Population - Biology

1. Fundamental Concepts

Definition: Population refers to the total number of individuals of a particular species living in a specific area at a given time.
Population Density: The number of individuals per unit area or volume, expressed as $$\text{{number}} \cdot \text{{unit area}}^{-1}$$.
Growth Rate: The rate at which the population size increases over time, calculated as $$\frac{\Delta N}{\Delta t}$$ where $$N$$ is the population size and $$t$$ is time.

2. Key Concepts

Exponential Growth: $$N_t = N_0 e^{rt}$$

Where $$N_t$$ is the population size at time $$t$$, $$N_0$$ is the initial population size, $$r$$ is the growth rate, and $$t$$ is time.

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

Where $$K$$ is the carrying capacity of the environment.

Carrying Capacity: The maximum population size that an environment can sustain indefinitely given the food, habitat, water, and other necessities available in the environment.

3. Examples

Example 1 (Basic)

Problem: A population of rabbits has an initial size of 50 ($$N_0 = 50$$). If the growth rate is 0.2 per month ($$r = 0.2$$), what will be the population size after 6 months?

Step-by-Step Solution:

Use the exponential growth formula: $$N_t = N_0 e^{rt}$$
Substitute the values: $$N_t = 50 e^{0.2 \cdot 6}$$
Calculate: $$N_t = 50 e^{1.2} \approx 182.2$$

Validation: Initial population: 50; After 6 months: approximately 182.

Example 2 (Intermediate)

Problem: A fish population grows logistically with a carrying capacity of 1000 ($$K = 1000$$) and an initial population of 100 ($$N_0 = 100$$). If the intrinsic growth rate is 0.5 per year ($$r = 0.5$$), what will be the population size after 5 years?

Step-by-Step Solution:

Use the logistic growth equation: $$\frac{dN}{dt} = rN \left(1 - \frac{N}{K}\right)$$
Solve the differential equation for $$N(t)$$ using the initial condition $$N_0 = 100$$.
The solution is: $$N(t) = \frac{K}{1 + \left(\frac{K - N_0}{N_0}\right)e^{-rt}}$$
Substitute the values: $$N(5) = \frac{1000}{1 + \left(\frac{1000 - 100}{100}\right)e^{-0.5 \cdot 5}}$$
Calculate: $$N(5) \approx 794$$

Validation: Initial population: 100; After 5 years: approximately 794.

4. Problem-Solving Techniques

Understanding Growth Models: Differentiate between exponential and logistic growth models based on the presence of a carrying capacity.
Using Differential Equations: For logistic growth, solve the differential equation using initial conditions to find the population size at any given time.
Graphical Interpretation: Plot population size against time to visualize growth patterns and understand the impact of different parameters.