Population Regulation and Logistic Growth 1. Logistic Growth Model The logistic growth model describes population growth that is initially exponential but slows as the population approaches the carrying capacity (K) of the environment. Equation: N dN = rN 1 − (1) dt K Variables: • N : Population size. • r: Intrinsic growth rate (per capita rate of increase). • K: Carrying capacity (maximum population size the environment can sustain). • dN : dt Rate of change of population size over time. 2. Key Characteristics of Logistic Growth • Growth is initially exponential when N is much smaller than K. • Growth slows as N approaches K due to increasing competition for limited resources. • At N = K, growth stops ( dN = 0). dt 3. Graphical Representation of Logistic and Exponential Growth 1,000 Population SizeGrowth (N ) (K = 1000) Logistic Carrying Capacity (K = 1000) 750 500 250 Time (t) 5 10 15 Figure 1: Logistic Growth Curve with Carrying Capacity (K) 1 20 1,000 Population Size (NGrowth ) Exponential 750 500 250 Time (t) 5 10 15 20 Figure 2: Exponential Growth Curve 4. Example Calculation of Logistic Growth Given: • r = 0.1 • K = 1000 • Initial population size, N0 = 100 Calculate population size after one time step (∆t = 1): N dN = rN 1 − dt K Step-by-Step Calculation: 1. Initial growth rate: dN 100 = 0.1 × 100 1 − dt 1000 2. Simplify: dN = 10 × (1 − 0.1) = 10 × 0.9 = 9 dt 3. Population size after one time step: N1 = N0 + dN × ∆t = 100 + 9 × 1 = 109 dt 2 (2) 5. Applications of Logistic Growth Model • Understanding how populations respond to environmental limits. • Managing wildlife populations and conservation efforts. • Predicting growth of human populations in specific regions. 6. Comparison with Exponential Growth Characteristic Exponential Growth Logistic Growth dN dN N Equation = rN = rN 1 − K dt dt Growth Pattern Unlimited growth Growth slows near K Carrying Capacity (K) Not considered Explicitly included Population Size (N ) Increases indefinitely Stabilizes at K Table 1: Comparison of Exponential and Logistic Growth 3