_3 = \lambda N_2 = \lambda (\lambda^2 N_0) = \lambda^3 N_0$
• General multi-step discrete projection equation:

$N_t = \lambda^t N_0$

Graphical Properties of Geometric Growth

• Plotting population size $N$ against discrete time $t$ produces a stair-step pattern.
• Because $\lambda$ remains constant while the baseline population size $N_t$ increases, the discrete jumps (absolute increase per step) grow larger over time, causing the population to grow progressively faster.

Exponential Population Growth Model (Continuous Time)

• Exponential growth models population change continuously over time without discrete steps, ideal for species with overlapping generations and continuous reproduction.

Derivation of Continuous Differential Form

• Shrinking time intervals $[(t+1) - t]$ to infinitely small increments (approaching zero, $\Delta t \to 0$):

$\frac{dN}{dt} = B - D$

• Expressing births and deaths as continuous per capita rates ($B = bN$, $D = dN$):

$\frac{dN}{dt} = (b - d)N$

• Defining $r$ as the instantaneous per capita growth rate ($r = b - d$):

$\frac{dN}{dt} = r N$

• Rearranging to express per capita growth rate directly:

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

• Parameter interpretation of $r$:
  • $r > 0$: Population size grows exponentially.
  • $r = 0$: Population size stays constant.
  • $r < 0$: Population size declines toward extinction.

Integrated Projection Form

• Integrating the differential equation $\frac{dN}{dt} = rN$ yields the continuous exponential projection model:

$N(t) = N(0) e^{r t}$

• $N(t)$: Population size at time $t$
• $N(0)$: Initial population size at time $t = 0$
• $r$: Instantaneous per capita growth rate
• $e$: Euler's number ($e \approx 2.71828$

Mathematical Intuition of Euler's Number ($e$) via Compounding

• Consider an initial population $N_0 = 1$ experiencing $100\%$ nominal growth ($r = 1$) over one time unit:
  • If growth occurs instantly in 1 step ($n = 1$):

$N_1 = \left(1 + \frac{1}{1}\right)^1 \times 1 = 2$

• If growth is split into 2 compounding intervals ($n = 2$):

$N_1 = \left(1 + \frac{1}{2}\right)^2 \times 1 = 2.25$

• If split into 3 intervals ($n = 3$):

$N_1 = \left(1 + \frac{1}{3}\right)^3 \times 1 \approx 2.37037$

• If split into 365 daily intervals ($n = 365$):

$N_1 = \left(1 + \frac{1}{365}\right)^{365} \times 1 \approx 2.71456$

• As the number of sub-intervals $n$ approaches infinity ($n \to \infty$):

$\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = e \approx 2.71828$

• Continuous compounding models instantaneous accumulation, analogous to financial accounts continuously compounding interest.

Relationship Between Geometric and Exponential Growth Models

• Geometric growth represents a discrete-time special case of continuous exponential growth.
• Equating discrete and continuous projections over a time step $t$:

$N_t = N_0 \lambda^t = N_0 e^{r t}$

$\lambda^t = e^{r t}$

• Mathematical equivalencies connecting $\lambda$ and $r$:

$\lambda = e^r$

$r = \ln(\lambda)$

• Parameter equivalence mappings:
  • Stable population: $\lambda = 1 \iff r = 0$
  • Growing population: $\lambda > 1 \iff r > 0$
  • Declining population: $\lambda < 1 \iff r < 0$
• Example Conversion: If $\lambda = 1.3$, the equivalent instantaneous per capita rate $r$ is:

$r = \ln(1.3) \approx 0.26236$

Doubling Time and Rate Analysis

Derivation of Doubling Time ($t_d$)

• Doubling time ($t_d$) is the time required for a population to double its initial size ($N(t_d) = 2 N(0)$).

$N(t_d) = N(0) e^{r t_d}$

$2 N(0) = N(0) e^{r t_d}$

$2 = e^{r t_d}$

$\ln(2) = r t_d$

$t_d = \frac{\ln(2)}{r}$

• Numerical constant: $\ln(2) \approx 0.693147$

Growth Rate vs. Per Capita Growth Rate

• Absolute Population Growth Rate ($\frac{dN}{dt}$):
  • Formula: $\frac{dN}{dt} = r N$
  • Dynamic behavior: Increases continuously as population size $N$ increases under exponential growth. Highest at the largest population sizes.
• Per Capita Growth Rate ($\frac{1}{N}\frac{dN}{dt}$ or $r$):
  • Formula: $\frac{1}{N}\frac{dN}{dt} = r$
  • Dynamic behavior: Remains perfectly constant across all population sizes under density-independent exponential growth.

Comprehensive Worked Examples and Practice Problems

Problem 1: Bacterial Proliferation (Escherichia coli)

• Scenario: A bacterial culture of Escherichia coli in exponential growth phase has an instantaneous per capita growth rate $r = 0.0347\,\text{min}^{-1}$. Initial colony size $N(0) = 0.25\,\text{billion cells}$.

Question 1A: Population Size at $t = 120\,\text{minutes}$

• Solution:

$N(120) = N(0) e^{r t} = 0.25 \times e^{0.0347 \times 120}$

$0.0347 \times 120 = 4.164$

$e^{4.164} \approx 64.3283$

$N(120) = 0.25 \times 64.3283 \approx 16.082\,\text{billion cells}$

Question 1B: Quadrupling Time ($t_q$)

• Solution: Set $N(t_q) = 4 N(0)$.

$4 N(0) = N(0) e^{r t_q}$

$4 = e^{r t_q}$

$\ln(4) = r t_q$

$t_q = \frac{\ln(4)}{r} = \frac{\ln(4)}{0.0347} \approx \frac{1.386294}{0.0347} \approx 39.951\,\text{minutes}$

Question 1C: Variable Initial Population Size

• Scenario: If initial population $N(0) = 0.1\,\text{billion cells}$, calculate size after $t = 120\,\text{minutes}$.
• Solution:

$N(120) = 0.1 \times e^{4.164} = 0.1 \times 64.3283 \approx 6.433\,\text{billion cells}$

Question 1D: Growth Rate Comparisons

• Fastest absolute growth rate ($\frac{dN}{dt}$): Point C (highest $N$ value).
• Highest per capita growth rate ($\frac{1}{N}\frac{dN}{dt}$): Equal across all points A, B, and C ($r = 0.0347\,\text{min}^{-1}$ is constant).

Problem 2: Annual Bird Reproduction

• Scenario: A population of $N_0 = 100$ birds breeds once per year. In one year, $B = 80$ eggs are counted, and $D = 42$ adult birds die. All eggs survive.

Question 2A: New Population Size ($N_1$)

• Solution:

$N_1 = N_0 + B - D = 100 + 80 - 42 = 138\,\text{birds}$

Question 2B: Annual Finite Growth Rate ($\lambda_{\text{annual}}$)

• Solution:

$\lambda_{\text{annual}} = \frac{N_1}{N_0} = \frac{138}{100} = 1.38$

Question 2C: Equivalent Monthly Finite Growth Rate ($\lambda_{\text{monthly}}$)

• Solution:

$N_{12} = N_0 (\lambda_{\text{monthly}})^{12} = N_0 \lambda_{\text{annual}}$

$(\lambda_{\text{monthly}})^{12} = \lambda_{\text{annual}} = 1.38$

$\lambda_{\text{monthly}} = (1.38)^{1/12} \approx 1.02721$

Problem 3: Amoeba Geometric Growth Determination

• Scenario: A population of amoeba exhibits geometric growth starting at $t = 0$ (start of year, Dec 31 / Jan 1) with $N_0 = 3$. End of January ($t = 1\,\text{month}$): $N_1 = 6$. End of February ($t = 2\,\text{months}$): $N_2 = 12$. Current count $N_t = 3072$.

Step 1: Calculate $\lambda$

$\lambda = \frac{N_1}{N_0} = \frac{6}{3} = 2$

Step 2: Solve for Elapsed Time $t$

$N_t = N_0 \lambda^t$

$3072 = 3 \times 2^t$

$1024 = 2^t$

$\ln(1024) = t \ln(2)$

$t = \frac{\ln(1024)}{\ln(2)} = 10\,\text{months}$

Step 3: Identify Calendar Month

• $t = 1$: End of January
• $t = 2$: End of February
• $t = 10$: End of October
• Month name: October